Proof. The manifold \(M\) is aspherical, since it is homotopy equivalent to \(Q_2\). Topological manifolds have CW homotopy type [21]. The classification of maps between Eilenberg–Mac Lane spaces therefore realizes \(\alpha\) by a map \(f\colon M\to M\), and realizes its inverse by a homotopy inverse to \(f\); see [16].

Suppose that a homeomorphism \(h\colon M\to M\) induces the outer class of \(\alpha\). We will construct closed manifold models for all sufficiently large odd-degree covers of \(Q\). Put \(K=\ker w\), and let \[X=K\backslash\widetilde M\] be the infinite cyclic cover of \(M\), where \(\widetilde M\) is its universal cover. We use left deck actions throughout. The element \(\gamma^2\in G_2\) induces a deck transformation \(T\) of \(X\), and \(T\) generates its deck group over \(M\).

Choose a lift \(u\colon\widetilde M\to\widetilde M\) of \(h\) that intertwines the deck action by exactly \(\alpha\): \[ u g u^{-1}=\gamma g\gamma^{-1} \qquad(g\in G_2). \tag{1}\] Indeed, any lift induces an automorphism in the prescribed outer class. Composing that lift with a deck transformation removes the inner discrepancy. Since \(\alpha(K)=K\), the map \(u\) descends to a homeomorphism \(h'\colon X\to X\). Moreover, \(\alpha(\gamma^2)=\gamma^2\), so (1) gives \[ h'T=Th'. \tag{2}\]

We next show that a sufficiently large translation composed with \(h'\) acts freely and cocompactly on \(X\). Represent the homomorphism \(w/2\colon G_2\to\mathbb Z\) by a map \(M\to S^1\) and lift that map to obtain a continuous height function \[\tau\colon X\longrightarrow\mathbb R, \qquad \tau(Tx)=\tau(x)+1.\] This height is proper. To see this, choose a compact set \(C\subset X\) whose \(T\)-translates cover \(X\), using compactness of \(M\), and choose \(a,b\in\mathbb R\) with \(\tau(C)\subset[a,b]\). A bounded closed height band \(\tau^{-1}([A,B])\) is a closed subset of \[\bigcup_{\substack{j\in\mathbb Z\\ A-b\leq j\leq B-a}}T^jC,\] which is a finite union of compact sets.

By (2), the continuous function \(x\mapsto\tau(h'x)-\tau(x)\) is \(T\)-invariant. It is therefore bounded, say in absolute value by \(c\geq0\). For an integer \(r>c+1\), put \[F_r=T^rh'.\] For every \(x\in X\) we have \[ 0<r-c\leq\tau(F_rx)-\tau(x)\leq r+c. \tag{3}\] Consequently no nonzero power of \(F_r\) fixes a point. Iterating the lower bound shows that only finitely many translates of any compact set can meet that set; thus the \(\mathbb Z\)-action generated by \(F_r\) is properly discontinuous. Finally, along every orbit the height tends to \(+\infty\) in positive time and to \(-\infty\) in negative time. The first nonnegative height on such an orbit belongs to \([0,r+c]\), by the upper bound in (3). The compact band \(\tau^{-1}([0,r+c])\) therefore meets every orbit. This proves cocompactness.

It follows that \[Y_r=X/\langle F_r\rangle\] is a closed connected topological \(n\)-manifold. Its universal cover is \(\widetilde M\), so \(Y_r\) is aspherical. To identify its fundamental group, lift \(F_r\) to \[\widetilde F_r=\gamma^{2r}u \quad\text{on }\widetilde M.\] Equation (1) gives \[ \widetilde F_r k\widetilde F_r^{-1} =\gamma^{2r+1}k\gamma^{-(2r+1)} \qquad(k\in K). \tag{4}\] The deck group over \(Y_r\) is generated by \(K\) and \(\widetilde F_r\). Its quotient by \(K\) is infinite cyclic, since its action on \(X\) is generated freely by \(F_r\). Hence (4) identifies it with \[\pi_1(Y_r) \cong K\rtimes_{\operatorname{Ad}_{\gamma^{2r+1}}}\mathbb Z \cong G_{2r+1}.\] For the second isomorphism, send the generator of \(\mathbb Z\) to \(\gamma^{2r+1}\) and act identically on \(K\). This is bijective because every element of \(G_{2r+1}\) has a unique expression \(k\gamma^{(2r+1)j}\) with \(k\in K\) and \(j\in\mathbb Z\).

Both \(Y_r\) and \(Q_{2r+1}\) are Eilenberg–Mac Lane spaces, so this group isomorphism is realized by a homotopy equivalence. Taking \(r\) sufficiently large contradicts the assumed absence of odd-degree manifold models. ◻

Proof. Since \(\mathsf A\leq R_X\), the inclusion of any \(a\delta\) with \(a\in\mathsf A\) in \(R_X\) would imply \(\delta\in R_X\). Hence \[N_{R_X}(\mathsf A) =R_X\cap\bigl(\mathsf A\times\langle\delta\rangle\bigr) =\mathsf A.\] Choose a Sylow \(2\)-subgroup \(P\) of \(R_X\) containing \(\mathsf A\). A proper subgroup of a finite \(2\)-group is strictly contained in its normalizer. For completeness, this follows by induction on the group order: an element of the center outside the subgroup enlarges its normalizer, while if the center is contained in the subgroup the claim follows from the induction hypothesis in the quotient by the center. Thus \(\mathsf A<P\) would contradict the displayed equality, and \(\mathsf A\) is a Sylow \(2\)-subgroup of \(R_X\).

If a homeomorphism \(X\to Y\) existed, (5) would place a conjugate of \(\mathsf B\) inside \(R_X\). Its order is \(|\mathsf A|\), so it too would be a Sylow \(2\)-subgroup of \(R_X\). Sylow conjugacy in \(R_X\) would make \(\mathsf A\) and \(\mathsf B\) conjugate in \(\mathop{\mathrm{Out}}(\Pi)\), contrary to the hypothesis. ◻

Proof. Propositions 38 and 40, together with Corollary 41, construct the base chamber and verify the chamber hypotheses of Theorem 6. The character is the one in (52). Lemma 42 proves the assertions about every lifted chamber. Proposition 44 gives the relative filling \(j\), and Proposition 52 gives the last assertion, using the marking in Definition 43. These results are proved in Sections 8–10 from the explicitly stated inputs of [22, 23]. ◻

Proof. A clique of vertices in \(\mathcal L_d\) projects to distinct pairwise adjacent vertices of \(\mathcal L\). Flagness downstairs supplies a simplex containing their images. Lift that simplex from one vertex of the clique. The clique edges incident to that vertex project to edges of this simplex; uniqueness of their lifts places all vertices of the clique in the chosen lifted simplex. Thus \(\mathcal L_d\) is flag.

Define a surjective homomorphism \[\pi_d\colon \mathsf A_d\longrightarrow\mathsf A_0\] by sending the coordinate of a lifted panel to the coordinate of its downstairs panel. In the chamber description of \(Q^0_d\) above, the formula \([a,x]\mapsto[\pi_d(a),x]\) is well-defined and surjective. At a point belonging to a set of meeting lifted panels, those panels project to distinct panels downstairs. The homomorphism \(\pi_d\) therefore restricts to an isomorphism on their coordinate subgroups. The sector identifications in (6) consequently give covering charts at this point; in chamber interiors the assertion is immediate. The map is finite because \(\mathsf A_d\) is finite. Finally, \(P_d\) is connected, and each coordinate generator of \(\mathsf A_d\) is realized by crossing its nonempty panel. All its chambers are therefore in one connected component. ◻

Proof. Suppose that \(Q^0_d\) has such a model. The finite cover of Lemma 8 determines, through the model equivalence, a finite cover of that closed manifold. Lifting the equivalence gives a closed topological four-manifold model for \(\widehat Q_d\).

Proposition 7 verifies all the hypotheses of Theorem 6 for \((D_d,P_d,S_d,H_d)\), with the lifted hyperbolic basis and boundary marking. The theorem therefore gives a marked filling of \(S_d\), contradicting the last assertion of Proposition 7. ◻

Proof. For \(J\subset\{0,\ldots,7\}\), let \(r(J)\) be the number of pairs \(\{2j-2,2j-1\}\), \(1\leq j\leq4\), contained in \(J\). Cancellation occurs only at the marked vertex of such a pair, so \[ \mathop{\mathrm{wt}}\left(\sum_{i\in J}c_i\right) =16\sum_{i\in J}2^i-2r(J),\qquad 0\leq2r(J)\leq8. \tag{9}\] For distinct \(J,J'\), the first terms on the right differ by at least \(16\), while their correction terms differ by at most \(8\). Thus all subset sums have different weights. Every nonempty subset sum has weight at least \(16-8=8\), so the \(c_i\) are independent.

A coordinate permutation preserves weight. Since each element of \(\mathsf C\) has a different weight from the other elements, any coordinate permutation preserving \(\mathsf C\) fixes every \(c_i\). It consequently fixes each singleton \[\mathop{\mathrm{supp}}(c_{2j-2})\cap\mathop{\mathrm{supp}}(c_{2j-1})=\{m_j\}.\] In particular, \(\sigma\) fixes the chosen tetrahedron pointwise. Every triangular face of a closed triangulated three-manifold belongs to exactly two tetrahedra. If one of them is fixed pointwise, an automorphism fixing that face must preserve the other tetrahedron and fix its fourth vertex. This propagates pointwise fixation across faces. The tetrahedron adjacency graph is connected: a path between interior points of two tetrahedra can be put in PL general position with respect to the triangulation, avoiding the one-skeleton and crossing triangular faces one at a time. Propagation along that graph shows that every vertex of \(\mathcal L\) is fixed. ◻

Proof. The group \(\mathsf C\) acts on \(Q^0\) by translation of its \(\mathsf A_0\)-labels. The stabilizer of \([b_0,x]\) in \(\mathsf A_0\) is \(\mathsf A_0(I(x))\). Equation (11) shows that its intersection with \(\mathsf C\) is trivial. Hence \(\mathsf C\) acts freely.

The orbit relation for this action, written on \(\mathsf A_0\times P\), identifies two points precisely when their chamber coordinates agree and their label difference lies in \(\mathsf C+\mathsf A_0(I(x))\). Its image under the label quotient is exactly the relation in (12). Thus \(Q^0/\mathsf C=Q\), with the quotient topologies. A free action of a finite group on a Hausdorff space is a covering action: for a point, choose disjoint neighborhoods of it and each of its nontrivial translates, then intersect the finitely many resulting neighborhoods to obtain one whose translates are pairwise disjoint. The space \(Q^0\) is Hausdorff because it is a finite CW complex, so this applies and proves the covering assertion.

For each open cell of \(P\), the set \(I(x)\) is constant on that cell, because every panel is a subcomplex. The relation in (12) therefore identifies entire copies of an open cell by the identity. The resulting finitely many open cells, with the attaching maps induced from those of \(P\), give a finite CW structure on \(Q\). The chamber \(P\) is connected. Each panel is nonempty, and crossing \(S_s\) joins a chamber labeled \(b\) to one labeled \(b+a_s\). Since the \(a_s\) span \(\mathsf A\), all chambers belong to one component. Covering maps induce isomorphisms on all higher homotopy groups. The asphericity of \(Q^0\) from Theorem 6 thus gives the asphericity of \(Q\). The formula for \(r\) is well-defined and continuous, and its restriction to each chamber is the identity on \(P\). ◻

Proof. The fold for \(Q^0\) also has a chamber section. Its connected cyclic cover is consequently \(Q^0_d=Q^0\times_P P_d\). Commutativity of the folding maps gives \[Q^0_d \cong Q^0\times_Q(Q\times_P P_d) =Q^0\times_Q Q_d.\] It follows that \(q\) pulls back to a finite covering \(q_d:Q^0_d\to Q_d\); its total space is connected by the preceding description of \(Q^0_d\).

Suppose that \(f:Z\to Q_d\) is a homotopy equivalence from a closed connected topological four-manifold. Under the isomorphism \(f_*\), the subgroup defining \(Q^0_d\) determines a connected finite cover \(Z'\to Z\). Choose compatible basepoints and lift \(f\) to \(f':Z'\to Q^0_d\). A homotopy inverse to \(f\) and the two inverse homotopies lift to the corresponding covers, so \(f'\) is a homotopy equivalence. The finite cover \(Z'\) is again a closed connected topological four-manifold. This contradicts Proposition 9 for \(Q^0_d\). ◻

Proof. The dual panels cover \(S\), and panels meet precisely over simplices, which gives \(1\leq q\leq4\). In a geometric simplex, the closed dual block of a vertex consists of the points at which that vertex’s barycentric coordinate is maximal. Let \(\sigma\) be the simplex of \(K\) whose relative interior contains \(\bar y\). At \(\bar y\) the \(q\) coordinates indexed by \(J_K(y)\) are tied for the maximum; all other coordinates are strictly smaller. After shrinking a neighborhood, only these \(q\) coordinates can be maximal.

The PL local product about a point of the relative interior of \(\sigma\) has the directions in \(\sigma\) and the transverse cone on its link. Because \(K\) triangulates a PL three-manifold, that cone supplies PL Euclidean transverse directions. In this product the barycentric coordinates belonging to \(\sigma\) acquire a common positive scale factor in a transverse direction. Their comparisons are therefore determined by the coordinates within \(\sigma\). Take the deviations of the \(q\) relevant coordinates from their mean, denoted \(d_1,\ldots,d_q\), so that \(\sum_i d_i=0\). These are \(q-1\) independent local coordinates in \(\sigma\); the other simplex directions and the transverse directions together give \(4-q\) coordinates. In this chart the \(u_i\)-panel is given by \(d_i=\max_j d_j\). Lifting a sufficiently small neighborhood to \(S_2\) gives the same description there.

