Cup and cap products in intersection (co)homology (Q390741)

Let \(F\) be a field and \(IH^{\bar{p}}_* (X;F),\) \(IH^*_{\bar{p}} (X;F)\) Goresky-MacPherson's intersection homology and intersection cohomology of a stratified pseudomanifold \(X,\) with coefficients in \(F\). The underlying chain model is here taken to be singular intersection chains. Given a perversity function \(\bar{p}\), let \(D\bar{p} = \bar{t} - \bar{p}\) be the dual or complementary perversity, where \(\bar{t}\) is the top perversity. The paper under review does not require perversities to obey Goresky-MacPherson's conditions, rather they are simply functions from the set of strata to the integers which are zero on the top strata. Consequently, the above intersection (co)homology groups will in general depend on the stratification. After reviewing when cross-products on intersection chains induce Künneth-isomorphisms on intersection homology, these cross products together with maps induced by the diagonal map on intersection homology (for certain perversity configurations), are used to define cup products NEWLINE\[NEWLINE\cup: IH^*_{\bar{p}} (X;F) \otimes IH^*_{\bar{q}} (X;F)\longrightarrow IH^*_{\bar{s}} (X;F)NEWLINE\]NEWLINE and cap products NEWLINE\[NEWLINE\cap: IH^*_{\bar{q}} (X;F) \otimes IH_*^{\bar{s}} (X;F)\longrightarrow IH_*^{\bar{p}} (X;F)NEWLINE\]NEWLINE when \(D\bar{s} \geq D\bar{p} + D\bar{q}\). It is important to note that it is well-known that no perversity-internal cup-products can generally exist on intersection cohomology. Thus, the intersection cochain complex \(IC^*_{\bar{p}}\) is in general not a differential graded algebra. For certain classes of pseudomanifolds \(X\), a cohomology theory \(HI^*_{\bar{p}} (X)\) which \textit{is} a ring for fixed \(\bar{p}\) has been constructed in [\textit{M. Banagl}, Intersection spaces, spatial homology truncation, and string theory. Dordrecht: Springer (2010; Zbl 1219.55001)].NEWLINENEWLINEFor oriented stratified \(n\)-dimensional pseudomanifolds \(X\) and compact \(K\subset X,\) the paper under review constructs a fundamental class in \(IH^{\bar{0}}_n (X,X-K)\). This class is then used to reprove generalized Poincaré duality for intersection homology: The above cap product with the fundamental class defines an isomorphism \(IH^i_{\bar{p},c} (X;F)\cong IH^{D\bar{p}}_{n-i}(X;F)\) when \(X\) is oriented over \(F\).NEWLINENEWLINENote that this paper is a prerequisite for the authors' work on the symmetric signature of Witt spaces.