Gysin map and Atiyah-Hirzebruch spectral sequence

TL;DR

This paper explores the relationship between the Atiyah-Hirzebruch spectral sequence and the Gysin map in the context of multiplicative cohomology theories on finite CW-complexes, providing conditions under which spectral sequence classes correspond to Gysin images.

Contribution

It establishes a connection between spectral sequence classes and Gysin map images for submanifolds in smooth manifolds within a multiplicative cohomology framework.

Findings

Spectral sequence classes representing submanifolds correspond to Gysin map images.

Survival of classes in the spectral sequence indicates their geometric origin.

Results apply to classes on submanifolds under orientability conditions.

Abstract

We discuss the relations between the Atiyah-Hirzebruch spectral sequence and the Gysin map for a multiplicative cohomology theory, on spaces having the homotopy type of a finite CW-complex. In particular, let us fix such a multiplicative cohomology theory h* and let us consider a smooth manifold X of dimension n and a compact submanifold Y of dimension p, satisfying suitable hypotheses about orientability. We prove that, starting the Atiyah-Hirzebruch spectral sequence with the Poincar\`e dual of Y in X, which, in our setting, is a simplicial cohomology class with coefficients in h^{n-p}(one-point), if such a class survives until the last step, it is represented by the image via the Gysin map of the unit cohomology class of Y. We then prove the analogous statement for a generic cohomology class on Y.