Proof. It is enough to describe one surface; the constructions have disjoint collars. Write \(B=\partial X\) and \(K=\partial F\). Before rounding corners, put the pushed surface at a horizontal collar level, with vertical walls along \(K\) returning to the boundary. The inner part of the exterior has homotopy type \(X\), the outer part has homotopy type \(B\setminus K\), and their overlap has homotopy type \(B\setminus F\). Product slices, thickened by a variable positive width off the deleted surface, give this homotopy-pushout decomposition.

In \(B\setminus K\), take a two-sided neighborhood of \(F^\circ=F\setminus K\) whose width decreases to zero near \(K\). Relative to \(B\setminus F\), it is a bridge over \(F^\circ\), joining the two sides of the cut. Replace \(F^\circ\) in this diagram by a slightly shrunken copy \(F_0\); this is an objectwise homotopy equivalence. The actual deleted surface and \(K\) are unchanged. In the rectangular normal coordinates to its horizontal copy, split the tube circle over \(F_0\) into its outer and inward semicircles. The outer semicircle is the bridge. The inward semicircle lies in the inner \(X\) piece and gives a specified homotopy between the bridge’s endpoint maps.

Using that homotopy to identify the endpoints replaces the bridge by \(F_0\times S^1\), attached along its inward section. Its circle is precisely the outer bridge followed by the inward return, hence the actual normal circuit. Finally \(F_0\to F\) and \(F_0\times S^1\to F\times S^1\) are homotopy equivalences. This also accounts for the proper edge collars and proves the assertion about the side maps, not just the absolute homotopy type of \(E\). ◻

Proof. Duality and the graph 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 former is free abelian on the cosets of the image of \(\pi_1S\), and the latter is \(\mathbb Z\); there can be only one coset. With ordinary coefficients the pair sequence and duality give \(H_2(P,S)=H_1(P,S)=0\) and \(H_3(P,S)\cong H^1(P)\cong\mathbb Z^r\). The assertions follow. ◻

Proof. Regard these forms as a sheaf on \(V\). Its stalk at a puncture is \(\mathbb C\) in degree zero and zero in higher degrees; elsewhere it is the ordinary de Rham resolution. Partitions of unity can be chosen constant on smaller neighborhoods of the finitely many punctures. The complex is therefore a fine resolution of the constant sheaf, so computes the filled cohomology. A based condition in degree zero removes only \(H^0\). Linear splittings onto cohomology and contracting homotopies may consequently be chosen inside this complex, exactly as for the ordinary vertex de Rham complex.

Wedges, Lie brackets, the curvature recursions, and differentiation in the formal parameters preserve this algebra. Each fixed positive-degree coefficient vanishes near \(Z\), so integration and Stokes take place on a compact smooth subdomain containing its support. The additional end boundaries contribute zero. An entire formal series need not have one common vanishing neighborhood: every calculation at a fixed coefficient uses only finitely many terms. The source’s formal lifts and divisions likewise use finite linear combinations coefficientwise, and extensions from face collars can be supported away from \(Z\).

Removing interior points in dimension four does not change the fundamental group. At each finite nilpotent order the flat systems in question vanish near the punctures in their local trivializations; gauges between them can be separately constant at those ends. Thus the based holonomy and exact-gauge arguments use the same marked group as the filled vertex. Allowing separate constants at distinct ends is important here. These are the uses of vertex smoothness in the proof of Proposition 6.1 of [16]. Its edge-algebraicity theorem, [16], is still applied only to the unchanged compact smooth three-manifolds \(Y_i\). ◻

Proof. The part of a new chamber \(D_c\) already present is exactly \[ A_c=\bigcup_{s\in\operatorname{Des}(c)}S_s. \tag{8}\] To verify the possible equal-length overlaps, consider a point with panel set \(T\). Its incident chambers are the special coset \(cC_T\). Write \(a\) for the shortest element of that coset. Its members have the form \(a\prod_{s\in J}s\), with \(J\subseteq T\), and length \(|a|+|J|\). If a distinct incident chamber occurs no later than \(c\), then \(c\) is not the shortest member, so it has a descent in \(T\). This places the point in (8). The converse follows from the shorter adjacent chamber across each descent panel.

The set in (8) is a three-ball. More generally a union of dual panels indexed by a nonempty simplex is a ball. One proves this by adding its panels one at a time. A new panel is a ball, and its intersection with the preceding panels is the corresponding union of dual panels in its boundary link. Induction on dimension makes that intersection a ball of one lower dimension. The dual-block coordinates give the product collars needed to glue the two balls. This argument also applies to panels in any simplicial cover of \(S\).

It remains to check that \(A_c\) is a ball in the boundary of the preceding union. Contractibility alone would not establish the claim. At a face incident to \(q\) panels, label the \(2^q\) local orthants by subsets of those panels, starting at the shortest incident chamber. The already present orthants form a Boolean downset \(I\): every proper subset of a present subset has smaller word length. This remains true with arbitrary choices within a length tie.

