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arXiv · 0904.4103

Gysin map and Atiyah-Hirzebruch spectral sequence

Fabio Ferrari Ruffino
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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 Poincarè 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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Fabio Ferrari Ruffino. 2011-03-30. Gysin map and Atiyah-Hirzebruch spectral sequence. https://arxiv.org/abs/0904.4103

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