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Steenrod squares on Intersection cohomology and a conjecture of M. Goresky and W. Pardon

Abstract : We prove a conjecture raised by M. Goresky and W. Pardon, concerning the range of validity of the perverse degree of Steenrod squares in intersection cohomology. This answer turns out of importance for the definition of characteristic classes in the framework of intersection cohomology. For this purpose, we present a construction of ${cup}_{i}$-products on the cochain complex of filtered face sets, built on the blow-up of simplices and introduced in a previous work. We extend to this setting the classical properties of the associated Steenrod squares, including Adem and Cartan relations, for any generalized perversities. In the case that the filtered face set is the singular filtered face set associated to a pseudomanifold, we prove that our definition coincides with M. Goresky's definition. Several examples of concrete computation of perverse Steenrod squares are given, including the case of isolated singularities and, more especially, we describe the Steenrod squares on the Thom space of a vector bundle, in function of the Steenrod squares of the basis and the Stiefel-Whitney classes. We detail also an example of a non trivial square, $\sq^2\colon H_{\overline{p}}\to H_{\overline{p}+2}$, whose information is lost if we consider it as values in $H_{2\overline{p}}$, showing the interest of the Goresky and Pardon's conjecture.
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Contributor : Martintxo Saralegi-Aranguren Connect in order to contact the contributor
Submitted on : Wednesday, February 13, 2013 - 2:47:58 PM
Last modification on : Friday, May 6, 2022 - 6:04:02 PM

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Chataur David, Martintxo Saralegi-Aranguren, Tanré Daniel. Steenrod squares on Intersection cohomology and a conjecture of M. Goresky and W. Pardon. Algebraic and Geometric Topology, Mathematical Sciences Publishers, 2016, 16 (4), pp.1851-1904. ⟨10.2140/agt.2016.16.1851⟩. ⟨hal-00787988⟩



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