When \(I\) is nonempty and proper, its frontier is a PL hypersurface. For an explicit verification, write local coordinates as \(x=u+t(1,\ldots,1)\) with \(\sum u_i=0\), and label an orthant by its positive coordinates. Its union is described by \[t\leq f(u),\qquad f(u)=\max_{A\text{ maximal in }I}\ \min_{i\notin A}(-u_i).\] The function \(f\) is finite, continuous, and piecewise linear. Hence the frontier is locally a graph. Along the incoming faces some sectors are present and the new chamber sector is absent, so these faces lie on this frontier. The panel identifications are PL, and the embedded incoming three-ball has locally flat boundary in this three-manifold frontier. Gluing the new chamber along it is therefore a boundary connected sum, after rounding. Induction proves the manifold and boundary assertions. Van Kampen gives the labelled free product, and interior stabilizations do not affect any collar argument. ◻

Proof. Use the same universal development with \(\widetilde D\) in place of \(\widetilde P\). The preceding covering argument is unchanged. The first chamber is simply connected; attaching subsequent simply connected chambers along balls shows that this development is the universal cover \(\widetilde N\). Its reduced homology is the direct sum of the chamber reduced homologies, each concentrated in degree two. There are finitely many chamber orbits in \(N\). For each one, induction from its chamber subgroup gives \[\mathbb ZG\otimes_{\mathbb ZH}\pi_2(D) \cong (\mathbb ZG)^{2k}.\] The spheres lie in chamber interiors, so different chambers are orthogonal. Within a chamber the form is its given \(\mathbb ZH\)-form, extended to \(\mathbb ZG\). Reversing one vector in a plane absorbs any chamber orientation sign. This gives the asserted basis and form.

The construction of \(P\) attaches exactly the indicated three-cells away from the boundary. Consequently \(Q\) is obtained from \(N\) by attaching three-cells on this entire middle basis. It follows at once that \(f\) induces the displayed identification of fundamental groups. These attachments split off a nonsingular middle summand in both absolute and relative duality: on homology they quotient by the sphere summand and on cohomology they restrict to its annihilator. With \([Q]=f_*[N]\), naturality of cap product identifies the surviving duality maps with those of \(Q\). The finite free chain criterion therefore gives Poincaré duality for \(Q\), just as for the chamber pair. Equivalently one may glue the chamber duality maps across the manifold collar sectors, where the orientations cancel in pairs. The top fundamental class is unchanged, so \(f\) has degree one.

For clarity, the regular-coefficient calculation is particularly short. In the cohomology sequence for these three-cell attachments, the only potentially new map is evaluation \[H^2(N;\mathbb ZG)\longrightarrow \operatorname{Hom}_{\mathbb ZG}(\pi_2(N),\mathbb ZG).\] Under duality it is the adjoint of the nonsingular hyperbolic form, and hence an isomorphism. The middle cohomology thus disappears, whereas \(H^4(Q;\mathbb ZG)=H^4(N;\mathbb ZG)=\mathbb Z\); all other degrees vanish. The right action on this last group is trivial because the orientation chosen above is preserved by \(G\).

Finally, finiteness and asphericity of \(Q\) give a finite presentation and a finite free resolution of \(\mathbb Z\) from the chains of \(\widehat Q\). They also give finite integral cohomological dimension, which excludes torsion: restriction to a nontrivial finite cyclic subgroup would give a finite projective resolution for that group, contrary to its nonvanishing cohomology in arbitrarily high degrees. This proves all the assertions, including (10). ◻

Proof. Choose a locally finite tree \(T\) on which the free group \(H\) acts freely and cocompactly, without inversions. For the trivial group one may take a single vertex. Assign to each lifted-nerve vertex \(s\) a point \(a_s\in T^{(0)}\), equivariantly under \(H\). There are finitely many vertex and edge orbits in the lifted nerve. Thus the distances \(d_T(a_s,a_t)\) for adjacent \(s,t\) have a common finite bound. Choose an integer \(r\) large enough that the closed radius-\(r\) subtrees \(T_s\) about \(a_s\) intersect for every such pair.

For a vertex \(v\) and an edge \(e\) of \(T\), put \[A_v=\{s:v\in T_s\},\qquad A_e=\{s:e\subset T_s\}.\] These sets have uniformly bounded finite size: there are finitely many generator orbits, and a fixed-radius ball in the free cocompact tree contains uniformly finitely many orbit points. Let \(W_v\) and \(W_e\) be the special subgroups of \(\widehat C\) generated by \(A_v\) and \(A_e\). For \(e=[v,w]\) we have \(A_e=A_v\cap A_w\), so the edge maps are special subgroup inclusions.

The graph of groups over the tree \(T\) has colimit \(\widehat C\). Indeed, occurrences of a generator \(s\) are identified precisely over its connected support \(T_s\). Every defining commutation of \(\widehat C\) occurs in some vertex group, because the two supports intersect at a vertex. Conversely, each vertex group uses the induced subgraph of the lifted nerve, and so introduces no additional relation. The resulting presentation is exactly the right-angled presentation of \(\widehat C\). All the edge maps are injective, as are the maps of these special subgroups into \(\widehat C\).

