Proof. Fix a finite triangulation of \(B\). For a closed simplex \(\sigma\), let \(j_\sigma:\sigma^\circ\hookrightarrow\sigma\) and \(i_\sigma:\partial\sigma\hookrightarrow\sigma\) be the inclusions. The open-complement triangle is \[ (j_\sigma)_!\underline{\mathbb R}_{\sigma^\circ} \longrightarrow\underline{\mathbb R}_{\sigma} \longrightarrow(i_\sigma)_*\underline{\mathbb R}_{\partial\sigma} \longrightarrow (j_\sigma)_!\underline{\mathbb R}_{\sigma^\circ}[1]. \tag{3}\] Induction on dimension expresses the first term using constants on closed simplices: apply the same argument to the finite triangulation of the boundary, starting with points. Filter \(B\) by its skeleta. On each skeleton the complement of the previous skeleton is a finite disjoint union of open simplices. Its constant sheaf extended by zero is the finite sum of the first terms of (3), extended into that skeleton. The open-complement triangles for this filtration therefore build \(\underline{\mathbb R}_{B}\) from constants on the closed simplices of \(B\). Pushing these finite triangles along \(f\) proves the assertion. For the semialgebraic categories all restrictions and simplex inclusions just used are semialgebraic.

For the second assertion, the class of summands of objects in a full triangulated subcategory is itself triangulated. Indeed, suppose \(A\oplus A'=C\) and \(B\oplus B'=D\) belong to that subcategory, and let \(u:A\to B\). Fullness places the block map \(u\oplus0:C\to D\) in the subcategory. A cone on this map is \[\mathop{\mathrm{Cone}}(u)\oplus B'\oplus A'[1].\] Hence \(\mathop{\mathrm{Cone}}(u)\) is a summand of an object of the subcategory. Shifts and finite sums have the same property, and an iterated summand is a summand. This proves the claim and also proves thickness. ◻

Proof. We first calculate on a closed \(n\)-simplex \(\Delta\), choosing an orientation of its interior. With \(j:\Delta^\circ\hookrightarrow \Delta\), \[ \mathcal D_\Delta\underline{\mathbb R}_{\Delta} =\omega_\Delta \simeq j_!\underline{\mathbb R}_{\Delta^\circ}[n], \qquad \mathcal D_\Delta\bigl(j_!\underline{\mathbb R}_{\Delta^\circ}[n]\bigr) \simeq\underline{\mathbb R}_{\Delta}. \tag{6}\] Here is the local calculation, including the boundary. At an interior point the compactly supported cohomology of a sufficiently small open ball is \(\mathbb R\) in degree \(n\) and zero otherwise. At a boundary point a cofinal system of relative neighborhoods is homeomorphic to a half-ball with its flat face retained. All its compactly supported cohomology vanishes. For example, its one-point compactification is a closed ball and has zero reduced cohomology; this also applies at corners after a local homeomorphism. The local description of the dualizing complex therefore gives the first isomorphism in (6). The orientation identifies its interior local system with the constant sheaf. For the dualizing-complex formulation of orientation on a manifold without boundary, see (Schapira 2023, sec. 11.7); the relative half-ball calculation above establishes the boundary case needed here.

For the second isomorphism, open-image duality gives \[\mathcal D_\Delta j_!\underline{\mathbb R}_{\Delta^\circ} \simeq Rj_*\underline{\mathbb R}_{\Delta^\circ}[n].\] The unit \(\underline{\mathbb R}_{\Delta}\to Rj_*\underline{\mathbb R}_{\Delta^\circ}\) is an isomorphism: the intersection of each sufficiently small relative neighborhood with \(\Delta^\circ\) is nonempty and contractible, with no higher constant-sheaf cohomology. Under these identifications the biduality map of \(\underline{\mathbb R}_{\Delta}\) is this unit. Thus it is the canonical biduality map, and is invertible, including at the boundary. The case \(n=0\) is ordinary duality of the one-dimensional vector space.

Now take a simplex generator \(A=Rf_*\underline{\mathbb R}_{\Delta}\). Since \(f\) is proper, (5) and (6) give \[\mathcal D_X A\simeq Rf_*\omega_\Delta.\] The boundary triangle (3) expresses \(\omega_\Delta\) using constants on compact subpolyhedra of \(\Delta\). Consequently \(\mathcal D_XA\) lies in the same generated category.

We check explicitly that proper duality transports the canonical biduality map. Put \(F=Rf_*\) and \(B=\underline{\mathbb R}_{\Delta}\), and write \[\zeta_C:F\mathcal D_\Delta C\xrightarrow{\sim}\mathcal D_XFC \qquad(C=B,\mathcal D_\Delta B)\] for (5). The source calculation makes \(B,\mathcal D_\Delta B,\mathcal D_\Delta^2B\) bounded, as are their proper images. The two comparisons \(\zeta_B,\zeta_{\mathcal D_\Delta B}\) and the dual of \(\zeta_B\) therefore show that all terms in the following diagram are bounded: \[ \begin{tikzcd}[column sep=large] FB \arrow[r,"\eta_{FB}"] \arrow[d,"F\eta_B"'] & \mathcal D_X^2FB \arrow[d,"\mathcal D_X(\zeta_B)"] \\ F\mathcal D_\Delta^2B \arrow[r,"\zeta_{\mathcal D_\Delta B}"'] & \mathcal D_XF\mathcal D_\Delta B. \end{tikzcd} \tag{7}\] Under tensor–Hom adjunction, both composites correspond to the pushed evaluation pairing with its two arguments interchanged. The cup map respects tensor symmetry, and both composites use the same trace, so the diagram commutes. Its left, bottom, and right arrows are isomorphisms. Hence \(\eta_A=\eta_{FB}\) is an isomorphism.

The class of objects whose dual is bounded and belongs to the generated category is closed under shifts and triangles. On this class, the objects for which \(\eta\) is invertible are closed under these operations as well: naturality gives a morphism of triangles, and an isomorphism on two terms gives an isomorphism on the third. The same assertions pass to direct summands, by applying the dual functor to the splitting maps. 4 now proves the duality assertions for all four categories.

Each generator vanishes outside \(f(\Delta)\). Finite sums, triangles, shifts, and summands preserve vanishing outside a finite union of such compact sets. Choose one such compact \(K\) for \(A\). Open restriction of duality shows that \(\mathcal D_X A\) also vanishes on \(X\setminus K\). ◻

Proof. If \(A\) vanishes outside a compact \(K\subset X\), let \(i:K\hookrightarrow X\). The canonical map \(A\to i_*i^{-1}A\) is an isomorphism, as is checked on cohomology stalks. The map \(g i:K\to Y\) is proper. Composition of direct images therefore identifies the natural comparison \[Rg_!A\longrightarrow Rg_*A\] with the corresponding, invertible comparison for \(g i\). This argument requires only compactness of \(K\), not a polyhedral structure on \(K\). On a generator the image is \[Rg_*Rf_*\underline{\mathbb R}_{B}\simeq R(gf)_*\underline{\mathbb R}_{B},\] which is again an allowed generator. Exactness and preservation of splittings prove category membership and boundedness for every object. The semialgebraic case follows from the same composition check.

Both \(A\) and \(\mathcal D_XA\) have compact support by 5. Hence (5) gives the natural isomorphism \[ Rg_!\mathcal D_XA\xrightarrow{\sim}\mathcal D_Y Rg_!A. \tag{8}\] It preserves biduality because it is defined by evaluation and trace. More explicitly, a pairing \(\beta:A\otimes A\to\omega_X[-r]\) is sent to the pairing \[ \begin{split} Rg_!A\otimes Rg_!A &\longrightarrow Rg_!(A\otimes A) \xrightarrow{Rg_!\beta}Rg_!\omega_X[-r] \xrightarrow{\operatorname{Tr}_g[-r]}\omega_Y[-r]. \end{split} \tag{9}\] The first arrow is the cup map obtained from the projection formula and the adjunction map \(g^{-1}Rg_!A\to A\) (using \(Rg_!A\to Rg_*A\)). Under tensor–Hom adjunction, (9) is exactly (8) composed with the image of the adjoint of \(\beta\). The symmetry of tensor products and the adjunction identities imply compatibility with biduality. For composable maps, the cup maps compose and \[\operatorname{Tr}_{h g} =\operatorname{Tr}_h\circ Rh_!(\operatorname{Tr}_g)\] under the composition identifications. Thus the duality functors and their pairings compose coherently. Notice that the last two arrows of (9) use \(Rg_!\) on \(\omega_X\); compact support of \(\omega_X\) is not required.

For an open inclusion, \(Rj_!=j_!\) and \(j^{-1}j_!\) is the identity. The adjunction \((j_!,j^{-1})\) gives \[\mathop{\mathrm{Hom}}_{\mathsf D(X)}(j_!A,j_!B) \simeq\mathop{\mathrm{Hom}}_{\mathsf D(U)}(A,B).\] Thus extension is fully faithful. The already proved comparison with \(Rj_*\) applies because objects from the test category on \(U\) have compact support in \(U\). ◻

Proof. Dualize and rotate the triangle. Its first arrow is \(\delta\#u\). Symmetry of \(u\) makes its first two terms compatible with \(\delta\eta_L\) and the identity of \(\#L\), with fourth vertical map \(\delta[1](\eta_L)\). Complete these maps to a morphism of triangles with third map \(\psi_0\). The signed transpose of \(\psi_0\) is another such completion. Replacing \(\psi_0\) by \[\frac12\bigl(\psi_0-\delta[1](\#\psi_0)\eta_C\bigr)\] preserves the first two squares and imposes symmetry for the translated biduality \(-\delta\eta\). The resulting map \(\psi\) satisfies (12). It is an isomorphism by the two-out-of-three property for the morphism of triangles. By definition it is the symmetric boundary of \(u\), hence neutral. Equivalently, rotate the displayed triangle to start with \(\#L\to C\); its third object is \(L[1]\simeq[1]\#(\#L)\), and (12) gives the triangular lagrangian. All these operations take place in the given full triangulated category; averaging uses that \(2\) is invertible. Finally \([1]\mathcal D_{X,r+1}=\mathcal D_{X,r}\) with exactly the translated signs, proving the degree assertion. ◻

Proof. First suppose \(A=Rf_*\underline{\mathbb R}_{\Delta}\). For \(B=H\) the proper external-image formula gives \[H\boxtimes A \simeq R(1\times f)_*\underline{\mathbb R}_{I\times\Delta}.\] The source is a compact finite polyhedron. For \(B=E_t\) the product is the generator on the endpoint copy of \(\Delta\). The triangle (14) proves membership for \(B=J\) as well. Exactness and preservation of summands in each variable prove the membership statement for all \(B,A\). All source maps in this argument are semialgebraic when \(f\) is.

