Background prerequisites
External results the proof uses but does not prove. Each article states the fact as the paper needs it, with a pinpoint to the reference; the background map records the full citations and which articles use them.
Transformation groups
A finite-order homeomorphism of a connected topological manifold that fixes a nonempty open set pointwise is the identity.
Cited for the local chart reduction. Modern statement and proof as referenced in the paper.
Dependencies
A locally compact group acting faithfully and continuously on a connected manifold, if not a Lie group, contains a topologically embedded copy of \(\Zp\) for some prime \(p\). Restriction to the embedded subgroup retains joint continuity and faithfulness.
Cited for the final step from \(p\)-adic exclusion to the full Hilbert--Smith conjecture. The underlying structure theory is Gleason, Montgomery--Zippin, and Yamabe; the action-specific form is Pardon's with Lee's treatment of the reduction.
Dependencies
Every nonzero closed subgroup of \(\Zp\) is open, of the form \(p^j\Zp\): an element of smallest finite \(p\)-adic valuation has integer multiples dense in that subgroup. Consequently a quotient \(\Zp/H\) by a closed subgroup is a point, a finite quotient, or \(\Zp\) itself, and a nonzero kernel in \(\Zp\) makes the action factor through a finite quotient.
Standard fact; the paper sketches the proof where it is used.
A compact abelian Lie group has identity component a torus of finite index, and the exponential map of a torus is surjective.
Cited for the almost-periodic flow consequence: the compact closure of an almost-periodic homeomorphism, once known to be a compact abelian Lie group, puts some positive iterate on a one-parameter subgroup.
Dependencies
Sheaf and derived formalism
For locally compact spaces and maps of finite \(c\)-soft dimension: \(Rf_!\) has right adjoint \(f^!\); composition identifies \(f^!\omega_Y\) with \(\omega_X\); the counit gives the trace \(Rf_!\omega_X \to \omega_Y\); the internal duality isomorphism \(Rf_*\D_X A \xrightarrow{\sim} \D_Y Rf_!A\) holds, adjoint to evaluation followed by the trace and compatible with evaluation maps; for an open inclusion \(j\), \(j^! = j^{-1}\), so duality commutes with open restriction. None of this asserts reflexivity of every bounded complex.
Cited throughout the sheaf-category chapters for duality, pushforward, and product computations.
Proper base change, the projection formula, and compact-support Kunneth in their natural forms; the composition formula for \(Rf_!\); cup maps obtained from the projection formula and adjunction; and the trace composition identity \(\Tr_{hg} = \Tr_h \circ Rh_!(\Tr_g)\).
Cited for pushforward-duality compatibility, external products, and the product trace in the sheaf-category chapters.
Finite \(c\)-soft dimension and its consequences: \(R\Gamma_c(\R^N;-)\) has no cohomology above \(N\); soft resolutions and their acyclicity on compact Hausdorff spaces, so sheaf cohomology there commutes with direct sums and sums of soft sheaves are soft.
Cited for the finite-dimensionality hypotheses on allowed maps and for the derived character-sum computation.
Dependencies
Bounded-below proper base change for closed separated maps with compact fibers; countable sheaf sums are exact and termwise sums compute coproducts in the derived category; the inclusions of bounded and bounded-below derived categories into the ambient derived category are fully faithful.
Cited for the orbit pullback vanishing, the derived character sum, and coproduct detection of morphisms between bounded objects.
Dependencies
For a saturated full triangulated duality-stable subcategory with essentially small categories, bijective multiplication by \(2\) on morphism groups, and the enriched octahedral axiom, there is a long exact sequence of triangular Witt groups for every shifted duality, with the boundary given by compatible symmetric cones. A symmetric form in the quotient is isometric to the image of a symmetric morphism whose cone lies in the kernel category.
Cited to produce the Witt localization sequence from the support characterization; the paper verifies the hypotheses and the naturality of the boundary map itself.
Dependencies
Triangular Witt foundations
Let \(C\) be an additive category with a duality: a contravariant additive endofunctor \(D\) with a biduality natural isomorphism \(\varpi : \mathrm{Id} \Rightarrow D \circ D^{\mathrm{op}}\). A symmetric form on \(X\) is a morphism \(\alpha : X \to DX\) equal to its own transpose \(\varpi_X \gg D(\alpha)\); it is nonsingular when \(\alpha\) is an isomorphism. The orthogonal sum \(\alpha \perp \beta\) of symmetric forms on \(X\) and \(Y\) is the biproduct form on \(X \boxplus Y\), via the canonical identification \(D(X \boxplus Y) \cong DX \boxplus DY\). Orthogonal sums of (nonsingular) symmetric forms are (nonsingular) symmetric.
Cited as the foundation for the triangular Witt layer: the doubling and cone arguments manipulate symmetric forms and their orthogonal sums throughout.
Proof sketch. The transpose is involutive by naturality of \(\varpi\); the biproduct identification is natural, so the matrix of the transpose is the transpose of the matrix, and a diagonal of symmetric entries is symmetric. A diagonal of isomorphisms is an isomorphism, so nonsingularity is preserved.
Sources
Homotopy, semialgebraic, and combinatorial inputs
For an odd sphere \(S^d\): the higher homotopy groups \(\pi_j(S^d)\), \(j > d\), are finite; and if \(qx = 0\) in a homotopy group, then postcomposition with a degree-\(2q\) map kills \(x\), in every homotopy degree.
Cited for the mapping-telescope computation identifying the telescope with \(K(\Q,d)\) in the odd-sphere nullhomotopy argument.
Dependencies
A continuous semialgebraic map from a compact polyhedron admits a finite semialgebraic partition of the target over whose pieces the map is trivial; fibers are compact semialgebraic sets with the finite cohomology of a finite triangulation.
Cited to find a common dense open set where finitely many semialgebraic generators have finite locally constant cohomology, for the generic signature evaluation.
Dependencies
Simplicial approximation after subdivision, and the local-degree formula identifying the degree of a sphere map with the sum of local signs over a regular value.
Cited to compute the pushforward of the orientation form on the sheet ball for the degree action.
Sources
Dependencies
On compact Hausdorff spaces, the sheaf--Cech comparison is an isomorphism commuting with pullback; for finite covers, Cech cohomology is the direct limit of nerve cohomologies, and the map from a nerve is pullback by the nerve map, independent of the subordinate partition.
Cited for rational orbit-cohomology detection and for reducing the odd-sphere nullhomotopy to a finite complex.