×

Characterization of homology manifolds by help of intersection homology invariants. (Caractérisation des variétés homologiques à l’aide des invariants d’homologie d’intersection.) (French) Zbl 0855.55005

Janeczko, Stanisław (ed.) et al., Singularities and differential equations. Proceedings of a symposium, Warsaw, Poland. Warsaw: Polish Academy of Sciences, Inst. of Mathematics, Banach Cent. Publ. 33, 19-22 (1996).
In [M. Goresky and R. MacPherson, Topology 19, 135-165 (1980; Zbl 0448.55004)] the authors defined certain intersection homology groups \(I_{\mathbf p} H_* (X)\) for a pseudomanifold \(X\) and formulated the conjecture: If \(X\) is a normal pseudomanifold such that \(I_{\mathbf p} H_* (X) \to I_{\mathbf q} H_* (X)\) is an isomorphism for all perversities \({\mathbf q} > {\mathbf p}\), then \(X\) is a homology manifold. H. C. King [ibid. 21, 229-234 (1982; Zbl 0509.55004)] constructed a counterexample to this conjecture and gave some sufficient condition to save the conjecture (his assumptions about \(X\) are: \(X\) is an untwisted normal pseudomanifold with boundary, and \(I_{\mathbf p} H_* (X) \to I_{\mathbf q} H_* (X)\) and \(I_{\mathbf p} H_* (\partial X) \to I_{\mathbf q} H_* (\partial X)\) are isomorphisms for all loose perversities \({\mathbf q} > {\mathbf p})\).
In the paper under review the authors give necessary and sufficient conditions for \(X\) to be a homology manifold (four variants of the so-called local criteria are given).
For the entire collection see [Zbl 0840.00028].

MSC:

55N33 Intersection homology and cohomology in algebraic topology
57N65 Algebraic topology of manifolds
PDFBibTeX XMLCite