We describe the natural map in (15) before proving that it is invertible. For locally compact spaces \(Z,X\) there is a product trace map \[ t_{Z,X}:\omega_Z\boxtimes\omega_X\longrightarrow\omega_{Z\times X}. \tag{16}\] By the adjunction for \(a_{Z\times X}\), this is the map corresponding, under compact-support Künneth, to the tensor of the two traces to \(\mathbb R\). To define \(\mu_{s,r}\), pair \[(\mathcal D_{Z,s}B\boxtimes\mathcal D_{X,r}A) \otimes(B\boxtimes A)\] by moving the second dual factor past \(B\), evaluating the two pairs, combining their output shifts, and applying \(t_{Z,X}[-s-r]\). Take its tensor–Hom adjoint. This construction proves naturality in both variables and specifies the actual comparison, rather than merely an abstract isomorphism of its objects.

The construction is compatible with proper pushforward. We verify the two ingredients of this assertion. First, for a proper \(f:Z'\to Z\), the two composites from \(R(f\times1)_!(\omega_{Z'}\boxtimes\omega_X)\) to \(\omega_{Z\times X}\) agree: one uses \(\operatorname{Tr}_f\) and then the product trace; the other uses the source product trace and then \(\operatorname{Tr}_{f\times1}\). Under adjunction to the point and Künneth, both are \[\operatorname{Tr}_{a_{Z'}}\otimes\operatorname{Tr}_{a_X}.\] Their equality therefore follows from adjunction, including its composition identities. Second, the cup maps used to push evaluation in (9) commute with external products. Indeed the external-image isomorphism is obtained from proper base change and the projection formula, and after inverse-image adjunction its cup map is the tensor of the two counits. Tensoring these counits before or after evaluating gives the same map. These two equalities show that the natural \(\mu\) is carried to the natural \(\mu\) by proper-image duality. They also show its compatibility with biduality. The assertion for shifts follows by inserting the tensor shift comparisons, not by changing any trace.

It remains to prove invertibility on the objects in the lemma. First take \(s=r=0\); the specified shift comparisons give the other degrees. On \(I\times\Delta\), for \(B=H\) and \(A=\underline{\mathbb R}_{\Delta}\), (6) identifies both sides with the constant sheaf on \((0,1)\times\Delta^\circ\) extended by zero and shifted by \(1+\dim\Delta\). This calculation applies to the product polyhedron with its boundary, using its product orientation. The map on the interior is the product of the orientation evaluations and is an isomorphism; both sides vanish on the boundary. For \(B=E_t\), closed direct-image duality reduces the map to the identity duality comparison on \(\Delta\). Thus it is an isomorphism for \(H,E_0,E_1\). Triangles and summands prove the assertion for every \(B\in\mathcal C_I\) and for their shifts.

For \(A=Rf_*\underline{\mathbb R}_{\Delta}\), the external-image isomorphism and proper duality identify (15) with the proper image of the comparison just verified on \(I\times\Delta\). The preceding naturality check is what identifies it with that map. It is consequently an isomorphism even for an arbitrary continuous \(f\). Finally, for fixed \(B\) the locus of \(A\) for which this natural comparison is an isomorphism is closed under triangles, shifts, and summands. 4 completes the proof.

For the last assertion, write the product pairing as the tensor of its two pairings, preceded by the middle-factor permutation and followed by (16). Naturality of the Koszul symmetry shows that interchanging its two arguments is the same as interchanging the arguments of each factor pairing. Both factor pairings are symmetric, so the product is symmetric. ◻

Proof. We first determine the interval boundary itself. Compact-support cohomology of (14) is the exact sequence \[ 0\longrightarrow\mathbb R \xrightarrow{x\mapsto(x,x)}\mathbb R^2 \xrightarrow{(x_0,x_1)\mapsto x_1-x_0}\mathbb R \longrightarrow0, \tag{18}\] where the last identification uses the chosen orientation of \(I\). For example, it follows by taking the cellular cochains of the pair \((I,\{0,1\})\): the coboundary of endpoint values is their difference. Proper duality to the point therefore identifies the shifted dual of \(\partial\) with a map \(H\to E\) having opposite nonzero coefficients at the two ends. This determines the sheaf morphism, since \[\mathop{\mathrm{Hom}}_{\mathsf D(I)}(H,E_t) \simeq\mathop{\mathrm{Hom}}_{\mathsf D(\mathrm{pt})}(\mathbb R,\mathbb R)=\mathbb R\] by closed direct-image adjunction, and taking global sections detects these two coefficients. The cone identity (12) consequently gives opposite endpoint pairings for the symmetric cone of \(u\).

Now use the symmetric morphism supplied by the product of \(u\) and \(\alpha\): \[ J\boxtimes A \xrightarrow{u\boxtimes\alpha} H\boxtimes\mathcal D_{X,r}A \xrightarrow{\mu_{1,r}} \mathcal D_{I\times X,r+1}(J\boxtimes A). \tag{19}\] It lies in the required category by 9, and it is symmetric by that lemma. Identifying its middle object with \(H\boxtimes A\) using \(\alpha\), its cone triangle is \[J\boxtimes A\longrightarrow H\boxtimes A \xrightarrow{v\boxtimes1}E\boxtimes A \xrightarrow{\partial\boxtimes1}(J\boxtimes A)[1].\] 8 supplies a neutral nonsingular form \(\psi\) of degree \(r\) on \(E\boxtimes A\).

The two slices are disjoint closed subsets. Closed direct-image adjunction thus makes every morphism between their supported complexes zero, in every degree. In particular \(\psi\) is diagonal. On either diagonal its precomposition with \(v\boxtimes1\) determines it: restriction of \(H\boxtimes A\) to that slice is \(A\), and the restriction of \(v\boxtimes1\) is the identity. Apply (12) to compute this precomposition. By the natural product comparison, the shifted dual of \(\partial\boxtimes1\) is the product of the shifted dual of \(\partial\) with the identity of \(\mathcal D_{X,r}A\), up to a single shift sign independent of the endpoint, as calculated in 45. After precomposing with \(1\boxtimes\alpha\), the two coefficients are therefore a common nonzero scalar times \(\alpha\) and its negative, as asserted. 7 fixes the absolute normalization of this scalar. ◻

Proof. For a form \((A,\alpha)\), 10 gives \[[\,(i_0)_*A,c(i_0)_*\alpha\,] +[\,(i_1)_*A,-c(i_1)_*\alpha\,]=0\] in the degree-\(r\) Witt group of the cylinder. Multiplying a pairing by a positive real scalar does not change its isometry class: scalar multiplication of its underlying object by the inverse square root gives an isometry. If \(c<0\), the displayed relation is the negative of the one with \(|c|\). Thus in either case \[(i_0)_*[A,\alpha]=(i_1)_*[A,\alpha].\] Push this equality along \(h\) using 7. Its composite maps are \(h_0\) and \(h_1\), so it proves the desired equality on form classes and hence on the Witt group. Properness of the whole homotopy is unnecessary: every object in the relation has support in \(I\times K\) for a compact \(K\subset X\). The proof for \(\mathcal P\) uses exactly the same relation, whose compact-source maps and homotopy images are then semialgebraic. Apply the endpoint equality to the two compositions of a homotopy equivalence and its inverse to obtain the final assertions. ◻

Proof. First take \(A=Rf_*\underline{\mathbb R}_{\Delta}\), with \(\Delta\) a compact simplex. The compact set \(F=f^{-1}K\) lies in the relatively open set \(f^{-1}O\). Finite subdivision gives a relatively open polyhedral neighborhood \(E\) of \(F\) such that \[C=\overline E\subset f^{-1}O, \qquad C\text{ and }C\setminus E\text{ are compact finite polyhedra}.\] For completeness, choose the subdivision so that the closed stars of vertices of simplices meeting \(F\) all lie in \(f^{-1}O\); the positive distance from \(F\) to the complement and a sufficiently small mesh ensure this. Assign value zero to those vertices and value one to all other vertices, and extend linearly to a function \(h\). It is zero on \(F\) and one outside \(f^{-1}O\). Take \(E=\{h<1/2\}\) and \(C=\{h\leq1/2\}\), subdividing along the level set. Because all vertex values are zero or one, \(C=\overline E\). If \(F\) is empty, take \(E=C=\varnothing\).

For \(e:E\hookrightarrow\Delta\), the natural map \(e_!\underline{\mathbb R}_{E}\to\underline{\mathbb R}_{\Delta}\) induces \[s:A'=Rf_*e_!\underline{\mathbb R}_{E}\longrightarrow A.\] On \(C\), the open–closed triangle is \[e_{C!}\underline{\mathbb R}_{E}\longrightarrow\underline{\mathbb R}_{C} \longrightarrow i_*\underline{\mathbb R}_{C\setminus E}\longrightarrow e_{C!}\underline{\mathbb R}_{E}[1],\] where \(e_C:E\hookrightarrow C\) and \(i:C\setminus E\hookrightarrow C\). Push this triangle along the restrictions of \(f\). Both compact sources map into \(O\), and therefore \(A'\) is extended from \(\mathcal B^0(O)\), with support bounded by the compact set \(f(C)\subset O\). In the semialgebraic case these source subsets and restricted maps are still semialgebraic. Moreover, \[N=X\setminus f(\Delta\setminus E)\] is open and contains \(K\). Its inverse image lies in \(E\), so open restriction of proper direct image identifies \(s|_N\) with an isomorphism.

