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

TL;DR

This paper proves a conjecture about the behavior of Steenrod squares in intersection cohomology, which is crucial for defining characteristic classes in singular spaces, and provides explicit computations and extensions of classical properties.

Contribution

It establishes the validity range of Steenrod squares in intersection cohomology and extends classical properties to this setting, confirming Goresky's definition for pseudomanifolds.

Findings

Proved the conjecture on the range of Steenrod squares in intersection cohomology.

Extended classical Steenrod square properties to generalized perversities.

Provided explicit examples, including on Thom spaces and singularities.

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 of importance for the definition of characteristic classes in the framework of intersection cohomology. For this purpose, we present a construction of ${\rm cup}_{i}$-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 generalized perversities. In the case of a pseudomanifold, we prove that our definition coincides with M. Goresky's definition. 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, in function of the Steenrod squares of the basis and the Stiefel-Whitney classes. We detail also an example of a non trivial square, $\sq^2\colon H_{\ bar{p}}\to H_{\ bar{p}+2}$, whose information is lost if we consider it as values in $H_{2\ bar{p}}$, showing the interest of the Goresky and Pardon's conjecture.