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 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 $2\bar{p}$, 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, ${Sq}^2:H_{\bar{p}}\to H_{\bar{p}+2}$, whose information is lost if we consider it as taking values in $H_{2‘\bar{p}}$, showing the interest of the Goresky–Pardon conjecture.