The assertion is preserved by shifts and finite sums. We check closure under cones, including the compatibility of the morphisms. Suppose \(a:A\to B\) is a morphism between objects for which the assertion holds. First choose a cutoff \(b:B'\to B\) over \(O\), invertible on an open \(N\supset K\). Cut off \(A\) over the smaller open set \(O\cap N\), obtaining \(c:A'\to A\). Since \(A'\) is supported in a compact subset of \(N\), for \(j:N\hookrightarrow X\) we have \(A'\simeq j_!j^*A'\). The adjunction \[\mathop{\mathrm{Hom}}_{\mathsf D(X)}(A',B') \simeq \mathop{\mathrm{Hom}}_{\mathsf D(N)}(A'|_N,B'|_N)\] therefore lifts \((b|_N)^{-1}(a|_N)(c|_N)\) to a morphism \(a':A'\to B'\). The same adjunction with target \(B\) proves the equality \(ba'=ac\) on \(X\). Complete this commutative square to a morphism of distinguished triangles. It gives \[\mathop{\mathrm{Cone}}(a')\longrightarrow\mathop{\mathrm{Cone}}(a)\] which is invertible on the intersection of the neighborhoods where \(b\) and \(c\) are invertible. Full faithfulness of open extension identifies \(a'\) with a morphism in \(\mathcal B^0(O)\), so its cone belongs to that category. The union of the compact support bounds for \(A'\) and \(B'\) is a compact support bound inside \(O\) for this cone. The finite triangulated generation of \(\mathcal B^0(X)\) now proves the assertion. ◻

Proof. An extended object has such a support bound by 7. Conversely, let \(S\in\mathcal B(X)\) vanish off a compact set \(C\subset U\). Write it as an actual summand \[S\xrightarrow{i}A\xrightarrow{r}S, \qquad ri=\mathrm{id}_S,\qquad A\in\mathcal B^0(X).\] By 12, choose \(s:A'\to A\) over \(U\) which is invertible on a neighborhood \(N\) of \(C\). Since \(S\) is extended by zero from \(N\), open-extension adjunction lifts \(i\) to \(i':S\to A'\) and gives \(si'=i\) globally. Consequently \((rs)i'=\mathrm{id}_S\). Restricting this splitting to \(U\) makes \(S|_U\) a summand of the object \(A'|_U\in\mathcal B^0(U)\). Thus \(S|_U\in\mathcal B(U)\), and its extension is \(S\).

Finite unions of compact subsets of \(U\) are compact in \(U\). The long cohomology sequence of a triangle and the stalkwise splitting of a summand show that this support condition is thick. If \(S\) vanishes off \(C\), open restriction of Verdier duality shows that \(\mathcal D_X S\) also vanishes off \(C\). Duality stability follows from 5. ◻

Proof. An object of \(\mathcal B(U)\) vanishes outside a compact \(C\subset U\), and therefore vanishes on the neighborhood \(X\setminus C\) of \(K\). Consequently a morphism with cone in \(\mathcal B(U)\) becomes invertible in germs. Restriction thus factors through the Verdier quotient.

We prove fullness first for \(A\in\mathcal B^0(X)\). Represent a germ by \(v:A|_N\to B|_N\) on \(N\supset K\). Apply 12 with \(O=N\) to obtain \(s:A'\to A\), where \(A'\) is compactly supported inside \(N\) and \(s\) is invertible near \(K\). Adjunction extends \(v(s|_N)\) to a morphism \(t:A'\to B\) on \(X\). The cone of \(s\) has a compact support bound \(C\) and vanishes on some open \(M\supset K\). It is therefore supported in the compact subset \(C\setminus M\subset U\); 13 puts it in \(\mathcal B(U)\). The roof \[A\xleftarrow{s}A'\xrightarrow{t}B\] defines a quotient morphism with germ \(v\). For a general \(A\in\mathcal B(X)\), choose \(A\xrightarrow{i}A_0\xrightarrow{p}A\) with \(pi=\mathrm{id}_A\) and \(A_0\in\mathcal B^0(X)\). Lift the germ \(v(p|_N)\) by the case just proved, then compose its lift with \(i\). This proves fullness for arbitrary summand sources and arbitrary targets.

For faithfulness, first let \(t:A\to B\) be an actual morphism whose germ is zero. If \(A\in\mathcal B^0(X)\), take a neighborhood \(N\supset K\) on which \(t\) is zero and a cutoff \(s:A'\to A\) supported inside \(N\). Open-extension adjunction gives \(ts=0\) globally. Since \(s\) becomes invertible in the quotient, \(t\) becomes zero there. For a summand source as above, apply this assertion to \(tp:A_0\to B\) and use \(t=(tp)i\). Finally, every quotient morphism is represented by a roof \(A\xleftarrow{s}C\xrightarrow{t}B\) with cone of \(s\) in \(\mathcal B(U)\). Its denominator is invertible near \(K\). If the roof has zero germ, the numerator \(t\) has zero germ and hence is zero in the quotient by the actual-morphism case. This proves faithfulness for all roofs.

The functor used throughout was restriction. Its compatibility with composition, triangles, and open Verdier duality proves the remaining statements; the cutoff choices were used only to prove that this canonical functor is full and faithful. ◻

Proof. We apply (Balmer and Walter 2002, Theorem 2.1, pp. 132–134). Let \(\mathcal J=\mathcal B(U)\) and \(\mathcal K=\mathcal B(X)\). By 13, \(\mathcal J\) is a full thick subcategory stable under the exact duality. It is therefore a saturated full triangulated subcategory in the terminology of that theorem.

These categories are essentially small: compact finite polyhedra and their maps to a fixed target have a set of representatives; finite triangulated constructions use sets of derived morphisms; and adding summands still gives a set of isomorphism classes. We may thus work with small equivalent categories. Morphism groups in \(\mathcal K\) are real vector spaces, so multiplication by \(2\) is bijective. The enriched octahedral axiom \((\mathrm{TR4}^{+})\) holds in derived categories of exact categories and passes to full triangulated subcategories and localizations (Balmer and Walter 2002, sec. 2, p. 133). Hence it holds for \(\mathcal K\subset D^b(\operatorname{Sh}_{\mathbb R}(X))\).

Let \(S\) be the class of morphisms whose cones belong to \(\mathcal J\). The ordinary Verdier quotient \(S^{-1}\mathcal K=\mathcal K/\mathcal J\) has full kernel \(\mathcal J\), and the duality preserves \(S\): dualizing a cone triangle gives a shift of the corresponding dual cone. All the localization hypotheses are now satisfied. Applying the theorem to every shifted duality and rewriting its cohomological indices as \(W_r=W^{-r}\) gives (21).

For the naturality assertion, \(\mathcal P\to\mathcal T\) preserves the support subcategories and the duality. Similarly, if \(g(U)\subset U'\), a compact support bound \(C\subset U\) maps to the compact support bound \(g(C)\subset U'\). The exact duality-preserving pushforward therefore preserves the kernel subcategories and induces a functor of their localizations.

To check the boundary map, use its construction in (Balmer and Walter 2002, sec. 2, p. 133). For the chosen shifted duality, a symmetric form in the quotient is isometric to the image of a symmetric morphism whose cone belongs to the kernel category. Its boundary is the Witt class of a compatible symmetric cone. An exact functor with the stated duality comparisons carries that morphism and its symmetric-cone diagram to such a representative in the target. Both routes around the naturality square can therefore use this same transported diagram. Independence of the representative gives equality of the boundary classes. The remaining maps commute by construction, proving the stated naturality. ◻

Proof. Extension from \(V\) sends \(\mathcal B(U\cap V)\) into \(\mathcal B(U)\), so it induces the asserted exact functor with duality. Put \(K=X\setminus U\); then \(K\subset V\) and \(K=V\setminus(U\cap V)\). Open neighborhoods of \(K\) contained in \(V\) are cofinal among its open neighborhoods in \(X\): intersect any given neighborhood with \(V\). Applying 14 on \(V\) and on \(X\) therefore identifies the morphism groups on the two sides of (22). This proves full faithfulness.

For \(A_0\in\mathcal B^0(X)\), apply 12 with \(O=V\) and this \(K\). Its cutoff \(A'_0\to A_0\) has cone compactly supported inside \(U\), so it becomes an isomorphism in \(\mathcal B(X)/\mathcal B(U)\). The object \(A'_0\) comes from \(\mathcal B^0(V)\). Every \(A\in\mathcal B(X)\) is a summand of such an \(A_0\), proving density. Fullness and exactness show that the replete image is triangulated: a morphism in the image lifts to the source, and its cone is the image of a cone there. It is also duality stable. All functors used are the same open extensions for \(\mathcal P\) and \(\mathcal T\), which proves the final assertion. ◻

Proof. Fix the degree and denote its duality by \(\#\). We first record a consequence of density: \[ y\oplus y[1]\in\mathcal A \qquad(y\in\mathcal C). \tag{23}\] Indeed, write \(c\simeq y\oplus z\) with \(c\in\mathcal A\). If density is expressed by a split inclusion, its cone supplies such a complement by the split-triangle identity. The endomorphism \(0_y\oplus\mathrm{id}_z\) of \(c\) belongs to \(\mathcal A\) by fullness. Its cone is \(y\oplus y[1]\), which proves (23). Applying this to \(y[-1]\) also gives \(y\oplus y[-1]\in\mathcal A\).

For a nonsingular symmetric form \(q=(x,\alpha)\) in \(\mathcal C\), set \[ D(q)=q\perp q\perp H(x[1]), \tag{24}\] where \(H\) denotes the hyperbolic form for \(\#\). Its underlying object is \[x\oplus x\oplus x[1]\oplus\#(x[1]) \simeq (x\oplus x[1])\oplus(x\oplus x[-1]),\] using \(\alpha:x\simeq\#x\) and \(\#(x[1])\simeq(\#x)[-1]\). By (23) it belongs to \(\mathcal A\). Repleteness and fullness make \(D(q)\) a form in \(\mathcal A\). This assignment respects isometries and orthogonal sums: the direct sum of isometries acts on the two copies of \(q\), and the hyperbolic construction respects isomorphisms and direct sums. It remains to check neutral forms.

Suppose \(q\) has a triangular lagrangian \(a:l\to x\). Its lagrangian triangle has the form \[l\xrightarrow{a}x\longrightarrow\#l\longrightarrow l[1],\] with the duality compatibility that defines a neutral form. The object \(z=l\oplus l[1]\) belongs to \(\mathcal A\), so \(H(z)\) is neutral already in \(\mathcal A\). In \(\mathcal C\), the form \(D(q)\perp H(z)\) has lagrangian \[ L=l\oplus l\oplus x[1]\oplus\#l\oplus l[1]. \tag{25}\] Here the first two factors are the lagrangians of the copies of \(q\); \(x[1]\) is the first lagrangian of \(H(x[1])\); and, in \(H(z)\simeq H(l)\perp H(l[1])\), choose the second lagrangian \(\#l\) of \(H(l)\) and the first lagrangian \(l[1]\) of \(H(l[1])\). Orthogonal sum of these lagrangian triangles gives the indicated duality-compatible lagrangian triangle.