Adjoin the inward collar coordinate \(t\geq0\). On the factor \(\{(d_i):\sum_i d_i=0\}\times[0,\infty)\), define \[ x_i=t+\max_j d_j-d_i,\qquad 1\leq i\leq q. \tag{19}\] This is a homeomorphism onto \([0,\infty)^q\). Its continuous inverse is \[t=\min_i x_i,\qquad d_i=\frac1q\sum_jx_j-x_i.\] Moreover, \(x_i=0\) holds exactly when \(t=0\) and \(d_i=\max_jd_j\). Thus it identifies the panels with the indicated zero faces. The finitely many panels not containing \(y\) are closed, so the initial neighborhood can be chosen disjoint from all of them. ◻

Proof. For \(a\in\mathsf F\), the set of boundary coordinates allowing the label difference \(a\) is \[ \begin{split} \Omega_a &=\{y\in S_2:a\in\mathsf F(J_K(y))\}\\ &=\bigcup_{\substack{J\subset\mathcal U\\ \sum_{u\in J}\lambda_u=a}} \ \bigcap_{u\in J}p_2^{-1}(S^K_u). \end{split} \tag{20}\] The empty intersection is understood to be \(S_2\). The equality holds because an element of \(\mathsf F(J_K(y))\) is the sum of the labels of some subset of \(J_K(y)\). Each \(\Omega_a\) is a finite union of closed panel intersections. The relation between two fixed chamber boundaries in (17) is the equality of their \(h_b\)-coordinates over \(\Omega_{b-c}\), hence is closed. On a single chamber the relation is its full diagonal, because \(h_b\) is injective. The finitely many closed relations between distinct chambers, together with these diagonals, give a closed equivalence relation on the compact Hausdorff space \(\coprod_b V_b\). Its quotient \(\mathcal M_\epsilon\) is therefore compact Hausdorff, and each chamber embeds in it. The same argument with \(P_2\) in place of \(V\) shows that \(R_2\) is compact Hausdorff and that its chambers embed. The panel-compatible CW structures also give \(R_2\) a finite CW structure by the cell argument in Lemma 11.

The manifold \(V\) is connected, since \(j\) is a homotopy equivalence and \(P_2\) is connected. Every lifted panel is nonempty. Consequently, for each \(u\) the chambers labeled \(b\) and \(b+\lambda_u\) meet. Since the labels span \(\mathsf F\), these intersections connect all the chambers, proving connectedness.

Interior points already have four-manifold charts. Consider a boundary point with coordinate \(y\), and write \(J_K(y)=\{u_1,\ldots,u_q\}\). The labels of the chambers containing its equivalence class form \[b+\mathsf F(J_K(y)) =\left\{b+\sum_{i=1}^q\eta_i\lambda_{u_i}: (\eta_1,\ldots,\eta_q)\in\mathbb F_2^q\right\}.\] Local independence makes this parametrization bijective, so there are exactly \(2^q\) chambers at the point. Choose collars of their boundaries, transporting the boundary coordinate to \(S_2\) by the maps \(h_b\). Lemma 13 supplies the same chart in each collar. Shrink it to a relative-open product box \[(-\rho,\rho)^{4-q}\times[0,\rho)^q\] that meets no panel outside \(J_K(y)\). These boxes together form an open saturated subset of the disjoint union of chambers: every allowed label change within them belongs to \(\mathsf F(J_K(y))\).

In the chamber indexed by \(\eta\in\mathbb F_2^q\), map this box to \(\mathbb R^{4-q}\times\mathbb R^q\) by \[(z,x_1,\ldots,x_q)\longmapsto (z,(-1)^{\eta_1}x_1,\ldots,(-1)^{\eta_q}x_q).\] Two such images agree exactly on the faces on which the relevant coordinates are zero, and these are precisely the identifications in (17). The quotient of the boxes is therefore \((-\rho,\rho)^4\). More explicitly, the displayed map is a continuous bijection after taking the quotient, and its inverse is continuous by the pasting lemma on the finitely many closed orthants of \((-\rho,\rho)^4\). This gives a chart without boundary. Thus \(\mathcal M_\epsilon\) is locally Euclidean of dimension four. Compactness supplies a finite cover by such charts and interior charts, so their countable bases also give a countable basis for \(\mathcal M_\epsilon\). We have proved that it is a closed topological four-manifold.

It remains to prove that \(\varphi_\epsilon\) is a homotopy equivalence. Enumerate the finitely many labels, and in both \(\mathcal M_\epsilon\) and \(R_2\) take the images of their chambers one at a time. These images and their finite unions are compact, hence closed. When the new label is \(b\) and the earlier labels are \(c\), its attaching set in the boundary coordinate \(S_2\) is \[K_b=\bigcup_{\text{earlier }c}\Omega_{b-c}.\] Formula (20) expresses \(K_b\) as a finite union of lifted panel intersections. It is therefore a subcomplex of the lifted boundary CW structure. In the new source chamber the attaching set is \(h_b^{-1}(K_b)\); in the new target chamber it is \(K_b\). The comparison on them is exactly the homeomorphism \(h_b\). The quotient descriptions and the chamber embeddings show that adjoining the new chamber is the pushout along these attaching sets. For example, the natural map from that pushout to the corresponding union is a continuous bijection from a compact space to a Hausdorff space and hence is a homeomorphism.

The inclusions of \(K_b\) into \(S_2\), and of \(h_b^{-1}(K_b)\) into \(\partial V_b\), are cofibrations because they are, up to the boundary homeomorphism, inclusions of CW subcomplexes. The inclusions of the boundaries into \(P_2\) and \(V_b\) are cofibrations by their collars. Composing these cofibrations gives cofibrations from the attaching sets into the new chambers on both sides. Hence each ordinary pushout is a homotopy pushout: its comparison with the double mapping cylinder is a homotopy equivalence. Homotopy pushouts preserve a map of spans that is a homotopy equivalence at all three objects. Here the map on the attaching set is \(h_b\), the map on the new chamber is the homotopy equivalence \(j_b\), and the map on the preceding union is a homotopy equivalence by induction. The first chamber starts the induction. This proves that \(\varphi_\epsilon\) is a homotopy equivalence. If \(R_2\) is aspherical, lifting this equivalence to universal covers shows that the universal cover of \(\mathcal M_\epsilon\) is contractible. ◻

Proof. First suppose that the triangulation is PL relative to the fixed PL structure on \(S\). Take \(\mathsf F=\mathbb F_2^{\mathcal U}\), \(\lambda_u=e_u\), and \(\epsilon=0\) in Proposition 14. These labels span and are independent on every simplex. The target \(R_2\) is the pullback description of the double cyclic cover of \(Q^0(K)\) from Section 3. The proposition supplies its four-manifold model. The full reflection is aspherical by Theorem 6, so the cover and the model are aspherical.

For an arbitrary finite triangulation, its abstract complex is a combinatorial three-manifold [24], and hence induces a PL structure on \(S\). By [15], an isotopy \(h_t:S\to S\) from the identity ends at a PL homeomorphism from this structure to the fixed one. The triangulation \(h_1(K)\) has the same abstract flag complex and is PL for the fixed structure. For this comparison, use the panel-compatible collared CW representative of \(P\) for \(h_1(K)\) on both sides. Extend \(h_t\) across its collar, fixed away from that collar, to an isotopy with endpoint \(h_P\), and use the pulled-back CW structure on the \(K\) side. The relative comparison maps from Section 3 fix \(S\). Since \(\pi_1(S)\to H\) is surjective, they preserve the marking of \(H\) and the folding character, so the induced reflected homotopy equivalences lift to the corresponding cyclic covers. Identify the full label spaces by \(e_u\mapsto e_{h_1(u)}\). With this identification, the map \([a,x]\mapsto[a,h_P(x)]\) is a homeomorphism from \(Q^0(K)\) to \(Q^0(h_1(K))\), since it carries each panel to its corresponding panel. It also identifies the cyclic covers: the folding maps commute with \(h_P\), and an isotopy to the identity preserves the character \(w\) on fundamental groups, up to an inner automorphism. The PL case therefore supplies a model for the double cover of \(Q^0(K)\) as well. ◻

Proof. An element of \(\mathsf A\times\mathbb F_2\) acting trivially on \(Q_2\) must fix a point in a chamber interior. Interior coordinates are not identified between labels, and the nontrivial deck transformation of \(P_2\) fixes no point. The element therefore has both coordinates zero, proving faithfulness.

For \(a\in\mathsf A\), define a map on the copies of \(V\) by changing the label from \(b\) to \(b+a\) and using the identity on \(V\). Its boundary coordinates satisfy \[h_{b+a}(v)=\theta^{\epsilon(a)}h_b(v).\] Thus two previously identified boundary coordinates are both changed by \(\theta^{\epsilon(a)}\). Their projections to \(S\) stay the same, and their label difference stays the same. The map respects (17), and descends to a homeomorphism \[T^\epsilon_a:M_\epsilon\longrightarrow M_\epsilon,\qquad T^\epsilon_a[b,v]=[b+a,v].\] The maps compose according to addition in \(\mathsf A\). Their action is faithful because an interior point in a chamber labeled \(b\) is sent to an interior point in the distinct chamber labeled \(b+a\) when \(a\ne0\).

Finally, linearity of \(\epsilon\) gives the exact equality \[\begin{split} f_\epsilon T^\epsilon_a[b,v] &=[b+a,\theta^{\epsilon(b+a)}j(v)]\\ &=(a,\epsilon(a))[b,\theta^{\epsilon(b)}j(v)] =(a,\epsilon(a))f_\epsilon[b,v]. \end{split}\] This is the required equivariance. Passing to fundamental groups and suppressing the basepoint paths gives the stated equality of outer automorphisms. ◻

Proof. The local assertions about the triangulation and its links follow by lifting PL charts. The action is free because a deck transformation of a connected covering that fixes a point is the identity. Its quotient is \(\mathcal L\) because the covering is regular with deck group \(H\).

A clique upstairs projects to a clique of distinct vertices downstairs. Flagness gives a simplex on their images. Lift this simplex from one vertex of the clique. Uniqueness of the lifts of its incident edges places every other vertex of the clique in that lifted simplex. Thus \(\widehat{\mathcal L}\) is flag.

For a four-cycle upstairs, adjacent vertices have distinct images. Opposite vertices also have distinct images: otherwise, at an intervening vertex, two incident lifted edges would be lifts of the same downstairs edge. The projected four-cycle therefore has four distinct vertices. It has a diagonal because \(\mathcal L\) has no induced four-cycle. The diagonal and two consecutive sides span a triangle downstairs by flagness. Lifting that triangle from their common vertex lifts the diagonal to the original four-cycle. Consequently the lifted cycle is not induced. ◻

Proof. Killing every generator outside \(J\) defines a retraction onto the Coxeter group of the induced graph on \(J\). Its composition with the natural map of that group into \(W(\mathcal G)\) is the identity, so that natural map is injective and gives \(W_J\). If \(d\in W_J\cap W_K\), apply the retraction onto \(W_J\) to a word for \(d\) on \(K\). The result is a word on \(J\cap K\) for \(d\), proving the intersection formula.

Choose a finite set \(F\) supporting a word for \(d\). Intersect \(F\) with every supporting set. Because \(F\) is finite, the result is already the intersection of finitely many of those sets; repeated use of the intersection formula shows that it still supports \(d\). It is therefore the unique smallest support. Its characterization makes it equivariant under graph automorphisms. This is the right-angled form of the standard Coxeter support statement [9]. ◻

Proof. The contractibility assertion is the universal-development argument in [23]. Its chamber hypotheses are the ones verified for \((P,S)\) in the reflection construction. That argument filters the chambers of (26) by word length in \(W\). A new chamber is contractible and meets the preceding chambers in the nonempty ball of its descent panels. The argument treats an infinite word-length layer simultaneously and takes a CW direct limit, giving a contractible development even when the lifted generator set is infinite. This is the lifted-mirror construction of Davis [9]; Equation (9.3) there is (27).

Here is the quotient and covering check for the label groups in this paper. The set \(J(x)\) projects bijectively to the set of panels meeting \(\overline x\). The restrictions of \(\chi_0\) and \(\chi\) to \(W_{J(x)}\) are therefore injective: for \(\chi_0\) the images are distinct coordinate vectors, and for \(\chi\) they are the simplex-independent labels \(a_s\). Their images are exactly the label subgroups used in the respective reflection identifications. This proves that the three displayed maps are well-defined. They are onto because every label has a representative in \(W\) and every chamber coordinate lifts.

We check their fibers uniformly. Use \(\lambda=\chi_0\) or \(\chi\) and the chamber cover \(P_d\), where the three cases are \((\lambda,d)=(\chi_0,1),(\chi,1),(\chi,2)\), with \(P_1=P\), \(H_1=H\), and \(H_2=w^{-1}(2\mathbb Z)\). For these cases put \[K_{\lambda,d} =\{(d_0,h)\in\mathcal E:\lambda(d_0)=0,\ h\in H_d\}.\] Suppose \([c,x]\) and \([c',x']\) have the same image. Equality of their chamber coordinates gives \(x'=hx\) for some \(h\in H_d\), and equality of the reflected points says that \(\lambda(c')-\lambda(c)\) belongs to \(\lambda(W_{J(x')})\). Choose \(t\in W_{J(x')}\) with \[\lambda(t)=\lambda(c')-\lambda(c).\] Then \(d_0=c't\,h(c)^{-1}\) satisfies \(\lambda(d_0)=0\), since \(\lambda\) is \(H\)-invariant and the label groups have characteristic two. Formula (27) sends \([c,x]\) by \((d_0,h)\) to \([c't,x']=[c',x']\). Conversely, every element of \(K_{\lambda,d}\) preserves the displayed map. Thus its fibers are exactly the \(K_{\lambda,d}\)-orbits.

