Intersection homology and Poincaré duality on homotopically stratified spaces

Abstract

We show that intersection homology extends Poincaré duality to manifold homotopically stratified spaces (satisfying mild restrictions). These spaces were introduced by Quinn to provide "a setting for the study of purely topological stratified phenomena, particularly group actions on manifolds." The main proof techniques involve blending the global algebraic machinery of sheaf theory with local homotopy computations. In particular, this includes showing that, on such spaces, the sheaf complex of singular intersection chains is quasi-isomorphic to the Deligne sheaf complex.55N33, 57N80, 57P99

Intersection homology of stratified fibrations and neighborhoods

“…In [7], these computations will be utilized in order to relate singular chain intersection homology to the Deligne sheaf construction [1,15] on manifold homotopically stratified spaces and to demonstrate intersection homology Poincaré duality on such spaces.…”

Intersection homology with general perversities

“…Thus instead we will show that D X Q * p (E)[−n] satisfies the axioms AX1q ,X,D U 1 E[−n] , which suffice by Propositions 3.8. This will be similar to the proof of our main theorem of [23], though simpler since the context there called for much more general spaces.…”

“…Again, we can limit ourselves to a cofinal system of distinguished neighborhoods, and it suffices to find then isomorphisms H j(U ; D * (IpS * (Ẽ 0 ))[−n]) → H j (U − U ∩ Z ; D * (IpS * (Ẽ 0 ))[−n]) that are functorial in that they commute with further restrictions U → V . By[23, Appendix]…”

On the construction and topological invariance of the Pontryagin classes

“…The papers of [37] and of Woolf [35], and the book of Banagl [1] also shed light on the structure of self-dual sheaves; and selfdual sheaves are studied in homotopy stratified settings in Friedman [9]. See Brasselet et al [2] for an application of self-dual sheaves in algebraic geometry.…”