Let \(\Sigma_v\) and \(\Sigma_e\) be the Davis cube complexes of these finite-generator right-angled groups. They are CAT(0), and the special subcomplexes \(\Sigma_e\subset\Sigma_v\) are convex in the standard piecewise Euclidean cube metric. The CAT(0) assertion is [4]; special-subgroup metric convexity is [20]. Form a tree of spaces with vertex spaces \(\widehat C\times_{W_v}\Sigma_v\) and edge cylinders \(\widehat C\times_{W_e}(\Sigma_e\times[0,1])\). Its underlying tree is the Bass–Serre tree, whose vertices and edges are the cosets \(\widehat cW_v\) and \(\widehat cW_e\), with their indicated labels in \(T\); it is not simply the original support tree. Glue each cylinder at its endpoints by the special-subcomplex maps. Convex gluing gives a CAT(0) space over every finite subtree, preserving the factors as convex isometric subspaces [2]. One can apply this gluing first over finite subtrees and then pass to the union; the local finiteness verified next ensures completeness.

There is no local-finiteness problem from potentially infinite Bass–Serre valence. In a fixed vertex Davis complex, for any one incident support edge, translates of its special subcomplex have vertex sets equal to the cosets of \(W_e\) in \(W_v\). These cosets partition the Cayley vertices. A finite subcomplex therefore meets only finitely many of these translates. Each support vertex has only finitely many incident edges. Since each \(\Sigma_v\) is locally finite, the whole tree of cube complexes is locally finite. Its cubes have unit edge lengths and uniformly bounded dimension, so the induced length metric is proper and complete.

Left multiplication gives the \(\widehat C\)-action. The equivariance of the supports extends this to \(\widehat C\rtimes H\). An element stabilizing a vertex space projects to an element of \(H\) fixing its support vertex, and that element is trivial. Within the vertex space its point stabilizer is therefore a finite Coxeter stabilizer. The same reasoning applies to edge interiors. The cellular action on this locally finite complex is proper. There are finitely many \(H\)-orbits of support vertices and edges, and each corresponding Davis complex has compact quotient by its special group. Hence the full action is cocompact. Each clique of the lifted nerve has at most four vertices; the vertex spaces have dimension at most four and the edge cylinders dimension at most five. Finally \(G\) has finite index by (9), so its restricted action has all the same properties. ◻

Proof. The Davis chamber is the cone on the barycentric subdivision of \(\mathcal L\), with its usual dual panels on the base. Identify that base panelwise with \(S\). Map a boundary collar of \(D\) to this cone by that identification on its outer boundary and by coning to the apex on its inner boundary; map the remainder of \(D\) to the apex. The maps agree across reflected panels and hence define \(\eta\). Every chamber image lies in the corresponding compact Davis chamber, whose diameter is independent of its label. At the identity apex, radial projection of the cone base to a sufficiently small metric sphere gives exactly its link identification. Stabilization changes only the collapsed inner core, so does not affect the construction. ◻

Proof. Since \(A\) is finite free over \(\Lambda\), integral group duality gives \[ H^2(G;A)=H^3(G;A)=0. \tag{13}\] The second Postnikov invariant lies in the latter group, so it vanishes. The former removes the degree-two ambiguity in making the identification with the split stage for the prescribed module marking. We use a functorial Eilenberg–Mac Lane model for \(K(A,2)\), so the given action on \(A\) defines the displayed homotopy quotient and its section.

Separately, freeness gives \(H_i(G;A)=0\) for \(i>0\). In total degree four, the homology spectral sequence of \[K(A,2)\longrightarrow B\longrightarrow Q\] therefore has only \(H_4(G;\mathbb Z)\) and \(H_0(G;\Gamma(A))\) as possibly nonzero terms: the intervening term \(H_2(G;A)\) is zero, and the fiber has no homology in degrees one and three. The possible incoming differentials to \(H_0(G;\Gamma(A))\) have sources \(H_3(G;A)=0\) and \(H_5(G;\mathbb Z)=0\). The section preserves the base term. This proves (12), including its integral, rather than merely rational, exactness. ◻

Proof. Write \(e_1,\ldots,e_{2b}\) for the chosen \(\Lambda\)-basis of \(A\) and \(\phi_i\colon A\to\Lambda\) for the coordinate functionals. The low-degree cohomology sequence of the Postnikov fibration, with regular coefficients, contains \[0\longrightarrow H^2(G;\Lambda) \longrightarrow H^2(B;\Lambda) \longrightarrow\operatorname{Hom}_{\Lambda}(A,\Lambda) \longrightarrow H^3(G;\Lambda).\] Both outside groups vanish. Thus each \(\phi_i\) is the restriction of a unique class \(x_i\in H^2(B;\Lambda)\).

Their cup products may be evaluated without multiplying away either coefficient factor: \[ z\longmapsto \langle x_i\smile x_j,z\rangle \in(\Lambda\otimes_{\mathbb Z}\Lambda)_G, \qquad z\in H_4(B;\mathbb Z). \tag{15}\] The tensor product has the diagonal left action. Its coinvariants identify, as an abelian group, with \(\Lambda\) by \(g\otimes h\mapsto g^{-1}h\). For a closed oriented manifold with the indicated middle module, evaluations in (15) are the equivariant cup form on the coordinate functionals. By Poincaré duality this is the dual form of its equivariant intersection pairing. The prescribed isometry therefore makes these evaluations equal on \((u_N)_*[N]\) and \((u_L)_*[L]\).

For completeness, these tests detect the fiber term integrally. The underlying abelian group of \(A\) is free on the elements \(g e_i\). The group \(\Gamma(A)\) has a basis consisting of one square generator for each such element and one cross generator for each unordered pair of distinct such elements. The action of \(G\) permutes this basis. Hence its coinvariants are free abelian on the corresponding orbits. A nonidentity element stabilizing an unordered pair would exchange its two members and have order two; torsion-freeness rules this out.

