Wu numbers of singular spaces

Abstract

THEOREM. The cobordism group of orientable 4k-dimensional LSF spaces is isomorphic to z/(4) and the invariant is given by 7t. (d) i-duality spaces. Of course, LSF spaces may be combined with local orientability to produce a host of characteristic numbers involving u '. Instead, we prefer to strengthen the LSF condition by demanding not just that Sq' vanish on the intersection homology of links of even codimension strata, but that the whole group vanish: Definition. A normal pseudomanifold X is an S-duality space if, for each stratum of odd codimension c = 2k + 1, we have ZHr(L; i?/(2)) = 0 and for each stratum of even codimension c = 2k we have ZH:(L; Z/(2)) = ZHF-1 (L; Z/(2)) = 0 where L is the link of the stratum. For such spaces the middle intersection homology groups admit a Bockstein operation /I: ZH,"(X; Z/(2)) + ZHF-1 (X; Z) whose mod 2 reduction is a Steenrod operation, Sq': ZHF(X; iZ/(2)) + ZHFi (X; Z/(2)). It even turns out that there is a universal coefficient theorem for I-duality spaces, and that the intersection homology of F-duality spaces satisfies Poincare duality over the localization Z,,,. THEOREM 16.5. An orientable n dimensional i-duality space X of odd dimension n = 2k + 1 is the boundary of an orientable S-duality space Y ifand only if the characteristic number vk Sq' t.'(X) vanishes.

Stiefel–Whitney numbers for singular varieties

“…These Stiefel-Whitney numbers have arisen before, in Goresky-Pardon's calculation of the bordism ring of locally orientable F 2 -Witt spaces [GP89]. There they appear in the guise v n−2i 1 v 2 i for 1 i n/2 and are defined by using local orientability to lift v 1 to cohomology and by using the F 2 -Witt condition to lift the Wu class v i to intersection cohomology where it can be squared to obtain a homology class (see [Gor84]).…”

Cobordism of maps of locally orientable Witt spaces

“…In this section we use definitions and notations of M. Goresky [6] and M. Goresky and W. Pardon [9]. First of all, let us fix notations in the smooth case.…”

“…In the singular case, the Steenrod square operations are defined in intersection cohomology by M. Goresky [6, §3.4] as follows: For such spaces, the middle intersection homology group satisfies the Poincaré duality over Z 2 . In the following, we will use the notion of locally orientable Witt-space that we recall: Let X be a Z 2 -Witt space, then the Wu classes v i (X) lift canonically to IH ī m (X) = IH ī n (X) (see [9] §10). We denote by v a−i (X) ∈ IHn a−i (X) the (homology) (a − i) th -Wu class of X, in intersection homology, dual to the Wu class v i (X) (denoted by Iv i ∈ IH i (X) in [6]).…”

Cohomologie d’intersection modérée. Un théorème de de Rham

“…Cette homologie partage plusieurs proprietes avec Γhomologie singuliere: description axiomatique [10], theorie de Morse [11], cobordisme [12], Theoreme de Lefschetz [8], Theoreme de deRham [1], [2], [15] .... Elle coincide avec Γhomologie ordinaire pour les varietes lisses, compactes et orientees. L'homologie d'intersection a ete definie en premiere instance dans la categorie lineaire par morceaux [9] (pseudovarietes stratifiees); elle s'etend naturellement aux complexes simpliciaux a Γaide d'une triangulation en drapeaux [13,Appendice] (pseudovarietes stratifiees simpliciales), cadre oύ se developpe [5].…”