We verify \(L\in\mathcal A\) using only fullness and triangles. Both objects of the morphism \[a\oplus0:l\oplus l[1]\longrightarrow x\oplus x[1]\] belong to \(\mathcal A\) by (23). Its cone, and hence the following object, belongs to \(\mathcal A\): \[C=\#l\oplus x[1]\oplus l[2].\] Add \(l\oplus l[1]\in\mathcal A\) and rearrange to obtain \[C\oplus l\oplus l[1] \simeq (\#l\oplus x[1]\oplus l)\oplus(l[1]\oplus l[2]).\] The second displayed summand \(l[1]\oplus l[2]\) is itself in \(\mathcal A\). Its split inclusion into the displayed total object is therefore a morphism in \(\mathcal A\); taking its cone puts \(\#l\oplus x[1]\oplus l\) in \(\mathcal A\). Finally, \[L\simeq(l\oplus l[1])\oplus(\#l\oplus x[1]\oplus l)\] also belongs to \(\mathcal A\). This removal used a split triangle between objects already in \(\mathcal A\).

The underlying object of \(D(q)\perp H(z)\), its lagrangian \(L\), and \(\#L\) now belong to \(\mathcal A\). Fullness supplies all arrows of the lagrangian diagram in \(\mathcal A\); its inherited triangulation and duality make the same diagram a lagrangian diagram there. Thus \(D(q)\perp H(z)\) is neutral in \(\mathcal A\), and since \(H(z)\) is neutral there, \([D(q)]=0\) in \(W_r(\mathcal A)\). The assignment (24) consequently descends to a homomorphism \(d:W_r(\mathcal C)\to W_r(\mathcal A)\).

In \(\mathcal C\), \(H(x[1])\) is neutral, so \(\iota d[q]=2[q]\). If \(q\) starts in \(\mathcal A\), then \(x[1]\) is in \(\mathcal A\) and the same hyperbolic form is neutral there, giving \(d\iota[q]=2[q]\). The first identity annihilates the cokernel of \(\iota\) by \(2\), and the second annihilates its kernel by \(2\). ◻

Proof. Apply 17 to the replete image in 16. ◻

Proof. A generator over a point is the finite dimensional bounded cohomology complex of a compact finite polyhedron. Conversely, the point constant and its shifts generate every bounded complex of finite dimensional real vector spaces: split a complex into its cohomology and a contractible complex. Both point categories are therefore precisely this derived category.

For a degree \(r\) form, the cohomology pairing pairs \(H^i\) perfectly with \(H^{r-i}\). The complex splitting identifies the form with the orthogonal sum of these cohomology pairings. Terms with \(i\ne r-i\) occur in hyperbolic pairs: choose one member as a lagrangian, the other being its shifted dual. If \(r\) is odd there is no remaining term. If \(r=2k\), the remaining form is on \(H^k\), with symmetry sign \((-1)^{k^2}=(-1)^k\) from tensor symmetry. For odd \(k\) it is alternating; a symplectic basis exhibits it as hyperbolic. For even \(k\) it is a real symmetric form. Diagonalization and cancellation of a positive and a negative line give its class as its signature times the positive line.

Signature respects all neutral relations. Indeed, in a lagrangian triangle \(L\to A\to\mathcal D_{\mathrm{pt},r}L\to L[1]\), at the middle degree \(k\) the second map is the adjoint of the first for the middle pairing. Thus exactness identifies the image of \(H^k(L)\) with its own perpendicular in \(H^k(A)\). Such a symmetric form has signature zero. The positive line has signature one, proving the claimed infinite cyclic group. Finally, sheaves and their dualities on the two-point space split by the two open and closed components, giving the last assertion. ◻

Proof. Denote the arrows in the first rows by \(a_i\) and \(b_i\). If \(x\in\ker f_3\), then \(a_3(qx)=0\), so choose \(v\in A_2\) with \(a_2v=qx\). Since \(b_2f_2v=0\), choose \(w\in B_1\) with \(b_1w=f_2v\). Lift \(qw\) to \(z\in A_1\), so that \(f_1z=qw\). Now \(qv-a_1z\in\ker f_2\), hence \(q(qv-a_1z)=0\). Applying \(a_2\) gives \(q^3x=0\).

For \(y\in B_3\), choose \(v\in A_4\) with \(f_4v=q b_3y\). Then \(a_4v\in\ker f_5\), so choose \(w\in A_3\) with \(a_3w=qv\). The element \(q^2y-f_3w\) lies in \(\ker b_3\); write it as \(b_2z\) with \(z\in B_2\). Choose \(t\in A_2\) with \(f_2t=qz\). The equality \[f_3(qw+a_2t)=qf_3w+qb_2z=q^3y\] proves the cokernel assertion.

In the square, if \(g(x')=0\), lift \(2x'\) to \(x\in A\). Then \(jf(x)=0\), so \(f(2x)=0\) and \(2qx=0\). Applying \(i\) gives \(4qx'=0\). For \(y'\in B'\), lift \(2y'\) to \(y\in B\) and then lift \(qy\) to \(x\in A\). This gives \(2qy'=g(ix)\), which even annihilates the cokernel by \(2q\). ◻

Proof. The assertion for \(j=0\) follows from 19, with \(e_0=0\). Suppose it holds for \(j-1\). Put \(X=S^j\), let \(U,V\) be the complements of two distinct poles, and put \(I=U\cap V\). The spaces \(U,V\) are semialgebraically contractible, and \(I\) is semialgebraically homotopy equivalent to \(S^{j-1}\) (for \(j=1\) this has two components). By 11, the comparisons on \(U,V\) are isomorphisms and those on \(I\) have the induction bound in every degree. These homotopies need not be proper: the pushforward functors of 7 use compact supports.

For \(\mathcal B=\mathcal P\) or \(\mathcal T\) write temporarily \[Q_r^{\mathcal B}(Z,A)=W_r\bigl(\mathcal B(Z)/\mathcal B(A)\bigr)\] when \(A\) is open in \(Z\). Naturality of 15 gives two exact rows of the form \[W_r(\mathcal B(I))\longrightarrow W_r(\mathcal B(V)) \longrightarrow Q_r^{\mathcal B}(V,I) \longrightarrow W_{r-1}(\mathcal B(I)) \longrightarrow W_{r-1}(\mathcal B(V)).\] Their four outer comparisons have the uniform induction bound. The first chase in 20 therefore bounds the comparison on \(Q_r(V,I)\) in all degrees.

By [prop:open-cover,thm:dense-doubling], the two horizontal maps in the natural square \[\begin{array}{ccc} Q_r^{\mathcal P}(V,I)&\longrightarrow&Q_r^{\mathcal P}(X,U)\\ \downarrow&&\downarrow\\ Q_r^{\mathcal T}(V,I)&\longrightarrow&Q_r^{\mathcal T}(X,U) \end{array}\] have kernels and cokernels annihilated by \(2\). The square chase therefore gives a uniform finite exponent for the right-hand comparison. Finally use the two localization rows \[Q_{r+1}^{\mathcal B}(X,U)\longrightarrow W_r(\mathcal B(U)) \longrightarrow W_r(\mathcal B(X)) \longrightarrow Q_r^{\mathcal B}(X,U) \longrightarrow W_{r-1}(\mathcal B(U)).\] A further five-term chase gives the desired bound on the middle comparison. Each step used only a finite exact diagram, and the same bounds work for every \(r\). ◻

Proof. Write \(\widetilde G_r(Z)=\ker(G_r(Z)\to G_r(\mathrm{pt}))\) for the kernel of projection to a point. A point inclusion splits this projection whenever \(Z\) is nonempty. Write \(\mathcal B\) for the category giving \(G\), and use the notation \(Q_r^{\mathcal B}\) from the preceding proof. For \(j\ge1\), use its pole cover \(X=U\cup V\) and \(I=U\cap V\). Since \(G_r(U)\to G_r(\mathrm{pt})\) is an isomorphism, the map \(G_r(U)\to G_r(X)\) is split injective in every degree. Localization therefore identifies \[Q_r^{\mathcal B}(X,U)\simeq\widetilde G_r(X).\] More explicitly, its kernel in \(G_r(X)\) is exactly the point summand, and its next boundary is zero because \(G_{r-1}(U)\to G_{r-1}(X)\) is injective.

Likewise \(G_r(I)\to G_r(V)\) is split surjective: project to the point, use contractibility of \(V\), and choose a point of \(I\) as a section. The other localization sequence consequently identifies its boundary as an isomorphism \[Q_r^{\mathcal B}(V,I)\simeq\widetilde G_{r-1}(I).\] The dense map between these two relative groups is an isomorphism after tensoring with \(\mathbb Q\) by 17. Thus \[\widetilde G_r(S^j)\otimes\mathbb Q \simeq\widetilde G_{r-1}(S^{j-1})\otimes\mathbb Q.\] For \(S^0\) the reduced group is the kernel of addition of two point groups, hence one point group. Induction and the split point summand prove the formula. At \((r,j)=(d,d)\) the two summands are \(0\) and \(\mathbb Q\), respectively, because \(d\) is odd and 19 gives \(G_d(\mathrm{pt})=0\) and \(G_0(\mathrm{pt})=\mathbb Z\). ◻

Proof. For a continuous semialgebraic map from a compact polyhedron, Hardt’s semialgebraic triviality theorem gives a finite semialgebraic partition of the target over whose pieces the map is trivial (Hardt 1980, main theorem); see also the precise finite-partition formulation in (Coste 2000, Theorem 4.1). Refining this partition into cells, the union of the top dimensional cells is open and dense in the sphere. Each fiber is a compact semialgebraic set, so has the finite cohomology of a finite triangulation; these semialgebraic stratification and triangulation facts are recalled in (Hardt et al. 2011, sec. 2.1, pp. 2481–2483). Proper base change and the trivializations show that a generator has finite dimensional locally constant cohomology on that open set.