These orbit maps are covering maps, including at the panels. At an interior point choose an evenly covered chamber neighborhood disjoint from the boundary. At a boundary point choose a small lifted collar neighborhood meeting only the panels in \(J(x)\) and mapping homeomorphically to its chamber neighborhood. The adjacent chamber sectors are indexed by \(W_{J(x)}\) upstairs and by \(\lambda(W_{J(x)})\) downstairs. The isomorphism between these two finite groups identifies their sector gluings, so each such assembled neighborhood maps homeomorphically onto the corresponding neighborhood in the reflection. The different \(K_{\lambda,d}\)-translates give the disjoint sheets. The point stabilizer in \(\mathcal E\) at \([c,x]\) is \(cW_{J(x)}c^{-1}\): a fixing element has \(h=1\) because \(H\) acts freely on \(\widetilde P\), and the relation (26) then gives this conjugate. The subgroup \(K_{\lambda,d}\) meets it trivially by the injectivity just proved. These are precisely the covering charts used in the cited universal-development argument.

For comparison with the notation of [23], let \(C_{\mathcal L}\) be the downstairs right-angled Coxeter group. The homomorphism forgetting the lift of each generator is \(q\colon W\to C_{\mathcal L}\), and that paper uses \(\Delta=\ker(C_{\mathcal L}\to\mathsf A_0)\) and \(\rho(d,h)=q(d)\). Its group \(\rho^{-1}(\Delta)\) is exactly \(\ker\chi_0\). Thus the first identification in (28) is also the group formula in the cited subsection of [23]. The fiber calculation gives the identifications for \(Q\) and \(Q_2\). Since \(\widetilde Q\) is contractible, these coverings are universal.

Finally, the map \([c,x]\mapsto x\) from \(\widetilde Q\) to \(\widetilde P\) is equivariant for the projection \(\mathcal E\to H\). It covers the folding map \(Q\to P\). Consequently the fold-induced homomorphism on \(G\) is that projection, which proves the assertion about \(w\). ◻

Proof. The subgroup \(\Pi\) is normal in \(\mathcal E\) by (28). Suppose that \((d,h)\) centralizes \(\Pi\). It commutes with \((1,k)\) for every \(k\in H_2\subseteq\Pi\). The semidirect-product formula gives \[hk=kh,\qquad d=k(d)\qquad(k\in H_2).\] The finite-index subgroup \(H_2\) of the nonabelian free group \(H\) is itself nonabelian. If \(h\ne1\), the first equality would put \(H_2\) inside the cyclic centralizer of \(h\) in \(H\), a contradiction. Thus \(h=1\).

The second equality and Lemma 18 make the finite set \(\mathop{\mathrm{supp}}(d)\) invariant under \(H_2\). Every \(H_2\)-orbit of vertices is infinite: \(H_2\) is infinite and its deck action is free. Hence \(\mathop{\mathrm{supp}}(d)\) is empty, so \(d=1\). This proves the injection and also proves that \(Z(\Pi)=1\).

Conjugation by the elements of \(\Pi\) gives exactly \(\mathop{\mathrm{Inn}}(\Pi)\), and it is injective because the center is trivial. Taking quotients now gives (30) through (25). To see that the preimage is full, let an automorphism of \(\Pi\) have the same outer class as conjugation by some \(e\in\mathcal E\). It differs from that conjugation by conjugation by an element \(\gamma\in\Pi\), and is therefore conjugation by \(\gamma e\in\mathcal E\). Taking the preimages of the two factors and using (29) gives the last assertion. ◻

Proof. In the covering map to \(Q_2\) from Proposition 19, Equation (27) changes the label by \(\chi(d,h)\) and the chamber coordinate by the action of \(h\) on \(P_2\). The latter action is \(\theta^{w(h)\bmod 2}\). This gives the displayed formula. Conjugation on the universal deck group is the induced outer automorphism, so (30) identifies these actions with the stated outer classes. The equivariance of the model maps in Proposition 16 then identifies the classes of the graph actions on the manifolds. ◻

Proof of Proposition 22 and Theorem 2. The quotient reflection \(Q\) is a connected finite aspherical complex. Its folding homomorphism is the epimorphism \(w\colon G\to\mathbb Z\) of Proposition 19. The space \(Q_2\) has the closed topological four-manifold model \(M\) by Proposition 14, whereas Proposition 12 excludes such a model for every odd \(d\geq3\). These facts verify all hypotheses of Theorem 3, with \(n=4\).

Choose \(h\in H\) with \(w(h)=1\). The element \(\gamma=(1,h)\) lies in \(G=\ker\chi\), has \(w(\gamma)=1\), and maps to \((0,1)\) under (25). Consequently conjugation by \(\gamma\) on \(\Pi\) represents \(\delta\) in (30). The cyclic-cover criterion gives a self-homotopy equivalence \(f\colon M\to M\) inducing this class, and asserts that no homeomorphism of \(M\) induces it. This proves Proposition 22. A map homotopic to \(f\) induces the same outer automorphism, so \(f\) is not homotopic to a homeomorphism. This proves Theorem 2. ◻

Proof. Consider the property that an outer automorphism has a finite-order lift to \(\mathop{\mathrm{Aut}}(\Pi)\). This property is invariant under conjugation in \(\mathop{\mathrm{Out}}(\Pi)\): if \(F\) represents the conjugating outer automorphism and \(\alpha\) is a finite-order lift, then \(F\alpha F^{-1}\) is a finite-order lift of the conjugate.

Every lift of an element \((a,1)\) in (30) belongs to \(\mathcal E\) by Proposition 20. Its projection to \(H\) has odd, hence nonzero, \(w\)-value. It therefore has infinite order in \(\mathcal E\) and, by the injection into \(\mathop{\mathrm{Aut}}(\Pi)\), infinite order as an automorphism. Thus no \((a,1)\) has the property. In contrast, each \((a_s,0)\) has a finite-order lift: any Coxeter generator above \(s\) is an involution in \(\mathcal E\) lifting it.

The graph subgroup \(\mathsf B_\ell\) and the horizontal subgroup \(\mathsf A\) have identical elements over \(\ker\ell\). Every element of \(\mathsf B_\ell\) over its complement has second coordinate \(1\) and lacks a finite-order lift. Some \(a_s\) lies in that complement, since the \(a_s\) span \(\mathsf A\) and \(\ell\ne0\); the corresponding element of \(\mathsf A\) does have such a lift. Hence \(\mathsf B_\ell\) has strictly fewer elements with the property than \(\mathsf A\). Conjugate finite subgroups have the same number, proving the claim. ◻

Proof. The group \(H\) is finitely generated: its free action on the connected locally finite graph \(\widehat{\mathcal L}^{(1)}\) has finite quotient, so lifting a spanning tree of that quotient leaves only finitely many edge identifications needed to generate the deck group. Choose the Cayley tree \(T\) of a finite free basis of \(H\). It is locally finite, and \(H\) acts freely and cocompactly without inversions.

Choose points \(a_u\in T^{(0)}\), equivariantly in \(u\in\widehat{\mathcal V}\); one chooses them on representatives of the finitely many \(H\)-orbits and then translates. Freeness of the deck action makes this assignment well defined. There are also only finitely many orbits of edges of \(\widehat{\mathcal L}\). Thus \(d_T(a_u,a_{u'})\), for adjacent \(u,u'\), has a common finite bound. Choose an integer \(r\geq 0\) at least this bound and let \(T_u\) be the closed combinatorial ball of radius \(r\) about \(a_u\). These balls are subtrees. Adjacent vertices have supports meeting at a vertex, and the number of vertices in each support is bounded uniformly by a finite number \(b_T\). In particular \(b_T\geq 1\).

For a vertex \(t\) and an edge \(e\) of \(T\), define \[\mathcal U_t=\{u:t\in T_u\},\qquad \mathcal U_e=\{u:e\subset T_u\}.\] Both kinds of set have uniformly bounded finite size. Indeed, a support containing \(t\) has its center in the radius-\(r\) ball about \(t\); that ball contains uniformly finitely many orbit points, and there are only finitely many generator orbits. The assertion for \(\mathcal U_e\) follows from the assertion for either endpoint. Since each \(T_u\) is a subtree, for \(e=[t,t']\) we have \[ \mathcal U_e=\mathcal U_t\cap\mathcal U_{t'}. \tag{32}\] Let \(W_t\) and \(W_e\) be the special subgroups of \(W\) generated by \(\mathcal U_t\) and \(\mathcal U_e\), respectively. The edge maps \(W_e\longrightarrow W_t,W_{t'}\) are special subgroup inclusions.

These groups form a graph of groups over \(T\) whose fundamental group is \(W\). To see the presentation explicitly, the copies of a generator \(u\) in the vertex groups are identified along precisely the connected subtree \(T_u\), by (32). Every commutation relation of \(W\) occurs in a vertex group, since the supports of its two adjacent generators meet at a vertex. Each vertex group is defined by the induced subgraph on \(\mathcal U_t\), so it introduces only relations of \(W\). The resulting presentation is therefore the right-angled presentation of \(W\). Special subgroups inject, so the Bass–Serre tree of this graph of groups has vertices \((aW_t,t)\) and edges \((aW_e,e)\), with the evident endpoint maps. This Bass–Serre tree, which can have infinite valence, is the tree over which the next spaces are assembled.

For a finite set \(J\subset\widehat{\mathcal V}\), let \(\Sigma_J\) denote the Davis cube complex of \(W_J\). Its vertices are the elements of \(W_J\). For each clique \(C\subset J\) and each coset \(kW_C\) in \(W_J\), it has a cube whose vertex set is \(kW_C\); its dimension is \(|C|\), since \(W_C\cong(\mathbb Z/2)^{|C|}\). For \(J=\varnothing\) this is a single vertex. These complexes are CAT(0), and inclusions of special subgroups induce closed convex cubical inclusions of their Davis complexes; see [9]. Set \(\Sigma_t=\Sigma_{\mathcal U_t}\) and \(\Sigma_e=\Sigma_{\mathcal U_e}\).

Following the support construction of [23], use the vertex spaces \(W\times_{W_t}\Sigma_t\) and the edge cylinders \[W\times_{W_e}\bigl(\Sigma_e\times[0,1]\bigr).\] Here \(W\times_K Z\) identifies \((ak,z)\) with \((a,kz)\) for \(k\in K\). Choose an ordering \(e=[t,t']\) of the endpoints of each edge of \(T\), and glue the cylinder at \(0\) and \(1\) into the corresponding vertex spaces by the special-subcomplex inclusions. Give \([0,1]\) its single-edge cubulation. The resulting cubical complex \(X\) is a tree of spaces over the Bass–Serre tree just described.

We can now identify its graph. A vertex of \(W\times_{W_t}\Sigma_t\) is represented by \([a,k]\) with \(k\in W_t\), and \([a,k]\mapsto ak\) is a bijection from these vertices to \(W\). An edge in \(\Sigma_t\) joins \(k\) to \(ku\) for \(u\in\mathcal U_t\), so the vertex-space edges are exactly the first line of (31). The same identification gives one vertex labeled by each element of \(W\) in \(W\times_{W_e}\Sigma_e\). Each endpoint inclusion preserves this label. Thus its cylinder has an edge from \((a,t)\) to \((a,t')\) for every \(a\in W\). The cylinder has no vertices over the interior of \([0,1]\); its remaining edges lie in its endpoint spaces and are already vertex-space edges. This proves that (31) lists every edge, including when \(\mathcal U_e=\varnothing\).

For completeness, the local and metric properties in the cited construction are compatible with this graph description. For a fixed edge \(e\) incident to \(t\), the translates of \(\Sigma_e\) in a vertex copy of \(\Sigma_t\) have as their vertex sets the cosets of \(W_e\) in \(W_t\). Those cosets partition the Cayley vertices. Hence a finite subcomplex meets only finitely many such translates, even if the Bass–Serre vertex has infinite valence. There are only finitely many edges of \(T\) incident to \(t\), and \(\Sigma_t\) is locally finite. It follows that \(X\) is locally finite. A vertex cube has dimension at most four and an edge cylinder at most five, by the clique bound. The unit-cube length metric is consequently proper and complete. Convex CAT(0) gluing first over finite subtrees of the Bass–Serre tree, and then over their union, proves that \(X\) is CAT(0): the finite-subtree gluings are CAT(0) and convex in larger ones, and completeness follows from the local finiteness just checked.

Left multiplication defines the \(W\)-action. Equivariance of the supports defines the \(H\)-action on the vertex and edge spaces, and on vertices the combined action is \((b,h)(a,t)=(b\,h(a),ht)\). This action is free on vertices: if it fixes \((a,t)\), then \(h\) fixes \(t\), whence \(h=1\), and then \(ba=a\) gives \(b=1\). Local finiteness makes the cellular action proper. There are finitely many vertex orbits, because \(W\) acts transitively on the first coordinate and \(T/H\) is finite. Local finiteness then gives finitely many cube orbits, so the action is cocompact. ◻

Proof. First consider any embedded four-cycle in \(X^{(1)}\). Projection to \(T\) shows that a cycle containing tree edges has either two or four of them. Four tree edges would give an embedded cycle in a tree, because the \(W\)-coordinate would be constant. With two tree edges, the projected closed walk traverses the same tree edge twice. They cannot be consecutive in the four-cycle, since two such consecutive steps would repeat a vertex. They are therefore opposite. The two remaining edges are also opposite and have the same Coxeter label, as their product in \(W\) is the identity.

