Gysin Map and Atiyah-Hirzebruch Spectral Sequence

Ferrari Ruffino, Fabio

Bollettino della Unione matematica italiana, Série 9, Tome 4 (2011) no. 2, pp. 263-273

Résumé

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 Poincaré dual of $Y$ in $X$, which, in our setting, is a simplicial cohomology class with coefficients in $h^{0}\{*\}$, if such a class survives until the last step, it is represented in $E^{n-p,0}_{\infty}$ 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 $Y$.