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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

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.

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