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2016 Steenrod squares on intersection cohomology and a conjecture of M Goresky and W Pardon
David Chataur, Martintxo Saralegi-Aranguren, Daniel Tanré
Algebr. Geom. Topol. 16(4): 1851-1904 (2016). DOI: 10.2140/agt.2016.16.1851

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 to be of importance for the definition of characteristic classes in the framework of intersection cohomology.

For this purpose, we present a construction of cupi–products on the cochain complex, built on the blow-up of some singular 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 loose perversities. In the case of a PL-pseudomanifold and range 2p̄, we prove that our definition coincides with Goresky’s definition. We also show that our Steenrod squares are topological invariants which do not depend on the choice of a stratification of X.

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 as a function of the Steenrod squares of the base space and the Stiefel–Whitney classes of the bundle. We also detail an example of a nontrivial square, Sq2: Hp̄ Hp̄+2, whose information is lost if we consider it as taking values in H2p̄, showing the interest of the Goresky–Pardon conjecture.

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David Chataur. Martintxo Saralegi-Aranguren. Daniel Tanré. "Steenrod squares on intersection cohomology and a conjecture of M Goresky and W Pardon." Algebr. Geom. Topol. 16 (4) 1851 - 1904, 2016. https://doi.org/10.2140/agt.2016.16.1851

Information

Received: 12 April 2014; Revised: 5 January 2015; Accepted: 24 December 2015; Published: 2016
First available in Project Euclid: 28 November 2017

zbMATH: 06627563
MathSciNet: MR3546453
Digital Object Identifier: 10.2140/agt.2016.16.1851

Subjects:
Primary: 55N33, 55S10, 57N80

Rights: Copyright © 2016 Mathematical Sciences Publishers

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