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

Chataur, David¹ ; Saralegi-Aranguren, Martintxo² ; Tanré, Daniel³

¹LAFMA, Université de Picardie Jules Verne, 33, Rue Saint Leu, Villeneuve d’Ascq, 80039 Amiens Cedex 1, France
²Laboratoire de Mathématiques de Lens, EA 2462, Université d’Artois, SP18, rue Jean Souvraz, 62307 Lens Cedex, France
³Département de Mathématiques, UMR 8524, Université de Lille 1, Villeneuve D’Ascq, 59655 Lille Cedex, France

Algebraic and Geometric Topology, Tome 16 (2016) no. 4, pp. 1851-1904

Résumé

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 H2‘p̄, showing the interest of the Goresky–Pardon conjecture.

Détail

DOI : 10.2140/agt.2016.16.1851

Classification : 55N33, 55S10, 57N80
Keywords: Intersection cohomology, Simplicial blow-up, Steenrod squares, Pseudo-manifold, Isolated singularity, Thom space, Stiefel-Whitney classes

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