If all four edges have Coxeter labels, their label word is trivial in \(W\). The abelianization of \(W\) is the direct sum of copies of \(\mathbb Z/2\) indexed by \(\widehat{\mathcal V}\), so each label occurs an even number of times. Embeddedness forbids consecutive equal labels, including at the closing vertex. Thus two distinct labels alternate. They commute: if they were nonadjacent, the special subgroup on them would be the infinite dihedral group, in which their alternating word of length four is nontrivial. We have proved that opposite edges of every embedded four-cycle have equal labels, adjacent edges cannot both have tree labels, and adjacent Coxeter labels are distinct and commute.

Let an isometric grid of side length \(n\) be given. Each of its elementary four-cycles is embedded. Equality of labels on opposite edges shows that the label at a fixed horizontal step is independent of the vertical coordinate, and the label at a fixed vertical step is independent of the horizontal coordinate. If each direction had a tree label, the elementary square where those steps meet would have adjacent tree edges. After interchanging the directions if necessary, all vertical labels are therefore Coxeter labels.

Suppose \(n\geq16\). The vertical labels include two that do not commute. Indeed, if they all commuted, their distinct labels would form a clique of size at most four. They would generate a group of order at most \(2^4=16\). Each column has constant tree coordinate, so its \(n+1\) distinct vertices would have at most 16 possible \(W\)-coordinates, a contradiction.

Every horizontal Coxeter label is distinct from and commutes with every vertical label, by the elementary-square calculation. If two horizontal Coxeter labels did not commute, they and the two noncommuting vertical labels would be four distinct vertices whose cross pairs are adjacent and whose two same-direction pairs are nonadjacent. They would form an induced four-cycle in \(\widehat{\mathcal L}\). Hence the horizontal Coxeter labels form a clique and generate a group of order at most 16. Along any one row, tree edges leave the \(W\)-coordinate unchanged, so all \(W\)-coordinates lie in a single left coset of this finite group.

Fix one vertical label \(u\). Each column has constant tree coordinate, and the \(u\)-edge occurs in every column at the same vertical step. The edge description (31) therefore places every tree coordinate of the row in \(T_u^{(0)}\). There are at most \(b_T\) such coordinates and at most 16 \(W\)-coordinates. The row has \(n+1\) distinct vertices, so \[n+1\leq16b_T.\] This proves the claimed bound when \(n\geq16\). If \(n\leq15\), the same bound follows from \(b_T\geq1\). ◻

Proof. Use the notation and hyperplane facts above for \(Y\). Fix \(v,v'\in I(x,y)\) with \(d_1(x,v)=d_1(x,v')\). Then \(J_x(v),J_x(v')\subset J_x(y)\) and these two finite sets have equal cardinality. Put \[B=J_x(v)\cap J_x(v'),\qquad n=|J_x(v)\setminus J_x(v')| =|J_x(v')\setminus J_x(v)|.\] For \(m=\operatorname{med}(x,v,v')\), Equation (34) gives \(J_x(m)=B\). Choose edge geodesics \[v_0=m,v_1,\ldots,v_n=v,\qquad v'_0=m,v'_1,\ldots,v'_n=v'.\] Their lengths are \(n\) by (33). Along each geodesic, the separating sets add exactly one hyperplane at each step: its endpoint set contains its initial set, and removing a hyperplane or adding one outside the endpoint set would require an extra crossing. Thus \[J_x(v_i)=B\cup A_i,\qquad J_x(v'_j)=B\cup A'_j,\] where \(A_i\) and \(A'_j\) are increasing chains of sets of cardinalities \(i\) and \(j\), respectively. All \(A_i\) lie in \(J_x(v)\setminus J_x(v')\), and all \(A'_j\) lie in the disjoint set \(J_x(v')\setminus J_x(v)\).

For \(0\leq i,j\leq n\), set \(z_{ij}=\operatorname{med}(y,v_i,v'_j)\). Since the two separating sets for \(v_i,v'_j\) are contained in \(J_x(y)\), the majority formula gives \[J_x(z_{ij})=J_x(v_i)\cup J_x(v'_j)=B\cup A_i\cup A'_j.\] The two chains add disjoint hyperplanes. Their nesting and (33) therefore give \[d_1(z_{ij},z_{kl})=|i-k|+|j-l| \qquad (0\leq i,j,k,l\leq n).\] In particular these vertices are distinct and form an isometrically embedded square grid of side length \(n\). Hence \(n\leq K\), and \[d_1(v,v')=|J_x(v)\mathbin{\triangle}J_x(v')|=2n\leq2K.\] This proves the uniform assertion about intervals.

For vertices \(a,b,c\), let \(m\) be their median and choose one edge geodesic from \(m\) to each corner. Concatenating the appropriate two chosen paths gives a geodesic between each pair of corners, because \(m\) belongs to every pairwise interval. Each of these three chosen sides lies in the union of the other two.

Now take arbitrary edge-geodesic sides of the triangle. A vertex on the arbitrary side from \(a\) to \(b\) and the vertex at the same distance from \(a\) on the chosen side both belong to \(I(a,b)\), so they are at distance at most \(2K\). The latter vertex lies on the chosen side from \(a\) to \(c\) or the chosen side from \(b\) to \(c\). Matching its distance from the corresponding endpoint on that arbitrary side costs at most another \(2K\). Thus every vertex on each arbitrary side is within \(4K\) of the other two sides. This vertex formulation is the usual thin-triangle criterion for the unit-edge graph. ◻

Proof. Lemmas 25 and 26, with \(K=16b_T-1\), show that \(X^{(1)}\) is a hyperbolic graph. Its unit-edge metric is proper because \(X\) is locally finite. The \(\mathcal E\)-action on this graph is proper and cocompact by Lemma 24. The Švarc–Milnor lemma [6] gives a quasi-isometry from a Cayley graph of \(\mathcal E\) to \(X^{(1)}\). Hyperbolicity is invariant under quasi-isometry [6], so \(\mathcal E\) is word-hyperbolic. The subgroup \(\Pi\) has finite index in \(\mathcal E\), and hence is word-hyperbolic as well. ◻

Proof. The finite aspherical complex \(Q_2\) is a finite-dimensional \(K(\Pi,1)\). Its lifted cellular chain complex gives \(\operatorname{cd}_{\mathbb Z}\Pi<\infty\). A group of finite integral cohomological dimension is torsion-free: if it contained a subgroup of prime order, restricting a finite projective resolution to that subgroup would contradict the nonzero integral cohomology of a cyclic group of prime order in arbitrarily high degrees. Thus \(\Pi\) is torsion-free. Also, \(\Pi\cap H\) has finite index in the nonabelian free group \(H\), so it is a nonabelian free group. In particular, \(\Pi\) is infinite and is not virtually cyclic.

Suppose that \(\Pi\) has a splitting as an amalgamated product or HNN extension with virtually cyclic edge group \(U\). Assume the splitting is nontrivial, meaning that the Bass–Serre tree \(\mathcal B\) has no global fixed vertex; the action has no inversions. Its unoriented edges form the \(\Pi\)-set \(\Pi/U\). Fix a vertex \(p\) of \(\mathcal B\), and define \[c(g)=\sum_{e\text{ on }[p,gp]}e \ \in\ \mathbb F_2[\Pi/U]\qquad(g\in\Pi),\] where the sum is empty when \(gp=p\). All sums have finite support. Concatenating the paths from \(p\) to \(gp\) and from \(gp\) to \(ghp\), and canceling edges traversed twice, gives the path from \(p\) to \(ghp\). Over \(\mathbb F_2\) this says \[c(gh)=c(g)+g\,c(h).\] Thus \(c\) is a group-cohomology cocycle.

The orbit of \(p\) is unbounded. Otherwise its bounded invariant subtree would have a center fixed by \(\Pi\); the center is a vertex or an edge midpoint, and in the latter case the absence of inversions also fixes its endpoints. Either case contradicts nontriviality. Consequently \[|\mathop{\mathrm{supp}}c(g)|=d_{\mathcal B}(p,gp)\] is unbounded as \(g\) varies. On the other hand, a coboundary \(gq-q\), for \(q\in\mathbb F_2[\Pi/U]\), has support size at most \(2|\mathop{\mathrm{supp}}q|\) for every \(g\). Hence \[ [c]\ne0\quad\text{in}\quad H^1(\Pi;\mathbb F_2[\Pi/U]). \tag{35}\]

Let \(M\) be the closed aspherical topological four-manifold with fundamental group \(\Pi\). Poincaré duality with local coefficients on \(M\) applies to arbitrary \(\mathbb F_2\Pi\)-modules, including the possibly infinite-dimensional permutation module \(\mathbb F_2[\Pi/U]\); see [27]. The orientation twist is trivial over \(\mathbb F_2\). Using the description of local coefficients by modules over the fundamental group in [16], asphericity and duality therefore give \[H^1(\Pi;\mathbb F_2[\Pi/U]) \cong H_3(\Pi;\mathbb F_2[\Pi/U]).\] The permutation module is the induced module \(\mathbb F_2\Pi\otimes_{\mathbb F_2U}\mathbb F_2\), with trivial \(U\)-action on the last factor. Homological Shapiro’s lemma [29] gives \[ H_3(\Pi;\mathbb F_2[\Pi/U])\cong H_3(U;\mathbb F_2). \tag{36}\] Indeed, if \(P_*\) is a free right \(\mathbb F_2\Pi\)-resolution of \(\mathbb F_2\), then \[P_*\otimes_{\mathbb F_2\Pi} (\mathbb F_2\Pi\otimes_{\mathbb F_2U}\mathbb F_2) \cong P_*\otimes_{\mathbb F_2U}\mathbb F_2.\] The restriction of \(P_*\) is a free \(\mathbb F_2U\)-resolution, so the right-hand complex computes the homology of \(U\). This also makes explicit why the induced, finite-support module is the one used here.

A torsion-free virtually cyclic group is either trivial or infinite cyclic [28]. Thus \(U\) has a classifying space that is a point or a circle, and \(H_3(U;\mathbb F_2)=0\); see [29]. Duality and (36) now contradict (35). This proves nonsplitting.

Finally \(\Pi\) is finitely generated by Proposition 27 and is infinite. Stallings’ ends theorem [28] says that a finitely generated group with more than one end has a nontrivial splitting over a finite subgroup. Such a subgroup is virtually cyclic, so the splitting just excluded would exist if \(\Pi\) had more than one end. Therefore \(\Pi\) has one end. ◻

Proof. Bestvina and Feighn, using Paulin’s construction of a limiting action on an \(\mathbb R\)-tree [25], prove that a word-hyperbolic group with infinite outer automorphism group splits nontrivially over a virtually cyclic subgroup [3]. The group \(\Pi\) is word-hyperbolic by Proposition 27, while Lemma 28 excludes every such splitting. The stated splitting theorem therefore implies that \(\mathop{\mathrm{Out}}(\Pi)\) is finite. ◻

Proof. Put \(L=\mathop{\mathrm{lk}}_{\mathcal G}(u)\). The presentations and the special-subgroup injections of Lemma 18 give \[W(\mathcal G) =W_{V\setminus\{u\}}*_{W_L} \bigl(W_L\times\langle u\rangle\bigr).\] If \(u\) is adjacent to every other vertex, the second factor is the whole group and the centralizer assertion is immediate. Otherwise consider the Bass–Serre tree of this amalgam. The element \(u\) fixes the vertex of the second factor. It fixes no edge incident to that vertex: every such edge stabilizer is a conjugate of \(W_L\) within \(W_L\times\langle u\rangle\), hence is \(W_L\), which does not contain \(u\). The fixed subtree of \(u\) is therefore that single vertex. Its centralizer must stabilize this vertex and so lies in the second factor. That factor centralizes \(u\), proving the first assertion.

For the second assertion, a finite-order element lies in a special subgroup on a finite supporting set by Lemma 18. Work in that finite induced graph and induct on its number of vertices. A finite-order element in the displayed amalgam fixes a vertex of its Bass–Serre tree, so is conjugate into one of the factors. The first factor has fewer generators. In the second factor its projection to \(W_L\) has finite order; induction conjugates that projection to a product on a clique in \(L\). Multiplying by the possible \(u\)-factor still gives a clique. This also covers the degenerate direct-product case, and starts with the trivial group.

Finally, an element is central exactly when it belongs to all the centralizers in the first assertion. By Lemma 18, its support must then lie in \(\bigcap_{u\in V}\mathop{\mathrm{st}}_{\mathcal G}(u)\), the set of universal vertices. Every generator in that set is central. This gives the asserted equality, also for an infinite graph. ◻

Proof. Let \(\mathcal T\subseteq W_{\mathrm{ab}}\) consist of the vectors represented by finite-order elements of \(W\). Lemma 30 shows that these are exactly the vectors supported on cliques, including the zero vector: the converse holds because a product of commuting involutions has finite order.

We first recover tetrahedra from this set. Suppose a vector subspace \(U\) is contained in \(\mathcal T\). Any two coordinates that occur somewhere in \(U\) occur together in one vector of \(U\): choose a witness for each coordinate; if neither witness has both coordinates, their sum does. That vector is supported on a clique. It follows that the union of all supports in \(U\) is a clique. Conversely, the full span of every clique lies in \(\mathcal T\). Hence a maximal subspace contained in \(\mathcal T\) is exactly the full span of a maximal clique. These spaces are \[V_\tau=\operatorname{span}_{\mathbb F_2} \{\mathbf e_u:u\text{ is a vertex of }\tau\},\] where \(\tau\) is a maximal clique. The triangulation is flag and is a pure three-manifold triangulation, so its maximal cliques are precisely its tetrahedra.