This property passes to the finite constructions defining \(\mathcal P\). Shifts and summands cause no difficulty. For a cone, the cohomology long exact sequence expresses each cohomology sheaf as an extension of a kernel and a cokernel of morphisms of finite local systems. Those kernels and cokernels are local systems, and such extensions are locally constant: on a small ball, positive extension groups between finite constant sheaves vanish. Indeed, for finite vector spaces \(V,W\) and a ball \(B\), \[\operatorname{Ext}^k_B(\underline{V}_{B},\underline{W}_{B}) =H^k\bigl(B;\underline{\mathop{\mathrm{Hom}}_{\mathbb R}(V,W)}_{B}\bigr)=0\qquad(k>0).\] Taking the finite intersection of the dense open subsets needed for a finite list of generators proves the first assertion for every finite list of objects. No condition is imposed on the derived morphisms in their constructions.

Choose a ball \(B\) in that common open subset. Every local system in question is constant on \(B\), and the same extension vanishing splits the successive truncation triangles of each bounded complex. Hence its restriction is isomorphic to the constant complex associated to its finite dimensional cohomology complex. In particular the constant-complex functor identifies the full subcategory of bounded complexes on \(B\) with finite locally constant cohomology with the bounded derived category of finite dimensional vector spaces. The orientation identification \(\omega_B\simeq\underline{\mathbb R}_{B}[d]\) identifies \(\mathcal D_{B,d}\) on this subcategory with ordinary derived linear duality. This identification is compatible with evaluation and biduality: on a finite constant complex it is the natural map \[\underline{\operatorname{RHom}_{\mathbb R}(C,\mathbb R)}_{B} \simeq \operatorname{R\mathcal Hom}_B(\underline{C}_{B},\underline{\mathbb R}_{B}),\] and the assertion passes through its finite shifts and sums. The orientation form of the constant line becomes the positive form \(xy\). This degree-zero local description holds in both odd congruence classes of \(d\) modulo four.

A nonsingular degree-\(d\) form on \(A\) thus gives an ordinary nonsingular symmetric form \(b\) on \(H^0(A)_x\), for \(x\in B\). Its signature is an integer, additive under orthogonal sums and invariant under isometry. To check neutrality, include a lagrangian triangle in the finite list before choosing \(B\). Writing \(\ell:H^0(L)_x\to H^0(A)_x\), its middle exact sequence is \[H^0(L)_x\xrightarrow{\ell}H^0(A)_x \xrightarrow{\ell^*b}\bigl(H^0(L)_x\bigr)^*.\] The last map can have the harmless common sign of the triangle convention. Exactness gives \(\mathop{\mathrm{im}}\ell=\ker(\ell^*b)=(\mathop{\mathrm{im}}\ell)^\perp\). This is an ordinary lagrangian; the middle signature is zero. Since each equality in a Witt group is witnessed by a finite combination of isometries, orthogonal sums, and neutral relations, this proves the assertion about any fixed relation.

The class \(u_{F,d}\) has signature one on every such oriented ball. If a nonzero multiple of it vanished, a finite relation witnessing that equality would contradict its signature. Thus \(u_{F,d}\) is nontorsion and spans the rational line of 23. For \(x\in F_d(S^d)\) write \(x\otimes1=(a/b)\overline{u}_{F,d}\), where \(a\in\mathbb Z\) and \(b>0\). The difference \(bx-au_{F,d}\) is torsion, so for some integer \(\nu>0\) there is an actual equality \(\nu bx=\nu au_{F,d}\) in the Witt group. Choose a common ball for this finite equality and put \(\sigma\) for the signature of a representative of \(x\) there. It gives \(\nu b\sigma=\nu a\), hence \(a/b=\sigma\in\mathbb Z\). All integral multiples of \(u_{F,d}\) occur, proving the asserted equality of images. ◻

Proof. By 21 the comparison is a rational isomorphism, so 24 identifies its nonzero rational orientation class with \(\overline{u}_d\). Take \(L_d=2^{e_d}\), which annihilates the comparison cokernel in degree \(d\). For every \(x\in E_d(S^d)\) there is \(y\in F_d(S^d)\) with \(L_dx\) equal to its image. The rational image of \(y\) is an integral multiple of \(\overline{u}_{F,d}\) by 24, so \(x\otimes1\in L_d^{-1}\mathbb Z\overline{u}_d\). ◻

Proof. Choose a finite polyhedral model for the source and a finite semialgebraic triangulation of the target sphere. The latter identifies its polyhedral model with \(S^d\) semialgebraically. On these models choose a simplicial approximation \(g\) to \(f\) after subdivision, as in (Hatcher 2002, Theorem 2C.1, pp. 177–179). The models and \(g\) are semialgebraic. Orient each model by its identifying homeomorphism; these identifications preserve the fundamental classes and the orientation forms used for degree and pushforward. The homotopy need only be continuous: 11 identifies \(f_*\) and \(g_*\) on the continuous theory, with the model identifications understood. The constant sheaf on the source model is an identity-map generator, so its orientation form belongs to the semialgebraic category.

Fix a top dimensional target simplex. In its interior choose a small ball \(B\) avoiding the images of lower dimensional source simplices and of collapsed top dimensional simplices. Each source simplex mapping onto this target simplex contributes one open sheet over \(B\), mapped homeomorphically onto \(B\). These are all preimages of \(B\), and there are finitely many. The restricted map is proper, as the base change of the map from the compact source. On \(B\) its direct image is the constant vector space with one basis vector for each sheet.

The pushed pairing is diagonal in this basis. Products between different sheets vanish; on one sheet the trace of the orientation pairing is \(+1\) or \(-1\) according as that homeomorphism preserves or reverses the chosen orientations. To see the sign directly, the trace for a homeomorphism identifies its source dualizing complex with the target dualizing complex; relative to the two orientation trivializations, the induced map on local top homology is exactly its orientation sign. Proper duality and its trace therefore give the same sign for the constant-line pairing. Its signature on \(B\) is the sum of these local signs. For a sphere source this is the degree \(s\) by the local-degree formula (Hatcher 2002, Proposition 2.30, pp. 135–136). For the stated general source, fix \(y\in B\) and write \(g^{-1}(y)=\{x_1,\ldots,x_t\}\). Excision identifies \[H_d(N,N\setminus g^{-1}(y);\mathbb Z) \simeq\bigoplus_{i=1}^t H_d(N,N\setminus\{x_i\};\mathbb Z).\] The fundamental class \([N]\) maps to the tuple of positive local orientation generators. The relative map induced by \(g\) sends the \(i\)th generator to \(\epsilon_i\) times the orientation generator of \(H_d(S^d,S^d\setminus\{y\};\mathbb Z)\), where \(\epsilon_i\) is the orientation sign of that sheet. Naturality of the absolute-to-relative maps, together with \(g_*[N]=s[S^d]\), gives \(s=\sum_i\epsilon_i\). This also covers an empty fiber, with sum zero. Only interiors of top dimensional simplices occur; these are open balls, so no PL manifold structure is required of the triangulation.

This local calculation determines the rational coefficient without assuming a globally defined generic signature. In fact, if the rational image of \(g_*u_{F,d}\) (or of the pushed class from \(N\)) is \(\rho\overline{u}_{F,d}\), clear denominators and torsion to obtain a finite Witt relation. Choose the signature ball for that relation inside the ball of sheets just used; this is possible since the relation’s common open set is dense. Its signature says \(\rho=s\). Apply the comparison to obtain the assertion in \(E\). Finally, \(\overline{u}_d\) spans the rational group by 25, so the assertion about the entire group follows by linearity. ◻

Proof. Homotopy invariance and 19 give \(E_d(U)=E_d(\mathrm{pt})=0\). Let \(i:U\hookrightarrow X\) and let \(p_Z:Z\to\mathrm{pt}\) denote projection. The composite \[E_{d-1}(U)\xrightarrow{i_*}E_{d-1}(X) \xrightarrow{(p_X)_*}E_{d-1}(\mathrm{pt})\] is \((p_U)_*\), an isomorphism. Thus \(i_*\) is split injective, with retraction \(((p_U)_*)^{-1}(p_X)_*\). If \(d\equiv1\pmod4\), the point group here is \(\mathbb Z\); if \(d\equiv3\pmod4\), it is zero. The split-injectivity argument applies in either case. All these maps exist by compact-support pushforward, whether or not the space maps are proper. The relevant part of 15 is \[0\longrightarrow E_d(X) \longrightarrow W_d\bigl(\mathcal T(X)/\mathcal T(U)\bigr) \longrightarrow E_{d-1}(U)\xrightarrow{i_*}E_{d-1}(X).\] Exactness and the last injectivity give the asserted isomorphism over \(\mathbb Z\). The functor \(g_*\) preserves the support subcategory because \(g(U)\subset U'\), so naturality follows from the natural localization sequence. No rationalization or splitting of quotient idempotents is involved. ◻

Proof. We use Newman’s theorem in the following form: a finite-order homeomorphism of a connected topological manifold which is the identity on a nonempty open subset is the identity everywhere (Newman 1931); for a modern statement and proof, see (Pardon 2019, author manuscript, pp. 2–3). The local reduction using this theorem also appears in (Pardon 2013, sec. 4.1).

There is a point \(x\in M\) such that no open subgroup of \(\mathbb Z_p\) fixes a neighborhood of \(x\) pointwise. To prove this, suppose otherwise, and write \[H_j=p^j\mathbb Z_p, \qquad F_j=\operatorname{int}_M\operatorname{Fix}(H_j).\] Some \(F_j\) is nonempty. We show that it is closed. If \(x\in\overline{F_j}\), then \(H_j\) fixes \(x\), since its fixed set is closed. By the supposition, some neighborhood \(N\) of \(x\) is fixed pointwise by \(H_\ell\). Compactness of \(H_j\) and continuity at \(H_j\times\{x\}\) give a coordinate ball \(B\ni x\) so small that \(H_jB\subset N\). The saturation \(N'=H_jB\) is open and connected: every translate of \(B\) contains \(x\). It is \(H_j\)-invariant, and the restricted action factors through the finite group \(H_j/(H_j\cap H_\ell)\). Each of its homeomorphisms is therefore periodic and fixes the nonempty open subset \(N'\cap F_j\) pointwise. Newman’s theorem makes each the identity on \(N'\). Thus \(x\in F_j\), proving closedness. Connectedness gives \(F_j=M\), contrary to faithfulness.