The square orbit represented by \(e_i\) evaluates to \(1\) in the \((i,i)\) test. For \(i<j\), the cross orbit represented by \(\{e_i,t e_j\}\) evaluates to \(t\) in the \((i,j)\) test. Finally, for \(t\ne1\), the orbit represented by \(\{e_i,t e_i\}\) evaluates to \(t+t^{-1}\) in the \((i,i)\) test. These last orbits are indexed by \(t\) modulo inversion. Since \(G\) has no elements of order two, their supports are disjoint two-element subsets of \(G\setminus\{1\}\). The displayed tests are therefore jointly injective on \(\Gamma(A)_G\). In particular, no integral square or hidden two-torsion is discarded.

The two fundamental classes have the same image \([Q]\) in \(H_4(Q;\mathbb Z)\). By Lemma 10, their difference lies in \(\Gamma(A)_G\). All its cup tests vanish, so the difference is zero. ◻

Proof. Replace \(u_L\colon L\to B\) by a fibration. Its fiber is two-connected, with third homotopy group \(\pi_3(L)\) carrying the usual \(G\)-action. The first obstruction to lifting \(u_N\) is consequently the pullback of a universal class \[\kappa\in H^4(B;\pi_3(L)).\] The pullback \(u_L^*\kappa\) is zero, since \(u_L\) lifts to its own fibration replacement. Naturality of cap product and Lemma 11 give \[\begin{align*} (u_N)_*\bigl((u_N^*\kappa)\mathbin{\cap}[N]\bigr) &=\kappa\mathbin{\cap}(u_N)_*[N]\\ &=\kappa\mathbin{\cap}(u_L)_*[L]=0 \quad\text{in }H_0(B;\pi_3(L)). \end{align*}\] The map \(u_N\) induces an isomorphism on fundamental groups, and hence on these degree-zero coefficient groups. Poincaré duality identifies \[H^4(N;\pi_3(L))\xrightarrow{\ \cap[N]\ } H_0(N;\pi_3(L))\cong\pi_3(L)_G.\] Thus the lifting obstruction vanishes.

The manifold \(N\) has a finite four-dimensional CW structure from the smooth chamber construction and the panel collars. There are no higher obstructions on this source. No triangulation or smooth structure on the hypothetical \(M\) is required. The resulting lift \(f\) has the stipulated maps on \(\pi_1\) and \(\pi_2\). The universal covers of \(N\) and \(L\) have reduced homology only in degree two, where the lift is an isomorphism. These manifolds have CW homotopy type [15]. The homological Whitehead theorem on their simply connected universal covers, followed by Whitehead’s theorem, shows that \(f\) is a homotopy equivalence [10]. Finally \(c_Lf\simeq q\) and both classifying maps have degree one, so \(f_*[N]=[L]\). ◻

Proof. The universal intersection forms are even, so \(N\) and \(L\) are almost spin. Indeed, evaluation of \(w_2\) on every spherical class is its self-intersection modulo two, and on the simply connected cover these evaluations detect \(w_2\).

We apply the proof of Kasprowski–Land’s Theorem 1.1, specifically Lemma 2.3, Lemma 3.3 and Theorem 3.1 of [11]. All hypotheses hold: \(Q\) is an aspherical Poincaré four-complex; \(G\) satisfies both Farrell–Jones conjectures; the two manifolds have the same orientation character and degree-one classifying maps; and they are almost spin. Their normal-invariant calculation, using assembly injectivity and the Atiyah–Hirzebruch spectral sequence, identifies the normal invariant of this particular \(f\) with a class in \[\ker\bigl(H_2(L;\mathbb Z/2)\longrightarrow H_2(Q;\mathbb Z/2)\bigr).\] The Serre spectral sequence for the universal-cover fibration shows that this kernel is represented by spherical classes. Since \(L\) is almost spin, their \(w_2\) evaluations vanish. Lemma 3.3 of [11] therefore supplies a target self-equivalence \(\phi\) with the same normal invariant as \(f\), of the form \[L\longrightarrow L\vee S^4 \xrightarrow{\operatorname{id}\vee\eta^2}L\vee S^2 \xrightarrow{\operatorname{id}+a}L.\] Here \(a:S^2\to L\) represents the spherical class. This pinch acts identically on \(\pi_1\) and \(\pi_2\), because its extra part factors through the four-sphere. Thus \(\phi^{-1}\circ f\) has trivial normal invariant and the same marking as \(f\). Surgery of its normal bordism over \(L\), as in Theorem 3.1 of [11], gives the required \(s\)-cobordism. Neither this conclusion nor the cited theorem requires \(G\) to be a good group. We have not asserted that \(W\) itself is a product. ◻

Proof. Apply the stable product theorem to \(W\) from Lemma 13. Denote its stabilized ends by \(N^*\) and \(L^*\) and the stabilized bordism by \(W^*\). We verify its marking, including the new summands.