We next recover each coordinate line from those subspaces. For a vertex \(u\), \[ \bigcap_{\tau\ni u}V_\tau=\langle\mathbf e_u\rangle, \tag{37}\] where the intersection ranges over tetrahedra containing \(u\). Indeed, if another vertex \(v\) lay in all those tetrahedra, then \(v\) would occur in every maximal triangle of \(\mathop{\mathrm{lk}}_{\widehat{\mathcal L}}(u)\). This link is a pure PL two-sphere. Every one of its simplices lies in a maximal triangle, so the link would equal the star of \(v\) and would be a cone. That contradicts its being a two-sphere. This proves (37). Conversely, every one-dimensional intersection of tetrahedron subspaces is a coordinate line: an intersection of subspaces spanned by subsets of a fixed basis is the span of the intersection of those subsets.

An automorphism of \(W\) preserves \(\mathcal T\) and hence permutes its maximal vector subspaces. Its induced invertible linear map preserves intersections, so (37) and the converse show that it permutes all coordinate lines. Over \(\mathbb F_2\) it therefore permutes the coordinate vectors themselves. For distinct \(u,v\), adjacency is equivalent to \(\mathbf e_u+\mathbf e_v\in\mathcal T\). The permutation preserves adjacency, and flagness makes it a simplicial automorphism.

The image of a Coxeter generator has finite order. By Lemma 30 it is conjugate to a clique product; its abelianization is the sum of the coordinates in that product. Since the abelianization just obtained is one coordinate vector, the product consists of the corresponding single generator. Composing with the inverse generator permutation gives the final assertion. ◻

Proof. The links in the statement are flag PL spheres of dimensions \(2\), \(1\), and \(0\). We prove the assertion for them in increasing dimension, and then for \(\widehat{\mathcal L}\). The link of a vertex at each induction step is one of the lower-dimensional links already treated.

For the zero-sphere, \(W(X)=\langle r,s\mid r^2=s^2=1\rangle\) is infinite dihedral. After an inner adjustment, arrange that \(\varphi(r)=r\). With \(t=rs\), every conjugate of \(s\), and hence its image, has the form \(t^j r\). The subgroup generated by \(r\) and \(t^j r\) is \(\langle r,t^j\rangle\); it is the whole group only if \(j=\pm1\). Since \(s=t^{-1}r\) and \(rsr=tr\), these two possibilities give the identity or conjugation by \(r\). This proves the base case.

Now let \(X\) have positive dimension and suppose the result is known for its vertex links. For each vertex \(u\), we claim that one element \(g_u\in W(X)\) implements \(\varphi\) on all the generators of its star. Choose \(b_u\) with \(\varphi(u)=b_u u b_u^{-1}\) and put \(\psi=c_{b_u^{-1}}\circ\varphi\), where \(c_g\) denotes conjugation by \(g\). The automorphism \(\psi\) fixes \(u\) and therefore preserves its centralizer. Flagness identifies that centralizer, by Lemma 30, as \[ C_{W(X)}(u) =\langle u\rangle\times W(\mathop{\mathrm{lk}}_X(u)). \tag{38}\] For a link generator \(v\), write \(\psi(v)=q_v v q_v^{-1}\). The image belongs to the star subgroup because it commutes with \(u\). Apply the star retraction from Lemma 18 to this equation. Writing the retracted conjugator as \(u^{\epsilon_v}k_v\) under (38), with \(k_v\in W(\mathop{\mathrm{lk}}_X(u))\), gives \[\psi(v)=k_v v k_v^{-1}.\] Thus \(\psi\) maps the link subgroup into itself and conjugates each of its generators inside that subgroup.

This restricted map is onto. Given \(y\in W(\mathop{\mathrm{lk}}_X(u))\), surjectivity of \(\psi\) on the centralizer supplies a preimage \(u^\epsilon k\) with \(k\in W(\mathop{\mathrm{lk}}_X(u))\). Because \(\psi(u)=u\) and \(\psi(k)\) is in the link subgroup, \(y=u^\epsilon\psi(k)\). Uniqueness of the direct-product coordinates forces \(\epsilon=0\). Hence the restriction of \(\psi\) is a conjugating automorphism of the link subgroup. By induction it is conjugation by some \(k\in W(\mathop{\mathrm{lk}}_X(u))\). That conjugation also fixes \(u\), so \(\psi\) is conjugation by \(k\) on the whole star. Undoing the adjustment gives the claimed element \(g_u=b_uk\).

We now synchronize the elements \(g_u\). If \(u,v\) are adjacent, put \(d_{uv}=g_u^{-1}g_v\). The two star conjugations agree on every generator in the intersection of the stars, so \(d_{uv}\) centralizes all those generators. In particular it centralizes \(u\) and \(v\). The centralizer formula and the intersection formula of Lemma 18 first put it in the special subgroup on the intersection of the star vertex sets. Flagness identifies the common neighbors with the vertices of the edge link, giving \[d_{uv}\in W_{\mathop{\mathrm{st}}_X(u)\cap\mathop{\mathrm{st}}_X(v)} =\langle u,v\rangle\times W(\mathop{\mathrm{lk}}_X(\{u,v\})).\] It is central in this subgroup. The link factor has trivial center. If the edge link is empty this is immediate. Otherwise it is a flag PL sphere. A universal vertex in its graph would extend every simplex to that vertex by flagness, making the sphere a cone. There is no such vertex, and the center formula of Lemma 30 applies. Consequently \[ d_{uv}\in\langle u,v\rangle,\qquad d_{uv}=u^{\lambda_{u,v}}v^{\lambda_{v,u}}, \quad \lambda_{u,v},\lambda_{v,u}\in\mathbb F_2. \tag{39}\] The subgroup \(\langle u,v\rangle\) is elementary abelian, so \(d_{vu}=d_{uv}^{-1}=d_{uv}\); the two coefficients in (39) depend on the underlying edge and its indicated endpoint, not on its orientation.

For a triangle with vertices \(u,v,z\), the exact identity \(d_{uv}d_{vz}d_{zu}=1\) follows by cancellation of the \(g\)’s. In the abelianization of \(W(X)\), its \(u\)-coordinate says \[\lambda_{u,v}=\lambda_{u,z}.\] When \(\dim X\geq2\), the link of every vertex is connected. Any two neighbors of \(u\) can therefore be joined by a link edge path; each edge of that path gives such a triangle with \(u\). The coefficient \(\lambda_{u,v}\) is thus independent of the neighbor \(v\).

The remaining positive-dimensional case is a link circle. Enumerate its vertices cyclically as \(u_0,\ldots,u_{m-1}\). Cancellation of the \(g\)’s gives \[d_{u_0u_1}d_{u_1u_2}\cdots d_{u_{m-1}u_0}=1.\] In its abelianization, the coordinate of \(u_i\) occurs only in the two incident edge differences. Their coefficients are equal. This gives the same independence at each vertex of the circle.

Write \(\lambda_u\) for the common coefficient at \(u\), and replace \(g_u\) by \(g'_u=g_u u^{\lambda_u}\). This still implements the star conjugation because \(u\) centralizes its star. For every edge \(uv\), Equation (39) gives \[(g'_u)^{-1}g'_v =u^{\lambda_u}d_{uv}v^{\lambda_v}=1.\] The graph of \(X\) is connected, so all \(g'_u\) are equal. This last step requires only a finite edge path between two vertices and therefore also holds for the infinite graph \(\widehat{\mathcal L}\). The common element conjugates every generator as \(\varphi\) does, proving that \(\varphi\) is inner. ◻

Proof. Lemma 31 gives the simplicial permutation and reduces the remaining automorphism to one that sends each generator to a conjugate of itself. Lemma 32, applied to \(X=\widehat{\mathcal L}\), makes that remaining automorphism inner. Moving an inner automorphism past a generator permutation, if necessary, gives the stated order of the two factors. ◻

Proof. Let \(\omega\in N_{\mathop{\mathrm{Out}}(\Pi)}(\mathsf A)\) and choose a representative \(F\in\mathop{\mathrm{Aut}}(\Pi)\). View \(\mathcal E_2\subseteq\mathop{\mathrm{Aut}}(\Pi)\) using Proposition 20. It is the full preimage of \(\mathsf A\), so conjugation by \(F\) restricts to an automorphism \[\Phi\colon\mathcal E_2\longrightarrow\mathcal E_2.\] It preserves the inner copy of \(\Pi\). More precisely, \(\Phi(c_\gamma)=c_{F(\gamma)}\) for \(\gamma\in\Pi\). Since \(\Pi\) is centerless, under its identification with \(\mathop{\mathrm{Inn}}(\Pi)\) this says that \(\Phi|_\Pi=F\).

The subgroup \(W\) is characteristic in \(\mathcal E_2\). Indeed, every finite-order element projects trivially to the torsion-free group \(H_2\), and so belongs to \(W\). Conversely, \(W\) is generated by its vertex involutions. It is therefore exactly the subgroup generated by all finite-order elements of \(\mathcal E_2\), a characteristic description. Proposition 33 applies to \(\Phi|_W\). After composing \(\Phi\) with conjugation by an element of \(W\), we may assume that this restriction is a simplicial permutation \(\sigma\) of \(\widehat{\mathcal L}\). This adjustment preserves \(\Pi\) because \(\Pi\) is normal in \(\mathcal E\).

The adjusted automorphism induces an automorphism \(\beta\) of \(\mathcal E_2/\Pi=\mathsf A\). For a vertex \(u\) above \(s\in\mathcal V\), it satisfies \[ \beta(a_s)=a_{\overline{\sigma(u)}}. \tag{41}\] The right side is therefore the same for all vertices over \(s\). Because the labels \(a_s\) are distinct, \(\sigma\) takes each fiber of the vertex projection into one fiber. Moreover, the target fibers for distinct \(s\) are distinct by injectivity of \(\beta\). The finite set \(\mathcal V\) thus acquires a permutation \(f\) with \(\overline{\sigma(u)}=f(\overline u)\); bijectivity of \(\sigma\) makes the maps of fibers onto. Every simplex of \(\mathcal L\) has a lift, whose image under \(\sigma\) projects to the simplex on the \(f\)-images of its vertices. Applying the same argument to \(\sigma^{-1}\) shows that \(f\) is a simplicial automorphism.

Let \(\pi\colon\mathsf A_0\to\mathsf A\) be the label quotient, and let \(f_*\) permute the coordinate basis of \(\mathsf A_0\). Equation (41) on each \(e_s\) says \(\beta\pi=\pi f_*\). Taking kernels, and using the same identity for the inverses, gives \(f_*(\mathsf C)=\mathsf C\). Lemma 10 excludes every nonidentity simplicial automorphism with this property. Hence \(f=1\).

It follows that \(\sigma\) is a simplicial automorphism over the identity of \(\mathcal L\). To see explicitly that it is a deck transformation, choose one lifted vertex \(u_0\). Regularity of \(\widehat S\to S\) supplies \(h_0\in H\) with \(h_0u_0=\sigma(u_0)\). Both \(h_0^{-1}\sigma\) and \(\mathrm{id}_{\widehat S}\) are lifts of the covering projection \(\widehat S\to S\), and they agree at \(u_0\). Uniqueness of lifts on the connected domain makes them equal. Thus \(\sigma\) is the deck permutation given by \(h_0\).

Both \(\mathcal E_2\) and \(\Pi\) are normal in \(\mathcal E\). We may consequently compose further with conjugation by \((1,h_0^{-1})\in\mathcal E\) and assume that \(\Phi\) fixes \(W\) pointwise. For \(h\in H_2\), write \(\Phi(1,h)=(d,h')\) with \(d\in W\) and \(h'\in H_2\). Applying \(\Phi\) to the conjugation action of \(h\) on \(W\) shows \[c_d\circ h'=h\quad\text{as automorphisms of }W.\] Inner automorphisms act trivially on \(W_{\mathrm{ab}}\), so \(h'\) and \(h\) induce the same permutation of its coordinate basis. The deck action on vertices is faithful, giving \(h'=h\). The displayed identity then places \(d\) in \(Z(W)\). This center is trivial by Lemma 30: the vertex set is infinite because the infinite group \(H\) acts freely on it, while local finiteness makes every vertex star finite, so there is no universal vertex. Hence \(d=1\). The adjusted \(\Phi\) fixes \(H_2\) as well as \(W\) and is the identity on \(\mathcal E_2\).

Undoing the adjustments, the original \(\Phi\) is conjugation on \(\mathcal E_2\) by an element \(e\in\mathcal E\). For \(\gamma\in\Pi\), its action on the inner copy of \(\Pi\) then gives \(c_{F(\gamma)}=c_{e\gamma e^{-1}}\). Centerlessness of \(\Pi\) implies \(F(\gamma)=e\gamma e^{-1}\) for every \(\gamma\). Thus \(\omega\) is the class of \(e\) and belongs to \(\mathcal E/\Pi=\mathsf A\times\langle\delta\rangle\). The reverse inclusion in (40) holds because that direct product is abelian and contains \(\mathsf A\). ◻

Proof of Theorem 1. The specialization of Proposition 14 to the chosen labels gives closed connected topological four-manifolds \(M,N\) with homotopy equivalences to \(Q_2\). Since \(Q_2\) is aspherical, both manifolds are aspherical. Their common fundamental group \(\Pi\) is word-hyperbolic by Proposition 27. It remains to exclude a homeomorphism between them.