Choose such an \(x\), a chart \(C\ni x\), and a coordinate ball \(B\ni x\) with \(\overline B\) compactly contained in \(C\). Joint continuity and compactness of \(\overline B\) give an open subgroup \(H\) such that \(H\overline B\subset C\). Shrinking \(H\) further gives \(Hx\subset B\). Then \[O=HB=\bigcup_{g\in H}gB\subset C\] is open and invariant, and is connected because every \(gB\) meets \(B\) at \(gx\). Its restriction kernel in \(H\) is closed. A nonzero closed subgroup of \(\mathbb Z_p\) is \(p^j\mathbb Z_p\) for some \(j\): choose an element of smallest finite \(p\)-adic valuation and use the density of its integer multiples in that subgroup. Thus a nonzero restriction kernel would be an open subgroup fixing the neighborhood \(O\) of \(x\), which is impossible.

The chart orients \(O\). Every homeomorphism of connected \(O\) either preserves or reverses this orientation, so every square preserves it. Replace \(H\) by \(G=2H\). This is an open subgroup isomorphic to \(\mathbb Z_p\) and its restricted action remains faithful. For odd \(p\) this replacement leaves \(H\) unchanged; for \(p=2\) it has index two. ◻

Proof. The inverse image of an open subset of \(W\) is open in \(S^d\). A basic neighborhood of \(\infty\) is \(W^+\setminus K\), where \(K\subset W\) is compact; its inverse image is the open set \(S^d\setminus K\). This proves continuity of \(c\) for an arbitrary open \(W\).

For compact \(K\subset W\), its saturation \(GK\) is compact, being the image of \(G\times K\). Every \(g\in G\) maps \(W^+\setminus GK\) into \(W^+\setminus K\). This gives joint continuity at all \((g,\infty)\), while continuity on \(G\times W\) was already established.

The space \(W\) is locally compact, noncompact, and second countable; its one-point compactification is compact metrizable. If \(d_0\) is a compatible metric on \(W^+\), then \[d_1(x,y)=\max_{g\in G}d_0(gx,gy)\] is a compatible invariant metric, by uniform continuity on the compact space \(G\times W^+\). For a closed subgroup \(H\subset G\), the metric \[d_H(Hx,Hy)=\min_{h\in H}d_1(x,hy)\] induces the quotient topology. Its positivity on distinct orbits follows from compactness, and the triangle inequality follows from invariance. This proves the last assertion. ◻

Proof. Fix \(\xi\in S^d\) and a map \(P:\mathbb R^d\to S^d\) which is constant with value \(\xi\) outside the unit ball and induces an orientation-preserving homeomorphism \(\overline B^d/\partial B^d\to S^d\). Choose \(z\in W\) and \(r>0\) with \(\overline B(z,r)\subset W\). The map \[P_{z,r}(w)=P((w-z)/r),\qquad P_{z,r}(\infty)=\xi,\] is continuous on \(W^+\) and \(P_{z,r}c\) has degree one.

Regard \(S^d\) as the unit sphere in \(\mathbb R^{d+1}\). By continuity and compactness, a sufficiently small open subgroup \(H\subset G\) satisfies \[\sup_{h\in H,\,w\in W^+} \|P_{z,r}(hw)-P_{z,r}(w)\|<\tfrac12.\] Let \(\mu_H\) be Haar probability measure and put \[A(w)=\int_H P_{z,r}(hw)\,d\mu_H(h).\] This is a continuous \(H\)-invariant \(\mathbb R^{d+1}\)-valued map and \(\|A(w)-P_{z,r}(w)\|<1/2\). Thus every vector \((1-t)P_{z,r}(w)+tA(w)\) has norm at least \(1/2\). Normalizing these vectors gives a homotopy from \(P_{z,r}\) to the invariant map \(A/\|A\|\). The homotopy fixes \(\infty\) and is constant with value \(\xi\) outside the compact subset \(H\overline B(z,r)\) of \(W\). Replace \(G\) by \(H\). The normalized map descends to \(a:Y\to S^d\), and the homotopy proves (28). Faithfulness and orientation preservation persist under this replacement. ◻

Proof. Every orbit is homeomorphic to \(G/G_x\) for a closed subgroup \(G_x\). Such a quotient is a compact zero-dimensional space: for \(\mathbb Z_p\) it is either a point, a finite quotient, or \(\mathbb Z_p\) itself. On a compact zero-dimensional space, global sections of sheaves of vector spaces form an exact functor. Indeed, local lifts through a sheaf epimorphism can be chosen on a finite cover and then glued over a finite disjoint clopen refinement. Thus the higher cohomology of each orbit vanishes.

The map \(r\) is closed and separated and has compact fibers. Bounded-below proper base change (The Stacks Project Authors 2026d, Lemma 20.18.1 and Theorem 20.18.2) therefore gives \[(R^jr_*\underline{k}_{E})_y =H^j(r^{-1}(y);\underline{k}_{r^{-1}(y)})=0 \quad(j>0).\] This use of proper base change imposes no covering-dimension bound on \(E\) or \(E/G\).

For an open \(U\subset E/G\) and a locally constant function \(\varphi:r^{-1}(U)\to\mathbb Q\), define \[ \mathcal A_U(\varphi)(r(x)) =\int_G\varphi(gx)\,d\mu_G(g). \tag{29}\] The expression is independent of the representative \(x\). For fixed \(x\), the locally constant function \(g\mapsto\varphi(gx)\) on \(G\) factors through some finite quotient \(G/p^\ell G\), so its average is rational.

It is also locally constant as a function of \(x\). For every \(g\in G\), continuity gives a product neighborhood of \((g,x)\) on which \(\varphi(hx')\) is constant. A finite collection of the group neighborhoods covers \(G\). Intersecting the corresponding neighborhoods of \(x\) shows that \(\varphi(hx')=\varphi(hx)\) for all \(h\in G\) and all \(x'\) in one neighborhood of \(x\). Their averages are equal there. An invariant locally constant function descends to a locally constant function on the orbit space, because \(r\) is an open quotient map. The operators (29) commute with restriction and fix functions pulled back from \(U\). They give the asserted sheaf retraction. Applying derived global sections, and using \(Rr_*\underline{\mathbb Q}_{E}=r_*\underline{\mathbb Q}_{E}\), makes the usual cohomology pullback split injective. The natural Čech–sheaf comparison gives the stated conclusion. ◻

Proof. The case \(K=\varnothing\) is immediate. Choose a finite open cover \(\mathcal U=\{U_i\}\) of \(K\) on each of whose members \(v\) has image of Euclidean diameter less than \(1/4\), and choose \(x_i\in U_i\). For every simplex \(\sigma\) of the nerve \(N\mathcal U\), choose a point in the corresponding intersection. All the vectors \(v(x_i)\) for vertices of \(\sigma\) are within \(1/4\) of its image. Their convex combinations are therefore nonzero. The formula \[b\left(\sum_i t_i i\right) =\frac{\sum_i t_i v(x_i)}{\|\sum_i t_i v(x_i)\|}\] defines a continuous map \(b:|N\mathcal U|\to S^d\). A subordinate partition of unity defines a nerve map \(p_{\mathcal U}:K\to|N\mathcal U|\). Normalized interpolation between \(v(x)\) and the corresponding convex combination gives \(v\simeq bp_{\mathcal U}\).

For finite covers of a compact space, Čech cohomology is the direct limit of the cohomologies of their nerves. The map to the limit from \(H^d(|N\mathcal U|;\mathbb Q)\) is the pullback by \(p_{\mathcal U}\); this follows from the vertex-star cover and is independent of the subordinate partition. Since the image of \(b^*\gamma_d\) is zero, some finite refinement \(\mathcal V\) and its simplicial refinement map \(\rho:|N\mathcal V|\to|N\mathcal U|\) satisfy \[(b\rho)^*\gamma_d=0\quad\hbox{in }H^d(|N\mathcal V|;\mathbb Q).\] Moreover \(p_{\mathcal U}\simeq\rho p_{\mathcal V}\): at each point, the vertices appearing in both barycentric expressions belong to cover members containing that point, so linear interpolation stays in one simplex of \(N\mathcal U\). We have reduced the assertion to the map \(b\rho:P\to S^d\) from the finite complex \(P=|N\mathcal V|\).

We now use that \(d\) is odd and \(d\geq3\). Using based cellular degree maps, consider the mapping telescope \[ \mathcal S=\operatorname{Tel} \bigl(S^d\xrightarrow{D_2}S^d \xrightarrow{D_3}S^d\xrightarrow{D_4}\cdots\bigr). \tag{30}\] We verify its homotopy type directly. Serre’s finiteness theorem gives finite \(\pi_j(S^d)\) for \(j>d\) (Serre 1951, V, §3, Proposition 3). We also use his precise degree-map assertion, valid in every homotopy degree of an odd sphere: if \(q x=0\) in a homotopy group, then postcomposition with a degree-\(2q\) map kills \(x\) (Serre 1953, V, §1, Corollary 2 to Proposition 1). The telescope has CW type, and its homotopy groups are the direct limits of those of its spheres: maps and homotopies from spheres have bounded height, and each finite subtelescope retracts onto its final sphere. Any element of \(\pi_j(S^d)\), \(j>d\), has some finite order \(q\). A sufficiently long subsequent product of the degrees \(2,3,4,\ldots\) is divisible by \(2q\); its sphere map is homotopic to a composite through a degree-\(2q\) map, and therefore kills that element. Thus all these higher direct limits vanish. In degree \(d\) the direct limit is \(\operatorname{colim}(\mathbb Z\xrightarrow{2}\mathbb Z \xrightarrow{3}\mathbb Z\to\cdots)=\mathbb Q\), and in degrees below \(d\) it is zero. Hence \(\mathcal S\) has type \(K(\mathbb Q,d)\). The same odd-sphere telescope and finite-stage argument are used in (Dranishnikov et al. 2003, proof of Proposition 4.5, p. 928).