The operation removes disjoint normal \(D^4\times[0,1]\) neighborhoods of arcs between the ends and glues \[\bigl((S^2\times S^2)\setminus\operatorname{int}D^4\bigr) \times[0,1]\] along the resulting \(S^3\times[0,1]\) interfaces. Removing these codimension-four arcs changes neither \(\pi_1\) nor second homology of the universal cover. Mayer–Vietoris across the lifted interfaces therefore gives a specified decomposition \[ \pi_2(W^*)\cong\pi_2(W)\oplus\Lambda^{2k}. \tag{16}\] The second summand is represented by the standard spheres in the added product pieces. Choose their whiskers along the stabilization arcs. The two end inclusions \(j_N\colon N^*\to W^*\) and \(j_L\colon L^*\to W^*\) carry each new pair to this same pair, while their old summands map to \(\pi_2(W)\) with exactly the original bordism identification.

A product structure \(F\colon N^*\times[0,1]\to W^*\) which is the identity on the incoming end restricts at the other end to a homeomorphism \(h\colon N^*\to L^*\). Use the product trace for the outgoing basepoint path. Any discrepancy from the original bordism path can be corrected by pushing the outgoing basepoint around a loop in \(L^*\), since \(j_L\) is an isomorphism on fundamental groups. The resulting based homotopy given by \(F\) implies \[(j_L)_*h_*=(j_N)_*.\] Both inclusions are isomorphisms on \(\pi_1\) and \(\pi_2\). Thus \(h_*=(j_L)_*^{-1}(j_N)_*\), with the same base paths. In the decomposition (16) it preserves the original marking and identifies the new standard basis element by element. There is no arbitrary automorphism mixing the old and new summands.

All \(b+k\) standard pairs on \(L^*=M\mathbin{\#}(b+k)(S^2\times S^2)\) have disjoint punctured \(S^2\times S^2\) neighborhoods. Give the two spheres of each pair the common whisker to their intersection used for its module marking, and pull the whole based neighborhood back by \(h\). A pair has its one prescribed transverse intersection; different pairs and their neighborhoods are disjoint. The marking just proved identifies their based homotopy classes with the original chamber basis and the added standard basis, respectively. ◻

Proof. The original immersed representatives stay in their chamber interiors, and the stabilization spheres can be put in their interior connected summands. Stabilization adds orthogonal standard planes and changes neither the chamber fundamental group nor the old pairings.

Proposition 14 supplies a finite collection of geometric pairs on the compact quotient \(N^*\), representing the specified based classes. Choose a based homotopy for each sphere between its two representatives. Lift the neighborhoods, sphere drawings, and homotopies to \(Y\), using the prescribed marking to choose the lift for each chamber label. Covering maps preserve \(\pi_2\) and lift these based homotopies, so they end at precisely the labelled geometric representatives. Disjoint embedded neighborhoods lift to disjoint embedded neighborhoods.

There are only finitely many labelled data on the quotient. Each chosen lift of a drawing, closed standard neighborhood, local basing, or homotopy has compact image. Local finiteness of the development implies that it meets only finitely many chambers. Take \(d_0\) to be the maximum gallery distance from its assigned label over this finite list of chosen lifts. All remaining data are translates by \(\Delta\), whose action preserves gallery distance. The same \(d_0\) therefore works everywhere. ◻

Proof. First recall the consequences of the graph-type Poincaré pair \((P,S)\) recorded in Corollary 4. Duality and its ordinary homology sequence give \[ H_1(S;\mathbb Z)\cong H_1(P;\mathbb Z)\cong\mathbb Z^r, \qquad H_2(S;\mathbb Z)\cong H^1(P;\mathbb Z)\cong\mathbb Z^r. \tag{17}\] The same corollary gives the surjectivity of \(\pi_1(S)\to H\).

Start with the oppositely oriented copy of \(D\). Represent a free basis of its fundamental group by disjoint embedded circles. Their normal bundles are trivial, and general position makes the circles disjoint from the finite sphere system. Interior surgery on these circles kills the free generators. The old sphere classes retain their hyperbolic intersection and framing data after augmentation to \(\mathbb Z\). In the now simply connected manifold, the sphere embedding theorem with geometric duals realizes a Lagrangian by disjoint framed embedded spheres. Surgery on these spheres preserves simple connectivity and removes the old hyperbolic planes. This use of the disc theorem is in the simply connected case only; see [17].

Let \(J\) be the result. Each circle surgery increases Euler characteristic by two, and each sphere surgery decreases it by two. Hence \[\chi(J)=(1-r+2k)+2r-2k=1+r.\] The boundary is connected and \(J\) is simply connected. Thus \(H^1(J,S;\mathbb Z)=0\) by the cohomology sequence of the pair, and Poincaré–Lefschetz duality gives \(H_3(J;\mathbb Z)=0\). Since \(J\) has nonempty boundary, \(H_4(J;\mathbb Z)=0\), and therefore \(b_2(J)=r\).

There is no integral index ambiguity in this calculation. The relevant part of the duality sequence is \[ 0\longrightarrow H_2(S;\mathbb Z)\longrightarrow H_2(J;\mathbb Z) \longrightarrow H^2(J;\mathbb Z)\longrightarrow H_1(S;\mathbb Z) \longrightarrow0. \tag{18}\] Here \(H_3(J,S;\mathbb Z)=H^1(J;\mathbb Z)=0\) accounts for the initial zero. The universal coefficient theorem makes \(H^2(J;\mathbb Z)\) free, because \(H_1(J;\mathbb Z)=0\). Every torsion element of \(H_2(J;\mathbb Z)\) would consequently belong to the image of the injected free group \(H_2(S;\mathbb Z)\), so there is no such element. The first two nonzero groups in (18) now have the same rank \(r\). The middle map has rank zero and takes values in a free group, hence is zero. Exactness proves that the first map is an integral isomorphism.