Use the realization subgroups \(R_M,R_N\leq\mathop{\mathrm{Out}}(\Pi)\) from Proposition 5, with the markings of these two models. The ambient group \(\mathop{\mathrm{Out}}(\Pi)\) is finite by Proposition 29. The graph groups \(\mathsf A\) and \(\mathsf B_\ell\) are \(2\)-subgroups of the same order. Proposition 21 gives \[\mathsf A\leq R_M,\qquad \mathsf B_\ell\leq R_N,\] and Proposition 22 gives \(\delta\notin R_M\). The element \(\delta\) is an involution and \(\mathsf A\times\langle\delta\rangle\) is an internal direct product by Proposition 20; this is the normalizer of \(\mathsf A\) by Proposition 34. Finally, Proposition 23 gives the required nonconjugacy. All hypotheses of Proposition 5 hold with \(X=M\), \(Y=N\), and \(\mathsf B=\mathsf B_\ell\). That proposition proves that \(M\) and \(N\) are not homeomorphic. ◻

Proof. We extend the marked three-tip model of [22]. Thicken a punctured torus to its product with an interval and put its two basis annuli on opposite faces. Two disjoint cutting arcs, each dual to one basis curve and disjoint from the other, give disjoint meridian disks in the product. To see these arcs explicitly, puncture the square torus near \((1/2,1/2)\), take the basis circles at \(x=1/4\) and \(y=1/4\), and cut along the arcs at \(y=1/2\) and \(x=1/2\). The puncture removes their only mutual intersection. Each product disk meets its own marked annulus in one essential arc and misses the other.

Cutting the thickening along these disks gives a ball. Each marked annulus becomes a rectangle joining the two faces of its own cutting disk. The rectangle and those two disk faces have a disk neighbourhood in the boundary sphere, disjoint from the analogous neighbourhood for the other annulus. A boundary homeomorphism, extended over the ball, identifies this marked configuration with the standard one. After reattaching the handles, the lower thickening is therefore a boundary connected sum of two longitudinal-annulus collars. This is a statement about the annuli themselves, not just their elements of the free group.

Attach an upper punctured-torus thickening along each of these two annuli. A longitudinal-annulus collar is a solid torus viewed as an annulus times an interval; gluing it on merely collars the attaching annulus of its upper thickening. The single boundary-sum tube of the lower model now joins the two upper thickenings. Extend the two meridian disks from each upper thickening through its own collar. The feet of the joining tube can be chosen arbitrarily small in the complement of the attaching annuli and moved away from the finitely many extending disk strips. The four extended disks are thus disjoint and dual to the four tip annuli. Cutting along them and repeating the preceding ball argument proves the asserted four-tip marked model. There is only one joining tube between the upper handlebodies, so no additional free generator is introduced. The initial boundary and all the surface markings are transported with this identification.

For placement we use the refinement of [23]: place the bodies in the complement of sufficiently thin attaching solid tori in the initial one-handlebody boundary, and lengthen only the returning tip collars to the attaching regions. The connectors and their basing paths then lie in \(K_v\) after plumbing. Reserve separated product parallels of the attaching curves, one for each occurrence of a tip, and join their thin longitudinal collars by tubes along a rooted tree. Let \(\eta\) run from the outside basepoint to its root and let \(p_\alpha\) be the tree path from that root to the \(\alpha\)th tip collar. Choose these paths so that \(\eta p_\alpha\) is homotopic relative to its endpoints to \(c_\alpha\kappa_{i_\alpha}\), using the product tracks to identify the parallel meridians. Then \[(\eta p_\alpha)m_{i_\alpha}^{\varepsilon_\alpha} (\eta p_\alpha)^{-1} =c_\alpha x_{i_\alpha}^{\varepsilon_\alpha}c_\alpha^{-1} \quad\text{in }\pi_1K_v.\] There is no compatibility restriction among the four chosen classes: finitely many arcs in a three-manifold can realize them disjointly away from their common root, with parallel tracks for repeated portions and with the reserved returning strips avoided. The tree paths fix the required basing; other paths inside the resulting handlebody add only words in its four tip generators. The initial boundaries remain in the outside pieces, and the bottom surfaces are proper pushes of disjoint boundary surfaces of \(X\).

We finally specify the framings. The surface-product model frames a surface by its oriented normal in the three-dimensional body and the collar direction. At a terminal annulus use the framing adjustment in the proof of [22], before fixing the returning collar. In a separated solid-torus neighbourhood this is the ribbon twist \[(\theta,z)\longmapsto(\theta,e^{in\theta}z).\] Replace \(e^{in\theta}\), if necessary, by a smooth degree-\(n\) circle map \(\phi(\theta)\) equal to \(1\) near the connector feet. The twist is then the identity at those feet and extends by the identity over the joining tubes. Apply it to the complete marked body, including the initial boundary and the lower and upper attaching annuli.

The returning collar is constructed after this transport. In coordinates \(S^1_\theta\times D^2_z\times[0,1]_u\), choose an embedded smooth regular plane arc \((r(s),u(s))\) from \((0,0)\) to \((0,1)\), departing in the positive \(r\) direction and vertical near its endpoint, with \(u\) strictly increasing for \(s>0\). The collar \[(\theta,s)\longmapsto (\theta,r(s)i\phi(\theta),u(s))\] leaves the transported surface ribbon in its outgoing normal direction and ends at the same product circle \(z=0,u=1\). Its normal quotient has frame \[v=\phi(\theta),\qquad w=r'(s)\partial_u-u'(s)i\phi(\theta).\] At the first end this is the surface-transverse and collar frame; at the last, up to a fixed constant change of frame, it has winding \(n\) relative to the core product frame. Choose \(n\) to match those framings. The four disjoint longitudinal neighbourhoods make the adjustments independent and leave the external basing paths fixed. The initial-boundary link and its surface framing are fixed after these transports. At each of the three symplectic pairs, the attachments depart on opposite normal sides. Put the bodies at an inner collar level and extend the tips into separated product parallels of the handle cores. The collar separation and the plumbing charts give the final assertions. ◻

Proof. The boundary of a punctured torus is the commutator of its based symplectic generators. Apply this first to the two upper tori and then to the bottom torus. Internal paths may conjugate a tip or an upper commutator by words in the four tips. Such conjugations change the weight-four expression only in weight at least five. If any one tip is set equal to the identity, the commutator at its upper torus becomes trivial, and then so does the bottom commutator. These are word identities in the abstract tips; substituting the markings (45) gives the corresponding initial-boundary words in \(\pi_1K_v\). ◻

Proof. The side-marked model of Lemma 37 gives \[D\simeq X\cup_{\coprod_jF_j}\coprod_j(V_j\times S^1).\] Because each attaching map \(F_j\to X\) is null-homotopic, the additional based summand is the homotopy cofiber of \(F_j\to V_j\times S^1\). Retract \(V_j\times S^1\) to the torus on its longitude \(b_j\) and normal circle \(z_j\). Since \(F_j\) retracts to \(a_j\vee b_j\), its cone adds one two-cell on the trivial loop \(a_j\) and another killing \(b_j\). Cancel the latter one/two-cell pair. The old torus two-cell, originally attached by \([b_j,z_j]\), now has trivial attaching map. The cofiber is therefore \[S^1_{z_j}\vee S^2\vee S^2.\] This proves (49). The model retains the normal-circle marking, so its quotient killing the new generators is the old \(X\)-marking. Equivalently, the classifying map to the graph of \(X\) extends over each replacement, since both \(a_j\) and \(b_j\) map trivially and the new normal circle is sent to the identity. Its restriction to the common boundary exterior is the original classifying map. ◻

Proof. Choose one terminal cap on each upper torus. Let \(f_j\) be the proper truncation of the compression disk on the \(a_j\)-branch, and let \(C_j\) be the compression disk on the \(b_j\)-branch. Near the tube side over \(b_j\), take the normal-circle torus and compress it with two parallels of \(C_j\). Denote the resulting sphere by \(g_j\).

Here is the local collar calculation, from [22], that also checks its applicability to the new \(a_j\)-branch. Use coordinates \((s,t,r,\theta)\), where \(s\) runs along \(b_j\), \(t\) is transverse to \(b_j\) in \(F_j\), and \((r,\theta)\) are polar coordinates normal to \(F_j\). The rim torus is \(t=0\), \(r=\rho\). The outgoing \(a_j\)-disk collar and the \(b_j\)-upper attachment have distinct angles \(\theta_A,\theta_B\). Cut the rim torus at a narrow band around \(\theta_B\) and keep the annulus containing \(\theta_A\). Place the interiors of its two compression disks outside \(r=\rho\). The retained annulus meets the \(a_j\)-disk collar once. Orient and base this point to contribute \(1\). This calculation only uses the outgoing framed collar of that disk, so it remains valid when the disk itself is obtained by upper-torus compression. The disjoint body neighbourhoods exclude collar intersections between different indices.

Every remaining intersection occurs in a cap–cap plumbing chart. Within \(g_j\), label the two copies of \(C_j\) by \(\epsilon\in\{+1,-1\}\) and the two cap sheets in a fixed copy by \(\eta\in\{+1,-1\}\). Their orientation coefficients are \(\epsilon\eta\). Fix \(\epsilon\) and an opposing sheet. The whisker on the opposing sheet is held fixed in this comparison. The two paths on \(g_j\) share the whole stem from the rim to the fixed copy of \(C_j\) and split only inside its upper torus. Their difference there is the other basis loop, which bounds the unused terminal cap in \(E\): no upper cap was removed with the bottom tubes. A small normal turn around the upper surface also bounds in \(E\), because that surface has not been deleted. The two intersection labels are consequently the same element \(h\in\pi_1E\), and their contributions cancel as \[\epsilon\eta h+\epsilon(-\eta)h=0.\] Figure 2 displays the two cancellations. The outer index \(\epsilon\) is fixed throughout this cancellation; we make no identification of paths that differ by a bottom normal circle. The opposing sheet may come either from another rim sphere or from the compressed \(a\)-branch. This proves (50).

A rim sphere uses only parallel copies of a single selected terminal cap. That core has no self-plumbing. The separated collars therefore make each sphere embedded, and the product compression framings give its trivial normal bundle. If the selected caps are all disjoint, there are no plumbing intersections at all; the single collar intersection in each pair is then the complete geometric intersection set. ◻

Proof. We give the middle-form verification used in [23]. The hermitian matrix of the \(u_i\) has even identity coefficient on its diagonal, since their normal framings give zero Stiefel–Whitney evaluations. The free group \(H\) has no nonidentity element of order two. Hence each diagonal entry can be written \(z+\overline z\): choose one coefficient from each nonidentity inverse pair and halve the even identity coefficient. Subtracting suitable \(\Lambda\)-linear combinations of the \(g_j\) from the \(u_i\), using a triangular half off the diagonal and these halves on the diagonal, makes all pairings among the modified \(u_i\) vanish. The cross-pairings and the \(g_i,g_j\) pairings in (51) are unchanged.

Represent these classes by immersed spheres using the usual tubing operations. Their even normal parity permits framed representatives. Write \(\mu\) for the full framed Wall invariant in the additive quotient \[\Lambda/\langle h-h^{-1}:h\in H\rangle_{\mathbb Z};\] in particular, its identity coefficient is retained. For a framed representative, the symmetrization identity \[\lambda(v,v)=\mu(v)+\overline{\mu(v)}\] and the absence of order-two elements show that its Wall quadratic self-intersection is zero when its self-pairing vanishes. This is the framed immersed-sphere calculation in the cited section; no embedded-surgery theorem is used. Write \(A_j,B_j\) for the resulting hyperbolic pairs.

They form a basis of the whole module. By Proposition 38, \(\pi_2D\) is free of rank \(2k\) over \(\Lambda\). Relative to any such basis, the square matrix with columns \(A_j,B_j\) has a left inverse supplied by its nonsingular hyperbolic pairing. Matrix rings over \(\mathbb ZH\) are directly finite: pass a one-sided inverse identity to every finite quotient of \(H\), use the finite dimensional regular representation over \(\mathbb C\) to obtain the reverse identity, and then use residual finiteness to separate the finite support of any nonzero group-ring entry. The matrix therefore has a two-sided inverse.

A map from \(BH\vee\bigvee^{2k}S^2\) to \(D\) realizing this basis is a homotopy equivalence: it induces the marked isomorphism on \(\pi_1\) and an isomorphism on the homology of the simply connected covers. Attaching the indicated three-cells cancels the sphere summands in that homotopy type. Thus \(P\simeq BH\).

For completeness, the duality argument of [23] applies to exactly this basis. The middle sphere module of \(D\) maps isomorphically to a split summand of \(H_2(D,S;\Lambda)\) by its nonsingular intersection form. The three-cell attachments quotient the absolute and relative middle homology by these summands; their attaching map is injective on the summands, so no new third homology appears. In cohomology the pullbacks identify the new groups with the annihilators of the sphere summands. If \(i:(D,S)\to(P,S)\) is the inclusion and \([P,S]=i_*[D,S]\), cap-product naturality gives \[\alpha\frown[P,S] =i_*\bigl(i^*\alpha\frown[D,S]\bigr).\] The duality isomorphisms for \((D,S)\) consequently induce duality on these annihilators and quotients. The finite free chain complexes over \(\Lambda\) give the regular-coefficient criterion for Poincaré duality with all local coefficients. This proves the assertion for \((P,S)\), with its original boundary collar and orientation. ◻

Proof. Duality for \((P,S)\) and the graph homotopy type of \(P\) give \[H_1(P,S;\mathbb ZH)=H_0(P,S;\mathbb ZH)=0.\] Therefore \(H_0(S;\mathbb ZH)\to H_0(P;\mathbb ZH)\) is an isomorphism. The source is free abelian on the cosets of the image of \(\pi_1S\), whereas the target is \(\mathbb Z\). There is only one coset. ◻

Proof. Corollary 41 gives connectedness of \(S_d\). The side-marked exterior model identifies the covering character on \(\pi_1E\) with the one inherited from \(X\), with each normal circle sent to zero. Its restriction to \(\pi_1(V_j\times S^1)\) is trivial: the longitude maps trivially in \(H\) and \(w(z_j)=0\). Thus the cover of each replacement is trivial, and performing the chamber construction on every lifted body in \(X_d\) gives \(D_d\). The cofiber calculation is unchanged.

