Ferrari Ruffino, Fabio:
Gysin Map and Atiyah-Hirzebruch Spectral Sequence
Bollettino dell'Unione Matematica Italiana Serie 9 4 (2011), fasc. n.2, p. 263-273, (English)
pdf (395 Kb), djvu (147 Kb). | MR 2840607 | Zbl 1241.55011
Sunto
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 $Y$.
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