The map \(H_2(S;\mathbb Z)\to H_2(D;\mathbb Z)\) is zero: the next map \(H_2(D;\mathbb Z)\to H_2(D,S;\mathbb Z)\) is the nonsingular hyperbolic intersection map. Also \(H_1(S;\mathbb Z)\to H_1(D;\mathbb Z)\) is the isomorphism in (17). Mayer–Vietoris, together with the isomorphism just proved for \(J\), therefore identifies \(H_2(D\cup_S J;\mathbb Z)\) with \(H_2(D;\mathbb Z)\). All products are unchanged, since their representatives lie in the interior of \(D\). Finally, the surjectivity of \(\pi_1(S)\to H\) and van Kampen show that \(D\cup_S J\) is simply connected. ◻

Proof. First project the outer immersed systems through the pinch and put them in general position. The deleted and retained blocks were orthogonal, since their classes agree with the correspondingly labelled standard pairs. The retained forms and quadratic self-intersection data are therefore still hyperbolic over \(\mathbb Z\pi_1(Y_R)\). Choose framed immersed representatives and cancel the algebraic intersection pairs. The framing corrections and, where necessary, local cusp changes can be chosen from finitely many local drawings. The group in question is a free product of free chamber groups, hence is torsion-free; there is no order-two ambiguity in these quadratic data.

We justify carefully the uniformity of the Whitney discs required at this and subsequent finite stages. The initial lifted data have finitely many types under a cocompact group of translations of \(Y\). A gallery ball of fixed radius contains a uniformly bounded number of chambers and of pieces of those data. There are thus finitely many local templates, even after specifying which of the standard neighborhoods meeting that ball have been peeled. Paths on participating spheres are taken inside their bounded carriers. Common long whiskers conjugate an intersection label but need not be included in the geometric Whitney circuit.

If a circuit enters a replacement ball, move its portions there to chosen arcs on the boundary three-sphere. This gives a circuit in the unpeeled space. Nullity is unchanged, since both the deleted punctured \(S^2\times S^2\) and the replacement ball are simply connected and have simply connected boundary. For every null circuit among the finitely many resulting templates, choose one compact filling in \(Y\). There is a finite maximum of their propagation bounds. Translate the chosen filling to the location of the circuit and apply the actual global pinch map. This constructs a filling in \(Y_R\).

A filling may encounter additional peeled neighborhoods. No further choice of fillings is then needed: the global pinch is already defined on them, and each such neighborhood has the same bounded carrier. Consequently this procedure is uniform in \(R\). It is a finite-template argument for the specified circuits, not a bound for arbitrary null loops. Repeating a fixed finite number of stages preserves the argument: enlarge the carrier radius and refine the finite template list at each stage.

Now apply the geometric Casson lemma to the two halves of the sphere system, removing the paired mixed intersections by regular homotopies. There is no iterative repair cascade here. Use the local construction of [17] with the two surface arguments taken to be the entire possibly disconnected families. Preselect all framed pairing Whitney discs with disjoint boundaries; their interiors miss the other Whitney boundary arcs by \(2+1<4\) general position. First push the first family off all these interiors by fingers along arcs in the Whitney discs. Choose the arcs away from their finitely many intersections with other Whitney discs and the second family. The finger supports then leave the other Whitney discs and attaching boundaries fixed. Next Whitney-move the second family using the same discs. The new intersections in each phase are only within that phase’s family; intersections between Whitney interiors cause only allowed second-family intersections. Thus the whole batch stays in the original bounded Whitney neighborhoods. No later disc is transported or chosen again through an expanding sequence of carriers. Local finiteness permits these bounded batches on the unbounded \(Y_R\). This produces the geometric duality in (1), while permitting self-intersections within either half. The remaining double points of the first half admit framed Whitney discs. Tube their interiors off the first-half spheres using the geometric duals. All the Whitney circuits involved are the null circuits just discussed.

For completeness, we verify the algebraic input to the grope preparation in the complement \(M_0\) of small plumbed neighborhoods of the \(a_i\). The punctured geometric dual \(b_i\) bounds an \(a_i\)-meridian in \(M_0\). Since these meridians normally generate the kernel, inclusion induces \(\pi_1M_0\cong\pi_1Y_R\). Thus the following calculation retains all intersection labels in the required complement group ring.

At one point of every paired double point take its Clifford torus. Cap its two basis curves by standard meridional discs and tube each meridional disc off its incident \(a_i\)-sheet into a punctured parallel of \(b_i\), with the product framing. Denote these caps by \(C_\ell\), and let \(i(\ell)\) be the index of the dual used. Their equivariant intersection calculation is the Clifford-cap calculation in the proof of [17]. The local meridional pieces have disjoint interiors. Choose the thin tubing tracks away from the finitely many other designated dual points; local changes of tracks introduce only oppositely signed points with the same group label. The remaining intersections are those between the corresponding based parallels of \(b_{i(\ell)}\). Hence their sums are obtained from the pairings of the \(b_i\) by the basing units and orientation signs. Likewise, a cap’s self-intersection sum is that of its framed \(b_i\) parallel, together with cancelling local terms. Since the mixed Casson regular homotopies preserved \(\lambda(b_i,b_j)=0\) and \(\mu(b_i)=0\), this proves \[\lambda(C_\ell,C_m)=0,\qquad \mu(C_\ell)=0 \quad\text{for every }\ell,m,\] including distinct parallel caps using the same dual sphere.