Each basis sphere has \(d\) lifts. Algebraically, the universal cover is the same and its free middle module is restricted from \(\mathbb ZH\) to \(\mathbb ZH_d\). Pairings of two chosen lifts are obtained by taking, in the pairing of their corresponding coset translates, the coefficients supported in \(H_d\). The lifted hyperbolic pairings are therefore standard. The Wall invariant of a chosen lift is obtained, after the corresponding change of whisker, by retaining precisely the self-intersection labels in \(H_d\). Because \(H_d\) is closed under inversion, this additive coefficient projection retains the identity and both members of every retained inverse pair. It therefore descends to the framed Wall quotient and carries the zero downstairs invariant to zero. The lifted sphere classes are a basis of the restricted module. The three-cell attachments and the orientation lift, giving the asserted pair. ◻

Proof. At vertex \(0\) of the plumbing double cover select all red ports, and at vertex \(1\) select all blue ports. Every plumbing edge joins these two vertices and has the same color at both ends. Exactly one end is selected. Consequently all product cap parallels belonging to selected ports are mutually disjoint. This also accounts for the two different lifts of each original plumbing type: each has one selected end, independently of its orientation in the two-vertex graph.

Each upper torus of each body has one red and one blue tip. Compress it using the cap selected at its vertex. Use these choices on both upper branches to construct \(u_j,g_j\) in \(D_2\), with \(j\) now ranging over the lifted bodies. Lemma 39 and the matching meridian-disk framings show that these are embedded framed pairs with a single transverse intersection inside each pair and no intersections between different pairs. Their hyperbolic pairing and the known free rank show, by the same direct-finiteness argument as in Proposition 40, that their classes form a basis of \(\pi_2D_2\).

A regular neighbourhood \(N_j\) of one such pair is the normal plumbing of two trivial disk bundles over \(S^2\) at one point. This is the standard punctured \(S^2\times S^2\): take tubular neighbourhoods of the two factor spheres in \(S^2\times S^2\), whose rounded complement is a ball. Reversing one sphere orientation if necessary matches the intersection sign. Thus \(\partial N_j\cong S^3\). Choose these neighbourhoods disjointly in the interior of \(D_2\), and set \[A=\overline{D_2\setminus\bigcup_jN_j},\qquad D'_2=A\cup_{\coprod_j\partial N_j}\coprod_jB^4.\] The complement \(A\) is connected: any segment of a path through \(N_j\) can be replaced by a path along its connected boundary. Since \(N_j\), \(\partial N_j\), and \(B^4\) are simply connected, van Kampen gives the same marked fundamental group \(H_2\) for \(A\), \(D_2\), and \(D'_2\).

We verify contractibility of the universal cover of \(D'_2\). Write \(\widetilde A\) for the common lifted complement. The universal cover of \(D_2\) is obtained by adjoining all lifts of the \(N_j\), whereas the universal cover of \(D'_2\) is obtained by adjoining balls at the same boundary spheres. These are locally finite families, and their homology contributions are direct sums. In degree two, Mayer–Vietoris gives \[H_2(\widetilde A;\mathbb Z)\oplus \bigoplus_{\widetilde N_j}H_2(\widetilde N_j;\mathbb Z) \ \cong\ H_2(\widetilde D_2;\mathbb Z).\] The second summand maps isomorphically onto the right-hand side because the lifted sphere pairs are a free middle basis. Hence \(H_2(\widetilde A;\mathbb Z)=0\). Since \(\widetilde D_2\) has the homotopy type of a wedge of two-spheres, the same exact sequence in the adjacent degrees gives \[H_4(\widetilde A;\mathbb Z)=0,\qquad \bigoplus_{\widetilde N_j}H_3(\partial\widetilde N_j;\mathbb Z) \xrightarrow{\ \cong\ }H_3(\widetilde A;\mathbb Z).\] Adjoining the balls kills exactly these third-homology classes and produces no fourth homology. It leaves the zero second homology unchanged. The resulting cover is simply connected, and its reduced homology vanishes in every degree. Manifolds have CW homotopy type, so the homological Whitehead theorem makes this cover contractible. Therefore \(D'_2\simeq BH_2\).

All modifications were interior, and the fundamental-group identifications above agree on \(S_2\). A classifying map \(D'_2\to BH_2\) consequently has the prescribed boundary marking, since maps to \(BH_2\) are determined up to homotopy by that group marking. This is the required marked filling, and (53) gives the relative homotopy equivalence to \(P_2\). ◻

Proof. An assignment supplied by Theorem 45 cannot assign two red and two blue edges to the same vertex. Hence every vertex receives at most one edge in at least one of the two colors. Call such a color deficient at that vertex. Two adjacent vertices cannot share a deficient color: each of the five edges of that color joining them must be assigned to one of the two vertices, whereas together the two vertices could receive at most two. Thus the nonempty sets of deficient colors at adjacent vertices are disjoint. Each such set is consequently a singleton, and the deficient colors alternate around the cycle. This is impossible when \(d\) is odd; see Figure 3. ◻

Proof. At each vertex choose a linear projection of the ambient coordinate space onto \(K^{a_v}\) that is invertible on the central tangent image. Its composite with \(M_v\) has a formal inverse over \(T_v\), with parameters fixed. The possible constant displacement lies in the nilpotent ideal \((t)\), so the substitution is well defined. Reparametrize to make this projection of \(M_v\) equal to \(y_v\).

For a subset \(A\subset E_+(v)\), write \(t_A=\prod_{e\in A}t_e\). Expand the graph and every coefficient witnessing (57) in the finite basis \(\{t_A\}_A\) of \(T_v\). All the resulting unknown coefficient series have the same independent variables \(y_v\). Represent the fixed edge functions on pointed \(\acute{e}\)tale neighborhoods. Implicit differentiation expresses their ambient derivatives in the same finite presentations, with inverse Jacobian determinants included among the units. Auxiliary variables for these presentations and units turn the following requirements into a finite polynomial system in the unknown coefficient series: the presentation equations, the fixed graph projection, the vanishing of \(\mathcal S_v\), and all the ideal-membership identities. No derivative of an unknown coefficient series occurs.

Artin approximation over \(K[y_v]_{(y_v)}^h\) [1] gives algebraic coefficient series that still solve this polynomial system exactly and agree with the original solution through first order. In particular, the central graph tangent and all chosen branch centers are preserved. Apply it separately at each vertex. These systems share only the fixed edge functions, and their mutual transversality is a condition on the preserved first jets. Thus no nested approximation is involved. When \(a_v=0\), the coefficient system already consists of equations over \(K\), and no approximation is needed there. The preserved branch centers and invertible Jacobians identify the auxiliary solutions uniquely with the compositions of the fixed edge germs and the approximated graph.

We now have finitely many algebraic germs and exact identities between them. Each is represented at a finite stage of a henselization, and every identity holds at a sufficiently large finite stage. Descend these presentations, compositions, branch centers, and identities to a finitely generated integral \(\mathbb Z\)-subalgebra \(A\subset K\). Include the inverse \(\acute{e}\)tale Jacobians and the inverse determinant in (56). The implicit differentiation identities descend as well, so specialization preserves the meaning of the gradient entries. The original presentations for \(F_e,b_e\) still use only \(h_e\).

There are specializations of \(A\) to finite fields in arbitrarily large characteristic. Indeed, a maximal quotient of the nonzero finite-type \(\mathbb Q\)-algebra \(A\otimes\mathbb Q\) is a number field \(E_0\). The images of a finite set of generators of \(A\), including all the specified inverses, lie in \(\mathcal O_{E_0}[1/d_0]\) for some positive integer \(d_0\). Reduction at a prime above any rational prime not dividing \(d_0\) gives the desired specialization. All declared units remain invertible. The pointed \(\acute{e}\)tale presentations therefore give formal germs satisfying the same identities and tangent conditions over the residue field. ◻

Proof. Every centered-modulo-parameters series \(c\) has \(c^q=0\) in the truncated ring. By Frobenius, a nonconstant internal monomial acquires an exponent at least \(q\), and a positive parameter monomial has zero \(q\)-th power. Consequently every formal coordinate change with such a center preserves the truncation ideal; its inverse proves equality of the ideals. This also justifies evaluating the formal series in the truncated ring.

Each factor \(c_s^{q-1}\mathrm dc_s\) is closed, and multiplication by \(c_s\) kills it. In graph coordinates, a one-form vanishing on the graph is a sum of graph-ideal multiples of one-forms and multiples of the \(\mathrm dc_s\). Both types wedge to zero with \(U_c\). The coordinate invariance just proved permits this calculation after truncation and over the full parameter ring. ◻

Proof. First perform the contraction on graph forms, identifying the two copies of every edge block, wedging, and integrating. Put \(N=\sum_eN_e\) and \(n_v=\sum_{e\in E(v)}N_e\). The isomorphism (56) gives \(\sum_va_v=N\), whereas \(\sum_vn_v=2N\). Thus the total number of graph equations is \(\sum_v(n_v-a_v)=N\).

After identifying edge coordinates, their linear parts are independent. Indeed, a vector annihilated by all these linear equations determines a tangent vector in the image of each \(A_v\). The tangent parameters are unique by injectivity, and their edge coordinates agree at opposite ends. They therefore lie in \(\ker\Lambda=0\). Let \(C\) be the resulting invertible \(N\)-by-\(N\) matrix of linear graph equations.

The polynomial coefficient of the wedge of all graph forms has degree at least \(N(q-1)\), the highest possible degree in the truncated polynomial ring. Only the linear terms of the graph equations and the constant terms of their differentials can contribute. A linear change with matrix \(C\) acts on the polynomial socle \(k\prod h_{e,r}^{q-1}\) by \(\det(C)^{q-1}\), and on the exterior top form by \(\det(C)\). The socle identity can be checked on diagonal matrices, permutations, and elementary transvections: for a transvection, each nonconstant term has an exponent at least \(q\) and vanishes. These matrices generate the general linear group. The contraction is therefore \(\pm\det(C)^q\neq0\). If \(N=0\), the same calculation uses the empty determinant, equal to one.

This contraction descends to the two homologies. For the first descent, replace the graph form at one vertex by a \(D_{v,0}\)-boundary and keep cycles at the other vertices. In the product of all incidence blocks, apply the sum of the vertex differentials to the product with that vertex’s primitive. Only the chosen vertex contributes. Upon edge identification the sum of its multiplying one-forms is zero, since every \(\mathrm dF_e\) occurs once with each sign. The contraction of the boundary is therefore zero. Under Künneth this is exactly the edgewise pairing on \(\mathcal H_e\). For the second descent, (58) makes that pairing a chain pairing for the \(\delta_e\); hence contracting a boundary against cycles gives zero. The resulting scalar is the contraction of the \(\omega_v\), and retains the computed nonzero value. ◻

Proof. Write \(Z=\ker\delta\) and \(B=\operatorname{im}\delta\). Choose a complement \(A_0\) to \(K_0\cap B\) in \(K_0\cap Z\), and extend it to a complement \(A\) to \(B\) in \(Z\). Choose a complement \(W_0\) to \(K_0\cap Z\) in \(K_0\); it meets \(Z\) trivially, so it extends to a complement \(W\) to \(Z\) in \(H\). Make every choice degree by degree. In the decomposition \[H=B\oplus A\oplus W,\] projection onto \(A\) sends \(K_0\) into \(A_0\). It is a chain map. The restriction \(\delta:W\to B\) is an isomorphism with degree shift. Its inverse on \(B\), extended by zero on \(A\oplus W\), is a homotopy \(s\) satisfying \(\operatorname{id}-P=\delta s+s\delta\). ◻

Proof. On \(\mathcal H_e\), the commutator \(\delta_eB_e-B_e\delta_e\) is induced by \(\mathrm db_e\wedge(-)\). Both terms kill \(x\), so \(\mathrm db_e\wedge a\) is a \(D_e\)-boundary. Choose \(a'\) with \(D_ea'=-\mathrm db_e\wedge a\). Expansion, using \(t^2=0\), proves (65). ◻

Proof of Theorem 45. Use Lemma 47 and the constructions above. Choose a homogeneous basis \((\beta_{e,s})_s\) of \(\mathcal W_e\) whose first members form a basis of \(\overline{\mathcal K}_e\). At the minus end use this basis, and at the plus end use its pairing-dual basis \((\beta_{e,s}^{\vee})_s\), normalized by \(\langle\beta_{e,s}^{\vee},\beta_{e,r}\rangle=\delta_{sr}\). The same index labels the two bases. Call an index a kernel index if its element \(\beta_{e,s}\) lies in the chosen basis of \(\overline{\mathcal K}_e\).

A term of the total contraction has one common index at the two ends of every edge. The assignment attached to such a term sends an edge to its plus endpoint for a kernel index, and to its minus endpoint otherwise. We will prove that a nonzero term obeys every local prohibition.

We claim that the coefficient of \(\omega_v\) at any tuple of indices is zero whenever, for some \(L\in\mathcal L_v\), all its plus edges \(I=L\cap E_+(v)\) have kernel indices and all its minus edges \(J=L\cap E_-(v)\) have nonkernel indices.

