Gysin map and Atiyah-Hirzebruch spectral sequence

Details Gysin map and Atiyah-Hirzebruch spectral sequence Gysin map and Atiyah-Hirzebruch spectral sequence

2009-04-27

arxiv.org/abs/0904.4103v3

Bollettino UMI (9) IV (2011)

Mathematics

Algebraic Topology

17 pages, no figures. The published version is shorter, since most of the review parts and of the overlaps with [6] have been

Scientific paper

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.

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