The Clifford bodies lie in a boundary collar of \(M_0\). Following [17], contract temporary parallel copies and push the Whitney discs off those contractions before using them as geometric dual spheres. At the remaining intersections with Whitney discs, tube the Clifford caps into parallel copies of these temporary duals. Lemma 4.1 of [17] now applies with its full cap-pairing hypothesis. Its two separation batches and height raising provide the prepared Clifford copies, whose entire capped surfaces are geometrically dual in the sense of [17]. These prepared copies, including all their caps, are the protected surfaces used from this point onward. The preparation in [17] produces height-two Whitney gropes, separates their caps from the protected Clifford caps, and raises the height to \(h\). The collar levels and the full dual-cap disjointness are part of this preparation, not an assertion made after the caps have been changed. There are exactly two cap-family separation batches before height raising. Each uses the same no-new-mixed-pairs invariant, with its protected bodies at fixed collar levels.

Each of the operations before height raising involves only a fixed number of kinds of tubing, push-off, Whitney-disc choice and regular homotopy. Their simultaneous use does not require traversing the whole list of sphere components. At each fixed stage, bounded carriers and finite drawings bound the number of other carriers meeting a given one. Divide the operations into finitely many classes with disjoint enlarged carriers and perform one class at a time. This gives a number of local stages independent of \(R\). New null circuits are handled by the finite-template procedure above.

For height raising there is also an explicit size estimate: [9], in the January 2013 numbering, bounds the increase of component-image diameter by a factor of seven at each step. A fixed requested height therefore has a fixed propagation bound. One can apply this estimate to the gallery map into the Davis complex: extend that map over each replacement ball by coning its boundary image inside a metric ball of uniformly bounded radius. This is possible by CAT(0) convexity. A point in a chamber is mapped within a fixed distance of its label, and properness of the Coxeter action turns a fixed metric bound into a fixed word-distance bound. The construction uses parallel transverse gropes and neighborhood operations; it preserves the protected Clifford capped surfaces. All these operations can be made locally finite in the unbounded development.

Finally, outside a fixed buffer of the peeling region the input is the original system in an individual chamber. Its intersection cancellations hold already over that chamber’s group ring. The same preparation, including the null discs, can consequently be chosen in that chamber. Only a bounded buffer of the switch between the peeled and unpeeled data requires the preceding translated templates. This proves the last assertion and permits later truncation without cutting off an unfinished preparation. ◻

Proof. Here is the radial estimate that supplies the first assertion. If \(x,y\in\Sigma_C\) have distance at least \(s>t\) from \(o\), put them first on the same metric sphere, of radius the smaller of their two radii. The distance between the resulting points is at most \(2d(x,y)\), by the triangle inequality and \(|d(o,x)-d(o,y)|\leq d(x,y)\). CAT(0) comparison on the two radial segments then gives \[ d(\rho(x),\rho(y)) \leq \frac{2t}{s}\,d(x,y). \tag{21}\] The gallery map takes each chamber into a set of uniformly bounded diameter. Thus a bounded gallery carrier has bounded image diameter. Properness of the Davis complex, together with its cocompact Coxeter action, makes its distance from \(o\) tend to infinity with the word length of its center. Equation (21) applies.

We only need the direction map outside the central core. Consider a peeled standard neighborhood that meets an outer object or any of its fixed enlarged carriers. Both the object and the whole standard neighborhood have bounded propagation. The label of this neighborhood therefore has length at least \(R\) minus a fixed constant. Its entire boundary three-sphere, not merely the arcs encountered by the object, has direction image of arbitrarily small diameter for large \(R\). Choose a slightly larger coordinate ball of \(S\) containing this image. Contracting that ball extends the boundary map across the replacement four-ball, with image still in the chosen ball. These extensions agree with the unchanged direction map on all their boundaries.

The near-central peeled neighborhoods need not admit such small extensions. They are disjoint from all the carriers under consideration once \(R\) is large, and no direction map on them is used. This proves the asserted far-out extension and its uniform smallness.

Before peeling, the composition with \(\beta\) extends across the central chamber by its given map to \(BH\). Its restriction to the central boundary agrees with that map: this is the boundary-direction identification in Lemma 9. In a noncentral chamber the gallery map collapses its inner part to the noncentral vertex, so every loop of its chamber group has constant direction there. The shelling description makes the chamber groups the free factors of the fundamental group of a finite ball of chambers. The induced homomorphism is thus exactly the projection retaining the central factor. By (19) this is also the homomorphism induced by passage to \(U_L^+\).