Fix those plus indices. For each \(i\in I\), choose a homogeneous representative \(x_i\in\mathcal K_i\cap\ker\delta_i\) of \(\beta_{i,s_i}\), and a homogeneous \(D_i\)-cycle \(a_i\) representing \(x_i\). By Lemma 51, choose a cycle \(a_i+t_i a_i'\) for \(D_i+t_i\mathrm db_i\wedge(-)\). Set all other parameters at \(v\) equal to zero in the actual boundary identity (60). Move its factors in \(I\) before the remaining ones, with the graded signs, and pair them with the tensor of these lifts, integrating in just those blocks. Because the perturbations act on separate blocks, the tensor of the lifts is a cycle for the full tensor differential over \(k[t_i:i\in I]/(t_i^2)\), including all mixed parameter terms.

Write \(\Phi_t\) for this partial contraction, and let \(D_R\) be the signed tensor differential on the remaining blocks. It is independent of the retained parameters. If \(m=\sum_{i\in I}(N_i-|a_i|)\), the homogeneous map \(\Phi_t\) has degree \(-m\). Equation (62), with the graded tensor convention, gives \(\Phi_tD_{v,t}=(-1)^mD_R\Phi_t\). In particular it takes \(D_{v,t}\)-boundaries to \(D_R\)-boundaries. Applying it to (60) and taking the coefficient of \(t_I=\prod_{i\in I}t_i\) gives \[ \left(\prod_{j\in J}b_j\right)\Phi_0(U_v) \in\operatorname{im}D_R. \tag{66}\] Indeed, multiplication by \(t_I\) kills every positive-degree parameter coefficient on the left. This coefficient extraction also applies when \(I\) is empty, with \(\Phi_t\) the identity.

In the first homology, let \(x\) denote the partial contraction of \(u_v\) with the classes \(x_i\). Equation (66) gives \[\left(\prod_{j\in J}B_j\right)x=0.\] The class \(x\) is a cycle for the remaining tensor differential \(\delta\): the edge pairings on \(\mathcal H_e\) satisfy (58), and every inserted class \(x_i\) is a \(\delta_i\)-cycle.

If \(J\) is empty, the equation already says \(x=0\). Otherwise, \[ x\in\sum_{j\in J} \left(\mathcal K_j\otimes \bigotimes_{e\in E(v)\setminus(I\cup\{j\})}\mathcal H_e\right), \tag{67}\] where the factors are placed in their fixed order. To verify this kernel description, quotient the tensor product by the displayed sum. It becomes the tensor product with \(\mathcal H_j/\mathcal K_j\) in each position \(j\in J\). Each induced map \(\mathcal H_j/\mathcal K_j\to\mathcal H_j\) is injective, and tensoring injections over a field preserves injectivity.

The decomposition in (67) need not be a sum of \(\delta\)-cycles. The projections of Lemma 50 replace it by cycle representatives without changing the second-homology class or losing the indicated kernel factors.

On every remaining \((\mathcal H_e,\delta_e)\), choose the projection of Lemma 50 for \(\mathcal K_e\). Their tensor product is chain-homotopic to the identity: expand \(\operatorname{id}-\bigotimes_eP_e\) by telescoping and use each individual homotopy with the graded signs. Applied to the cycle \(x\), this projection therefore represents the same second-homology class. By (67), that class belongs to the sum of tensor subspaces having a factor \(\overline{\mathcal K}_j\) for some \(j\in J\). The class is exactly the contraction of \(\omega_v\) against the selected \(\beta_{i,s_i}\) in its plus positions. Since all the \(J\)-positions are minus positions, their bases are the \(\beta_{j,s}\). A coefficient with nonkernel indices at every one of those positions must vanish. This proves the claim.

Expand the total contraction of the \(\omega_v\) in the chosen bases. It is a finite signed sum of products of vertex coefficients, with equal indices at the two ends of every edge. By Lemma 49, at least one such product is nonzero. Choose its indices and use the assignment attached to that term. If a law \(L\in\mathcal L_v\) had all its edges assigned to \(v\), its plus indices would all be kernel indices and its minus indices would all be nonkernel indices. The coefficient at \(v\) would vanish by the claim, a contradiction. The assignment has the required property. ◻

Proof. We use the dual-handle and cut construction of [23], keeping the full cyclic quotient instead of passing to its parity quotient. Attach to the proposed filling the dual two-handles that undo surgery on the initial law curves. The attaching circles represent, with their prescribed paths, the lifted normal-circle generators \(z_j\). Indeed \(w(z_j)=0\), so every replacement piece and each such generator has an individual lift. These are free-factor generators in the graph marking of the filling. Attaching the two-cells therefore cancels the corresponding one-cells in its homotopy type, leaving the marked graph of \(X_d\).

The boundary after these attachments is \(\partial X_d\). On the common link exterior its map to the remaining graph is the original marking: the quotient has killed exactly the normal-circle generators. The same holds on the restored solid tori, whose group elements come from their gluing tori. Equality of the resulting fundamental-group maps is equality up to homotopy of graph-valued maps. Thus the marking of \(W\) is the required one. The two-handle cocores are disjoint proper locally flat disks with the original law curves as boundaries.

For the cuts, pass to the cover associated to the quotient that forgets the extra loops. Its deck group is \(\pi_1\Gamma_d\), acting freely on the universal covering tree of \(\Gamma_d\). The fundamental group of this cover of \(W\) is freely generated by the named extra loops at tree vertices, with their whiskers. Prescribe an equivariant map to the tree to be constant at the appropriate vertex on each outside face and each cocore neighborhood, and to cross the appropriate edge on every join-torus collar. The prescriptions agree on overlaps. They extend over the remaining space because the associated tree bundle has contractible fiber, as in the cut construction of [23].

Choose the punctured smoothing used in that construction, with its punctures away from the prescribed disks and collars. Make the tree map constant near each puncture. Smooth approximation over the middle of the edges and regular values then give compact smooth cuts, disjoint from the disks. It remains to obtain connected cuts and connected vertex pieces; regular values alone do not ensure this.

Form the component graph of these cuts in the tree cover. Its fundamental group is a quotient of that of the cover: a graph path can be lifted by joining successive crossings inside connected pieces. Each named extra generator is conjugate to a loop lying in one piece, and hence has trivial image in the component graph. The component graph is therefore a tree. The outside faces and prescribed torus collars embed the original plumbing tree in it. In fact different tree-vertex labels cannot lie in the same component after the target midpoints have been removed, and the prescribed collars give exactly the edges joining these principal vertices.

Every component outside this principal subtree attaches at a unique principal vertex. A deck transformation stabilizing that component fixes its attaching vertex, and hence is the identity. There are only finitely many orbits of cut and complementary components in the finite quotient. Since an orbit meets the branches attached to a fixed principal vertex at most once, those branches have uniformly finitely many edges and vertices. Discard their closed cuts and merge their pieces into the principal pieces. This leaves exactly one connected cut with each prescribed torus as its entire boundary. The disk prescriptions and boundary markings are unchanged. Descending to \(\Gamma_d\) proves the lemma. ◻

Proof. The graph homotopy type of \(W\) gives \(H^2(W)=H^3(W)=0\). Mayer–Vietoris for the decomposition over \(\Gamma_d\) consequently makes \(\Delta^2\) an isomorphism and \(\Delta^1\) surjective. Since the pieces and cuts are connected, the kernel of \(\Delta^1\) is \[H^1(W)\big/H^1(\Gamma_d).\] The graph classes vanish on the named extra loops. By the retained marking, evaluation on those \(dp\) loops has rank \(dp\), and \(b_1(W)=b_1(\Gamma_d)+dp\). These evaluations therefore identify the displayed quotient with \(\mathbb C^{dp}\). Given any restrictions on the cuts, its unique kernel correction makes all extra evaluations zero, proving the second assertion.

For a connected oriented three-manifold with one torus boundary, half-lives/half-dies makes the boundary image in degree one a line. The restrictions from the two vertex ends span \(H^1(Y_e)\). At the torus, each end restriction annihilates its longitude, which bounds in its outside face \(K_v\). These two longitudes are the two different coordinate slopes, because plumbing interchanges meridian and longitude. A line different from both coordinate lines meets each of them trivially; if it were the boundary image, all vertex restrictions would vanish at the boundary, contradicting surjectivity onto that nonzero line. Thus it is one coordinate line. ◻

Proof. We give the boundary calculation to distinguish the present statement from the two-vertex conclusion of [22]. For a Lie-algebra-valued one-form \(a\) on an oriented three-manifold \(Y\), the action and its coefficient variation are \[ \begin{split} \mathcal C_Y(a)&=\int_Y\operatorname{tr} \left(\tfrac12a\wedge da+\tfrac16a\wedge[a,a]\right),\\ \dot{\mathcal C}_Y(a)&=\int_Y\operatorname{tr} (\dot a\wedge\mathcal F_a) -\tfrac12\int_{\partial Y}\operatorname{tr}(a\wedge\dot a), \qquad \mathcal F_a=da+\tfrac12[a,a]. \end{split} \tag{75}\] For the edge slice \(s_e\) with trace \(u_e\zeta_e+v_e\xi_e\) and \(u_e=t_eT_e\), the adjusted potential of [22] is precisely \[f_e+t_eb_e^{\mathrm{form}} =\mathcal C_{Y_e}(s_e) +\tfrac{\epsilon_e}{2}\operatorname{tr}(u_ev_e).\] This normalization will cancel the fixed-\(u_e\) boundary variation.

On the free outside face \(K_v\), normalize the flat quotient connection to its strip representative: a sum of the signed based holonomy logarithms times closed one-forms dual to the free generators, with disjoint supports. The disjoint meridian-dual disks and extra-loop-dual spheres in the standard outside face supply these strips. Each boundary torus meets only its own port strip. Thus the choices at different tori coexist in one representative: a plus port supplies a \(\zeta_e\) trace and a minus port a \(\xi_e\) trace. This representative has zero Chern–Simons action. Its torus trace is \(u_e\zeta_e\) at a plus end, where \(u_e=t_eT_e\), and is in the \(\xi_e\) polarization at a minus end. The strip forms and torus normalizations are those of [22].

After this normalization, gauges equal to identity on the tori put the flat restrictions on the cuts into their central or perturbed edge slices. The boundary traces agree exactly in the residual quotient: at a plus end \(G_{\beta_e}=1\) forces \(v_e=0\) there, while at a minus end both traces have zero \(\alpha_e\) coefficient and the same based \(\beta_e\) logarithm. The torus-identity slice gauge of [22] therefore applies at every end, and the gauges on intersecting faces agree. Lift the edge coordinates and gauge logarithms from \(R_v/\mathcal I_v\) to \(R_v\), and extend the logarithms through collars. The lifting and division argument in [22] uses a fixed finite residual list: coefficientwise formal division expresses each discrepancy as an actual sum \(\sum_a r_{v,a}\gamma_a\) of form-valued series. Every output coefficient uses finitely many input coefficients. This also works in the punctured form complex. Finally change the lifted connection by such ideal multiples so that its restriction to \(K_v\) is exactly the strip representative. Call the result \(\widehat a_v\).

The curvature of \(\widehat a_v\) lies in \(\mathcal I_v\), and its edge restrictions agree with the chosen slices modulo that ideal. The added ideal multiples change the leading curvature coefficient forms in (73) only by exact forms; the gauges have identity constant term. Stokes’ formula gives \[\mathcal C_{\partial V_v}(\widehat a_v) =\tfrac12\int_{V_v}\operatorname{tr} (\mathcal F_{\widehat a_v}\wedge \mathcal F_{\widehat a_v})\in\mathcal I_v^2.\] The outside-face contribution is zero. Interpolating an edge restriction to its chosen slice has an interior action variation in \(\mathcal I_v^2\): the displacement and the curvature are both in \(\mathcal I_v\). At a minus end the boundary variation is zero because both traces are in the \(\xi_e\) polarization. At a plus end the traces change from \(u_e\zeta_e\) to \(u_e\zeta_e+v_e\xi_e\), and the torus variation is \[-\tfrac{\epsilon_e}{2}\operatorname{tr}(u_ev_e).\] The adjustment in the perturbed edge potential cancels this term exactly. Reversing the orientation of a boundary face changes its action by a sign. Summing these per-end formulas proves the first identity of (74), including all mixed parameter terms.

The internal edge variation formula pairs slice curvature with a derivative of the slice. It gives the second identity with an actual coefficient matrix against the fixed list \(r_v\). At the joint origin its entry for an edge tangent representative \(\theta_{e,k}\) and a vertex residual representative \(\omega_{v,a}\) is, with the face sign, \[\int_{Y_e}\operatorname{tr} (\theta_{e,k}\wedge\omega_{v,a}|_{Y_e}).\] Here \(\theta_{e,k}\) is a closed relative one-form. Added exact two-forms do not change this pairing, by integration by parts and its zero boundary trace. Injectivity of the vertex \(H^2\) restriction and the perfect pairing (69) make this matrix have full column rank. Its selected minor is a unit. This argument computes the coefficient matrix from curvature, and so does not assume that the residual list has no syzygies. Finally, every lift has the same first derivatives as its residual-quotient restriction because \(\mathcal I_v\) has order at least two. These are exactly the cohomological tangent maps already identified. ◻

Proof of Proposition 52. A marked filling would give the cut pieces of Lemma 53, and hence the formal data of Proposition 55 on \(\Gamma_d\). Its law family consists of every choice of two red and two blue incident edges. Corollary 46 forbids exactly these data when \(d\) is odd. This contradiction proves the proposition. ◻