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Refined intersection homology on non-Witt spaces. (English) Zbl 1316.55002

Intersection homology has been successfully introduced to obtain Poincaré duality on stratified manifolds, despite their singularities. It has several equivalent formulations; in the present paper the approach is via sheaf cohomology and intersection sheaves. This introduces an additional variable: the intersection sheaf (determined by its perversity). Not in all cases does one obtain a self-dual intersection cohomology and associated signature invariants. Witt spaces have been introduced because they admit a self-dual intersection sheaf.
An analytic solution to the same problem is given by Cheeger’s \(L^2\) de Rham cohomology for suitably defined metrics on (the regular part of) the stratified space; again this works well for Witt spaces and there is a de Rham isomorphism to intersection homology in this case.
The paper at hand discusses aspects of the generalizations of this theory to non-Witt spaces.
In previous work, Banagl has introduced refined intersection homology sheaves which sit “between” the two middle perversity intersection homology sheaves. Albin, Leichtnam, Mazzeo, Piazza, on the other hand, have introduced a generalization of Cheeger’s \(L^2\) de Rham cohomology (using suitable boundary conditions, called mezzoperversities). The first part of the paper systematically studies these; in particular it derives the duality theory. In particular, the cases where one can find self-dual refined intersection homology sheaves or (equivalently) self-dual mezzoperversities is investigated: these spaces are called Cheeger spaces or L-spaces.
The main result of the paper is a de Rham isomorphism: starting with one of the boundary conditions above (a mezzoperversity), a canonically associated refined intersection cohomology sheaf is produced (and it is shown that this procedure is a bijection between the mezzoperversity and the refined intersection homology sheaves). Moreover, the authors prove a de Rham theorem establishing a canonical isomorphism between the resulting generalized \(L^2\) de Rham cohomology and refined intersection homology.
It should be pointed out that the topological constructions work for very general stratified spaces, whereas the analytic ones require a Thom-Mather stratification. Of course, only on Thom-Mather stratified spaces the comparison of the two theories can be carried out. The refined intersection homology is a stratified homotopy invariant, but might depend on the stratification (in constrast to the situation on Witt spaces).

MSC:

55N33 Intersection homology and cohomology in algebraic topology
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
57N80 Stratifications in topological manifolds
58A35 Stratified sets
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References:

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