#### Volume 16, issue 4 (2016)

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

### David Chataur, Martintxo Saralegi-Aranguren and Daniel Tanré

Algebraic & Geometric Topology 16 (2016) 1851–1904
##### 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 ${cup}_{i}$–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 $2\stackrel{̄}{p}$, 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, ${Sq}^{2}:{H}_{\stackrel{̄}{p}}\to {H}_{\stackrel{̄}{p}+2}$, whose information is lost if we consider it as taking values in ${H}_{2‘\stackrel{̄}{p}}$, showing the interest of the Goresky–Pardon conjecture.

##### Keywords
Intersection cohomology, Simplicial blow-up, Steenrod squares, Pseudo-manifold, Isolated singularity, Thom space, Stiefel-Whitney classes
##### Mathematical Subject Classification 2010
Primary: 55N33, 55S10, 57N80
##### Publication
Received: 12 April 2014
Revised: 5 January 2015
Accepted: 24 December 2015
Published: 12 September 2016
##### Authors
 David Chataur LAFMA Université de Picardie Jules Verne 33, Rue Saint Leu Villeneuve d’Ascq 80039 Amiens Cedex 1 France Martintxo Saralegi-Aranguren Laboratoire de Mathématiques de Lens, EA 2462 Université d’Artois SP18, rue Jean Souvraz 62307 Lens Cedex France Daniel Tanré Département de Mathématiques, UMR 8524 Université de Lille 1 Villeneuve D’Ascq 59655 Lille Cedex France