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On topological invariants of stratified maps with non-Witt target

Author: Markus Banagl
Journal: Trans. Amer. Math. Soc. 358 (2006), 1921-1935
MSC (2000): Primary 57R20, 55N33
Published electronically: December 20, 2005
MathSciNet review: 2197435
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Abstract: The Cappell-Shaneson decomposition theorem for self-dual sheaves asserts that on a space with only even-codimensional strata any self-dual sheaf is cobordant to an orthogonal sum of twisted intersection chain sheaves associated to the various strata. In sharp contrast to this result, we prove that on a space with only odd-codimensional strata (not necessarily Witt), any self-dual sheaf is cobordant to an intersection chain sheaf associated to the top stratum: the strata of odd codimension do not contribute terms. As a consequence, we obtain formulae for the pushforward of characteristic classes under a stratified map whose target need not satisfy the Witt space condition. To prove these results, we introduce a new category of superperverse sheaves, which we show to be abelian. Finally, we apply the results to the study of desingularization of non-Witt spaces and exhibit a singular space which admits a PL resolution in the sense of M. Kato, but no resolution by a stratified map.

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Additional Information

Markus Banagl
Affiliation: Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany

Keywords: Stratified maps, signature, characteristic classes, intersection homology, self-dual sheaves, perverse sheaves, t-structures, desingularization, cobordism of sheaves
Received by editor(s): February 11, 2003
Published electronically: December 20, 2005
Additional Notes: The author was supported in part by NSF Grant DMS-0072550
Article copyright: © Copyright 2005 American Mathematical Society

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