Let \(j:S^d\to\mathcal S\) include the first sphere. Choose \(\gamma_{\mathcal S}\in H^d(\mathcal S;\mathbb Q)\) with \(j^*\gamma_{\mathcal S}=\gamma_d\). Indeed, \(j_*\) in degree-\(d\) homology is the inclusion \(\mathbb Z\hookrightarrow\mathbb Q\), and universal coefficients identify this restriction map with the isomorphism \(\mathop{\mathrm{Hom}}(\mathbb Q,\mathbb Q)\to\mathop{\mathrm{Hom}}(\mathbb Z,\mathbb Q)\). Then \((jb\rho)^*\gamma_{\mathcal S}=(b\rho)^*\gamma_d=0\). Ordinary Eilenberg–MacLane representability on the finite complex \(P\) makes \(jb\rho\) nullhomotopic. A nullhomotopy has compact image. The continuous height function on the telescope is therefore bounded on that image, so the image lies in a finite subtelescope. Retracting this subtelescope onto its last sphere gives a nullhomotopy of \(D_s b\rho\), where \(D_s\) is the composite of finitely many of the displayed degree maps and \(s>0\) is their product. Composing with \(p_{\mathcal V}\) and using \(v\simeq b\rho p_{\mathcal V}\) proves the assertion. ◻

Proof. Some \(x\in W\) has stabilizer \(G_x\subset p^kG\). Otherwise, the classification of closed subgroups of \(\mathbb Z_p\) would imply \(p^{k-1}G\subset G_x\) for every \(x\in W\), contradicting faithfulness. Set \(H=p^kG\), \(Z=W/H\), and let \(z\) be the image of \(x\). This is a Hausdorff orbit space, and the stabilizer of \(z\) for the finite action of \(G/H\) is \((G_x+H)/H=0\). Write \(\tau\) for the element induced by \(1\in G\).

Choose pairwise disjoint open neighborhoods \(U_j\) of \(\tau^jz\), \(0\le j<m\), and put \(N=\bigcap_{j=0}^{m-1}\tau^{-j}U_j\). Then the translates \(\tau^jN\) are pairwise disjoint. If \(r:Z\to W/G\) is the finite quotient map and \(V=r(N)\), openness of \(r\) shows that \(V\) is nonempty and open, and \[r^{-1}(V)=\coprod_{j=0}^{m-1}\tau^jN.\] Each restriction \(r:\tau^jN\to V\) is an open continuous bijection, hence a homeomorphism. Pulling these sheets back under \(W\to Z\) partitions \(q^{-1}(V)\) into relatively clopen subsets. Give the pullback of \(\tau^jN\) the constant value \(\zeta^j\). The resulting function \(\beta\) is locally constant, and the cyclic permutation of the sheets proves (31). ◻

Proof. Put \(I=[0,1]\). Paste \(H\) with \(f\) on the closed subset \[C=(F_0\times I)\cup(Y_0\times\{0\})\subset Y_0\times I.\] Viewing the resulting map as \(\mathbb R^{d+1}\)-valued, extend each of its \(d+1\) coordinates by the Tietze extension theorem to obtain \(E:Y_0\times I\to\mathbb R^{d+1}\). The open set \(N=\{\|E\|>1/2\}\) contains \(C\), and \(E/\|E\|\) extends the pasted sphere map to \(N\). Compactness of \(I\) gives an open \(U\supset F_0\) with \(U\times I\subset N\); explicitly, take the complement of the projection of the compact set \((Y_0\times I)\setminus N\). Choose a continuous \(\lambda:Y_0\to[0,1]\) equal to one on \(F_0\) and zero on \(Y_0\setminus U\). Then \[\widetilde H(y,t) =\frac{E(y,t\lambda(y))}{\|E(y,t\lambda(y))\|}\] is defined everywhere: if \(y\in U\) its argument lies in \(U\times I\), and otherwise its argument is \((y,0)\in C\). It is continuous, starts at \(f\), and agrees with \(H\) on \(F_0\times I\). ◻

Proof. We first show \((a|_F)^*\gamma_d=0\). Choose a round ball with closure in the nonempty open set \(q^{-1}(V)\subset W\). The pinch in 30 can be moved to this ball by a homotopy on \(W^+\). Here is explicit control of that homotopy. Connect the two centers by a path in connected open \(W\), which is path connected. Its compact image has positive distance from \(\mathbb R^d\setminus W\). Choose a sufficiently small fixed radius for transport along the path; at the two ends shrink and enlarge radially as needed. Applying the formula \(P_{z(t),r(t)}\) from 30 gives a continuous family of pinches. All radii are positive and all closed balls lie in one compact subset of \(W\). Consequently the family is also continuous at \(\infty\), where it has constant value \(\xi\).

It follows that \(aq\) is homotopic to a map constant on \(q^{-1}(F)\). Thus the pullback of \((a|_F)^*\gamma_d\) to \(q^{-1}(F)\) is zero. This preimage is a closed invariant compact subset of \(W^+\), and its orbit space is \(F\). The restricted orbit map \(q^{-1}(F)\to F\) therefore meets the hypotheses of 31, so \((a|_F)^*\gamma_d=0\). By 32, some \(D_sa|_F\) is nullhomotopic. Apply 34 to this nullhomotopy and \(f=D_sa\) on \(Y\). Its endpoint is the required \(h\). Finally, \(y_\infty\in F\), and every point of \(F\) maps to \(b\), which gives the asserted preimage inclusion. ◻

Proof. Apply 31 to the compact metrizable \(G\)-space \(W^+\) with \(k=\mathbb R\). The coefficient sheaf \(\underline{\mathbb C}_{W^+}\) is the direct sum of two real constant sheaves, so additivity gives the first isomorphism in (32). This argument imposes no dimension bound on \(Y\).

The stalk of \(\mathcal H\) at an orbit is the complex vector space of locally constant functions on \(G/H\). The pullback of any such function to \(G\) factors through \(G/p^jG\) for some \(j\): a finite clopen partition on which the function is constant admits a refinement by cosets of a common open subgroup. The representation is consequently the union of finite cyclic representations. Each of the latter decomposes into the eigenspaces of \(T\), since \(z^{p^j}-1\) has distinct complex roots. Kernels and sheaf direct sums can be checked on stalks, proving the second isomorphism in (32). This asserts an algebraic sheaf direct sum; a section over a noncompact open set need not have globally finite character support.

We record explicitly the compactness argument needed for (33). If \(K\) is compact Hausdorff and \((\mathcal F_\lambda)\) is a family of sheaves, every section of \(\bigoplus_\lambda\mathcal F_\lambda\) on \(K\) involves only finitely many indices. Indeed it has this property locally by the definition of sheafification, and a finite subcover makes the set of indices finite on \(K\). Thus global sections commute with direct sums. The same argument on every closed subset of \(K\) shows that direct sums of soft sheaves are soft: extend the finitely many component sections separately. Soft resolutions and their acyclicity on compact Hausdorff spaces therefore give \[ H^r\!\left(K,\bigoplus_\lambda\mathcal F_\lambda\right) \simeq\bigoplus_\lambda H^r(K,\mathcal F_\lambda) \qquad(r\geq0). \tag{34}\] The soft-resolution facts used here are the compact case of (Schapira 2023, Propositions 11.2.4, 11.2.5, and 11.2.8).

Apply (34) to \(K=t^{-1}(x)\) and the restrictions of the eigensheaves. Proper base change identifies the stalk cohomology of (33) with these isomorphisms. All the derived images have the common lower bound zero, so their sum remains bounded below. Finally, countable sheaf sums are exact, and termwise sums compute coproducts in the derived category by (The Stacks Project Authors 2026c, Lemma 13.33.5). This proves both assertions. ◻

Proof. Write \(A=A_t|_{X_t^\circ}\) and \(\alpha=\alpha_t|_{X_t^\circ}\), and let \(R^\dagger=\alpha^{-1}\mathcal D_{X_t^\circ,d}(R)\alpha\) denote the adjoint of an endomorphism of \(A\). The orientation-preserving action of \(G\) on the proper manifold map above acts by isometries of \(\alpha\), by naturality of orientation and proper duality. Complex scalar multiplication by a unit also preserves \(B\). In particular, if \(J=M_{\mathrm i}\), then \[J^\dagger=-J,\qquad M_\lambda=(\operatorname{Re}\lambda)\mathrm{id}+ (\operatorname{Im}\lambda)J,\qquad T^\dagger=T^{-1},\qquad M_\lambda^\dagger=M_{\lambda^{-1}} \quad(\lambda\in\Lambda).\]

Fix \(\lambda\notin E\). On \(B_{\{\lambda\}}\) the endomorphism \(T-M_\lambda\) is zero. Let \(R_E\) denote the endomorphism of \(B_E\) induced by \(T^{-1}-M_{\lambda^{-1}}\) on \(\mathcal H_E\). The latter operator is invertible: on the \(\mu\)-summand its inverse is multiplication by \((\mu^{-1}-\lambda^{-1})^{-1}\). These componentwise inverses define one endomorphism of the algebraic sheaf direct sum, and hence induce an inverse \(R_E^{-1}\) on \(B_E\) after \(Rt_*\) and restriction. For the two inclusions \(i_\lambda:B_{\{\lambda\}}\to A\) and \(i_E:B_E\to A\), adjointness yields \[\begin{split} 0 &=\mathcal D_{X_t^\circ,d}(i_E)\alpha(T-M_\lambda)i_\lambda\\ &=\mathcal D_{X_t^\circ,d}(R_E)\, \mathcal D_{X_t^\circ,d}(i_E)\alpha i_\lambda. \end{split}\] The first factor in the last line is invertible, so the cross pairing from \(B_{\{\lambda\}}\) to \(\mathcal D_{X_t^\circ,d}B_E\) vanishes. This applies also when \(\mu=\overline\lambda\ne\lambda\): the common complex scalar operators distinguish these two real subrepresentations.