Replacing a simply connected piece by a simply connected ball does not alter this calculation. Loops through the replacement can be moved to the boundary of the piece, and the direction extension agrees there with the old map. The comparison therefore holds on the required region in \(Z\). If a whole image system maps into a contractible patch, every one of its loops, including loops passing through arbitrarily many intersections, has trivial image under \(\beta\). Since \(BH\) is a classifying space and \(\pi_1Z=H\), these loops are nullhomotopic in \(Z\). ◻

Proof. We first specify the relative setting. Remove a sufficiently small regular neighborhood of the five retained stages, truncating each upper attaching collar at its boundary. The upper components are now properly immersed disk-like capped gropes of height at least three. By the duality condition in Lemma 17, their entire images miss the protected Clifford capped surfaces. Choose a neighborhood of the upper system disjoint from those surfaces and from the retained stages except at the prescribed attaching regions. All the following modifications are neighborhood operations in this relative manifold. They therefore leave the lower stages, attaching framings, and whole Clifford systems unchanged, including the unused boundary-collar levels needed later.

We use the following precise content of Freedman–Quinn’s controlled separation lemma [9]. For a map from a four-manifold to a locally compact metric space, a properly immersed union of disk-like capped gropes of height at least three, with sufficiently small component images, admits new caps for the same bodies such that distinct components are disjoint and their images remain small. Its compact-case proof separates finitely many cover collections before separating the members of each collection [9]. We retain this two-phase construction because we need a bound on gallery propagation only in its first phase.

Choose compact sets \(K_1,\ldots,K_q\) covering \(S\), each contained in an open coordinate ball \(V_j\). Choose \(\epsilon>0\) so that the closed \(\epsilon\)-neighborhood of every \(K_j\) lies in \(V_j\). Assign each upper component to a \(K_j\) met by its direction image. If every initial image has diameter less than \(\delta\), that component lies within distance \(\delta\) of its assigned \(K_j\).

For the first phase, regard all components assigned to \(K_j\) as one collection. The construction in [9] raises height by one, contracts the added stages of the first collection and pushes the later collections off, then continues through the collections in order. The original bodies are retained. Its component-diameter estimate is \(3^{q-1}7^2\delta\). A point of each retained body stays within \(\delta\) of \(K_j\), so the entire resulting \(j\)th collection lies in the \(3^{q+3}\delta\)-neighborhood of \(K_j\). Choose \(\delta\) with \(3^{q+3}\delta<\epsilon\), leaving room for a small regular-neighborhood thickening. The number of height-raising and contraction passes depends only on \(q\). The same size estimates apply to the gallery map into \(\Sigma_C\), extended over the replacement balls as in Lemma 17. Since each original body is retained, its image anchors the resulting component within a bounded metric distance of the original label. Properness of the Coxeter action converts this into a gallery bound independent of the number of components and of \(R\).

The collections now have pairwise disjoint images. Choose disjoint regular neighborhoods of their entire images, with the \(j\)th neighborhood still mapping into \(V_j\). Inside each such neighborhood, apply the collection-separation procedure again, this time taking each individual upper component as a collection. There may be arbitrarily many of them. The height is raised once more inside that neighborhood before contracting the added stages, so this second phase again retains the original bodies. No estimate depending on their number is needed: all operations stay in the already chosen neighborhood. The resulting upper components have pairwise disjoint neighborhoods. Totally contract each in its own neighborhood to obtain the claimed mutually disjoint framed immersed cap discs.

Every loop in a resulting cap image lies in its color neighborhood, whose entire direction image is contained in the contractible \(V_j\). Lemma 18 therefore makes that loop nullhomotopic in \(Z\). The second phase need not bound the gallery diameter of an individual cap. It preserves instead the containment of its whole image in a direction patch, which is exactly the condition needed for nullity. ◻

Proof. Perform surgery on the disjoint framed Lagrangian spheres just constructed. Their geometric duals kill the meridians in their complement, so the surgeries preserve the fundamental group. All changes are in the interior; the boundary and its marking are unchanged.

For clarity, let there be \(m\) remaining planes and let \(W\) be the five-dimensional surgery trace. With regular \(\mathbb ZH\) coefficients, the relative chain complex of \((W,Z)\) is concentrated in degree three, with module \((\mathbb ZH)^m\). Its boundary map to \(H_2(Z;\mathbb ZH)\) sends the basis to the Lagrangian basis. By (20) this is a split injection. Consequently \[H_3(W;\mathbb ZH)=0,\qquad H_2(W;\mathbb ZH)\cong(\mathbb ZH)^m,\] the latter module represented by the dual half, and the other positive-degree homology groups vanish. Read the same trace from \(V\). Its relative chain complex is concentrated in degree two, again with module \((\mathbb ZH)^m\). The map \[H_2(W;\mathbb ZH)\longrightarrow H_2(W,V;\mathbb ZH)\] is the pairing of the surviving dual classes with the surgered Lagrangian, hence is an isomorphism. The sequence of \((W,V)\) therefore gives \(H_i(V;\mathbb ZH)=0\) for every \(i>0\).

The universal cover of \(V\) is simply connected and acyclic. A topological manifold has CW homotopy type [15]; in the boundary case one may first append an exterior open collar, which does not change homotopy type. The homological Whitehead theorem therefore makes this universal cover contractible [10]. Thus \(V\) is a \(K(H,1)\) and is homotopy equivalent to the finite graph \(BH\). Under this equivalence its boundary map is the retained central marking, as asserted. ◻