Blown-up intersection cohomology

TL;DR

This paper develops the theory of blown-up intersection cohomology, establishing its fundamental properties, product structures, functoriality, and invariance, extending previous work and connecting it with existing intersection cohomology theories.

Contribution

It introduces and analyzes the main properties of blown-up intersection cohomology, including cup and cap products, functoriality, and topological invariance, broadening the framework for intersection cohomology theories.

Findings

Existence of cup and cap products at the cochain level.

Functoriality of blown-up intersection cohomology under stratified maps.

Topological invariance for pseudomanifolds without codimension one strata.

Abstract

In previous works, we have introduced the blown-up intersection cohomology and used it to extend Sullivan's minimal models theory to the framework of pseudomanifolds, and to give a positive answer to a conjecture of M. Goresky and W. Pardon on Steenrod squares in intersection homology. In this paper, we establish the main properties of this cohomology. One of its major feature is the existence of cap and cup products for any filtered space and any commutative ring of coefficients, at the cochain level. Moreover, we show that each stratified map induces an homomorphism between the blown-up intersection cohomologies, compatible with the cup and cap products. We prove also its topological invariance in the case of a pseudomanifold with no codimension one strata. Finally, we compare it with the intersection cohomology studied by G. Friedman and J.E. McClure. A great part of our results involves general perversities, defined independently on each stratum, and a tame intersection homology adapted to large perversities.