By 37 and exact open restriction, \(B_D\) is the derived coproduct of the \(B_{\{\lambda\}}\), \(\lambda\in D\). A morphism out of a coproduct is determined by its composites with the coprojections. The cross pairing from \(B_D\) therefore vanishes. Symmetry gives the cross pairing in the opposite order. In a finite decomposition these equalities say precisely that every projector is self-adjoint. ◻

Proof. We work in \(\mathcal Q_U(X)\) and abbreviate \((A_t,\alpha_t)\) to \((A,\alpha)\). On \(A^{\oplus m}\) let \[Q=\operatorname{diag}(2e_0-\mathrm{id},\ldots,2e_{m-1}-\mathrm{id}), \qquad Q_0=\operatorname{diag}(\mathrm{id},-\mathrm{id},\ldots,-\mathrm{id}).\] Let \(P_j\) be the permutation matrix interchanging the zeroth and \(j\)th copies, with \(P_0=\mathrm{id}\), and let \(E_j\) be the diagonal matrix having \(e_j\) in every diagonal entry. Define \[S=\sum_{j=0}^{m-1}E_jP_j.\] All the \(E_j\) commute with the permutation matrices. Using only (37), we have \(S^2=\mathrm{id}\) and \(S^\dagger=S\), where the adjoint is for \(\alpha^{\oplus m}\). Moreover, \[S^\dagger Q_0 S =\sum_{j=0}^{m-1}E_jP_jQ_0P_j =Q.\] For the last identity, its \(i\)th diagonal entry is \(e_i-\sum_{j\ne i}e_j=2e_i-\mathrm{id}\) and all off-diagonal entries are zero. Thus \(S\) is an isometry from the sum of the twisted forms to \((A,\alpha)\perp(A,-\alpha)^{\perp(m-1)}\). The sum of their Witt classes is \((2-m)[A,\alpha]\). Adding the \(m\) untwisted terms in (38) gives \(2[A,\alpha]\). The injectivity of \(\ell_U\) proves (39). ◻

Proof. Choose a contractible open \(U'\ni g(b_t)\), and then a contractible open \(U\ni b_t\) with \(U\subset g^{-1}(U')\). Pushforward takes an object supported in a compact subset of \(U\) to one supported in a compact subset of \(U'\). Consequently [prop:test-pushforward,thm:witt-localization] give a duality-preserving quotient functor \[\overline g_*:\mathcal Q_U(X)\longrightarrow\mathcal Q_{U'}(X')\] and a commuting square on Witt groups with the maps \(\ell_U,\ell_{U'}\). Composition of proper direct images identifies the pushed total form with \((A_{gt},\alpha_{gt})\).

We check the endomorphisms in this identification. Represent a projector \(e_i\) by a quotient roof. Its restriction agrees with the character projector on some open neighborhood \(N\) of \(X\setminus U\). Shrinking \(N\) if necessary, arrange \(N\subset X\setminus\{b_t\}\) and that the denominator of the roof is invertible there. Define \[ N'=X'\setminus g(X\setminus N). \tag{41}\] Since \(X\) is compact, \(g(X\setminus N)\) is closed. Since \(X\setminus N\subset U\) and \(g(U)\subset U'\), the set \(N'\) contains \(X'\setminus U'\). Also \(g^{-1}(N')\subset N\), and \(g(b_t)\notin N'\). Over \(N'\) the pushed roof can therefore be computed entirely where the original projector agrees with its character description.

Open base change and composition identify that description with the projector on \(R(gt)_*\mathcal H\) for \(D_i\). Every upstairs preimage in this computation avoids \(\infty\), so (36) identifies it with the projector for \(gt\) on \(A_{gt}\). The germ theorem gives equality of these endomorphisms in \(\mathcal Q_{U'}(X')\). The quotient functor thus carries the two forms in (38) to their counterparts for \(gt\). Applying the integral localization isomorphisms proves (40). ◻

Proof. The value at \(y_\infty\) may move during the homotopy. We will combine two sufficiently close stages into a single compact test and recover them by homotopic projections. Consider the compact test \[ C_d=\{(x_0,x_1)\in S^d\times S^d: \langle x_0,x_1\rangle\geq0\}. \tag{42}\] It is compact semialgebraic, hence homeomorphic to a finite polyhedron. It has a contractible open neighborhood basis at each point. To check this also at its boundary, the differential of \(F(x_0,x_1)=\langle x_0,x_1\rangle\) on \(S^d\times S^d\) is nonzero where \(F=0\): its derivative in the tangent direction \(x_0\) at \(x_1\) equals \(1\). Thus \(C_d\) is a smooth \(2d\)-manifold with boundary, with local Euclidean or half-space charts; sufficiently small balls in these charts give the required basis. Its test forms have degree \(d\), inherited by pushforward from \(S^d\); the target dimension is \(2d\).

The projections \(\pi_0,\pi_1:C_d\to S^d\) are homotopic by \[(x_0,x_1,r)\longmapsto \frac{(1-r)x_0+rx_1}{\|(1-r)x_0+rx_1\|},\qquad 0\leq r\leq1.\] The denominator is nonzero, since its square is \((1-r)^2+r^2+2r(1-r)\langle x_0,x_1\rangle\geq1/2\).

Let \(H:Y\times[0,1]\to S^d\) be the given homotopy. Uniform continuity on its compact domain gives a finite subdivision \(0=r_0<\cdots<r_N=1\) such that each pair \[t^{(j)}=(H(-,r_j),H(-,r_{j+1})):Y\longrightarrow C_d\] lands in \(C_d\). Its distinguished value has the neighborhood basis just verified. By [prop:class-covariance,thm:homotopy], \[z_i(H(-,r_j)) =\pi_{0*}z_i(t^{(j)}) =\pi_{1*}z_i(t^{(j)}) =z_i(H(-,r_{j+1})).\] Iteration proves the assertion. ◻

Proof. Let \(b\) be that constant value. Since \(y_\infty\notin V\), it equals \(b_h\). The source of the restricted test over \(S^d\setminus\{b\}\) lies in \(q^{-1}V\subset W\). Multiplication by \(\beta\) on its coefficient sheaf is defined because \(\beta\) is locally constant, and is a real isometry because \(|\beta|=1\). With our convention for \(T\), \[T M_\beta=M_{\zeta^{-1}}M_\beta T.\] It therefore takes a \(\lambda\)-eigensheaf to the \(\zeta^{-1}\lambda\)-eigensheaf and cyclically permutes the subsums in (43). After the proper restricted manifold pushforward it remains an isometry of \((A_h,\alpha_h)|_{S^d\setminus\{b\}}\) and conjugates \(e_i\) to \(e_{i-1}\), with indices modulo \(m\).

The isometry and its inverse define quotient morphisms by 14. Its conjugation identity makes the twisted forms for \(e_i\) and \(e_{i-1}\) isometric in the quotient. Their untwisted terms agree as well. The integral isomorphism in 27 and 39 now give \(z_i(h)=z_{i-1}(h)\). ◻

Proof. Suppose such an action is faithful. Choose an odd integer \(d>n\) with \(d\geq3\), and form the data at the beginning of this section by [lem:collapse,lem:averaged-test]. The latter may replace \(G\) by an open subgroup; its action remains faithful and orientation preserving.

First fix the lattice. 25 gives a positive power of \(2\), denoted by \(L_d\), such that \[E_d(S^d)\otimes\mathbb Q=\mathbb Q\overline{u}_d,\qquad \mathop{\mathrm{im}}\bigl(E_d(S^d)\longrightarrow E_d(S^d)\otimes\mathbb Q\bigr) \subset L_d^{-1}\mathbb Z\overline{u}_d.\] This bound depends only on the chosen sphere comparison. Choose \(k\) so large that \(m=p^k>4L_d\), and then choose \(V,\beta\) and the partition (43). These choices give integral classes \(z_i(a)\in E_d(S^d)\).

Write \(\overline{z}_i(a)=z_i(a)\otimes1\) and \(\overline{u}_d=u_d\otimes1\). We now compare these rational images. By 35 there are an integer \(s>0\) and a map \(h\), constant on \(Y\setminus V\), such that \(h\simeq D_s a\), where \(D_s:S^d\to S^d\) has degree \(s\). Thus [prop:equal-parts,prop:class-homotopy,prop:class-covariance] give \[D_{s*}z_i(a)=z_i(D_s a)=z_i(h)=z_j(h)=D_{s*}z_j(a).\] On the rational line \(\mathbb Q\overline{u}_d\), the map \(D_{s*}\) is multiplication by \(s\), by 26. Since \(s>0\), all the \(\overline{z}_i(a)\) are equal. This cancels \(s\) only as a nonzero rational scalar: the original classes \(z_i(a)\) remain integral, and their rational images lie in the same fixed lattice. In particular, the bound \(L_d\) is unchanged, regardless of the size or prime divisors of \(s\).

Finally compute their sum. The coefficient form \(B\) is the orthogonal sum of two positive real lines. Therefore \([A_a,\alpha_a]=2(aqc)_*u_d\), and the degree-one property of \(aqc\) gives \(\overline{[A_a,\alpha_a]}=2\overline{u}_d\). By 40, \[\sum_{i=0}^{m-1}\overline{z}_i(a)=4\overline{u}_d, \qquad \overline{z}_i(a)=\frac4m\overline{u}_d \quad\text{in }E_d(S^d)\otimes\mathbb Q.\] Each \(z_i(a)\) is an integral class, so \(\overline{z}_i(a)\in L_d^{-1}\mathbb Z\overline{u}_d\). The lattice inclusion would force \(4L_d/m\in\mathbb Z\). But \(0<4L_d/m<1\). This contradiction applies to every prime, including \(p=2\). ◻

Proof of 2. Let \(M\) have any positive finite dimension \(n\), and suppose that its given action were faithful. 28 produces a faithful orientation-preserving action of an open subgroup, itself isomorphic to \(\mathbb Z_p\), on a connected invariant chart subset \(O\subset\mathbb R^n\). This contradicts 44. Hence the original kernel is nonzero. It is closed, since it is the intersection of the closed stabilizers of the points of \(M\). Every nonzero closed subgroup of \(\mathbb Z_p\) is open: if it contains \(x\) of valuation \(j\), it contains \(\overline{x\mathbb Z}=p^j\mathbb Z_p\), the closure of the integer multiples of \(x\). The action therefore factors through a finite quotient. The chart reduction requires none of compactness, global orientability, or triangulability of \(M\), and permits all stabilizers, as established in 5. ◻

Proof of 1. The classical Hilbert–Smith reduction says that a locally compact group which acts faithfully and continuously on a connected manifold and is not a Lie group contains a topologically embedded copy of \(\mathbb Z_p\) for some prime \(p\); see (Pardon 2013, Introduction, p. 1). Apply this reduction to the given locally compact Hausdorff second-countable group. Restriction to the embedded subgroup retains joint continuity and faithfulness on the same \(n\)-manifold, contrary to 2. The group is therefore a Lie group. ◻

Proof. Keep \(l,n\) as the degrees of the two shifted dual elements. The Hom shift comparison on the first factor contributes \((-1)^{l-s}\), while the comparison on the total dual contributes \((-1)^{l+n-s-r}\). In (48), replacing \(b\) by \(b[1]\) changes the braiding exponent by \(-n\). After the first Hom comparison, the degree of the corresponding dual element changes along with that of \(b\), so the output-shift term is the same along the two routes. The discrepancy is therefore \[(l+n-s-r)-(l-s)-n\equiv r\pmod2.\] All dependence on the individual degrees cancels. This is the stated scalar, and it is the suspension discrepancy used when dualizing \(\partial\boxtimes1\) in 10. ◻