Gysin map and Atiyah-Hirzebruch spectral sequence

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.

💡 Research Summary

The paper investigates the precise relationship between the Atiyah‑Hirzebruch spectral sequence (AHSS) and the Gysin map for a multiplicative cohomology theory h* on spaces that have the homotopy type of a finite CW‑complex. After fixing such a theory, the authors work with a smooth n‑dimensional manifold X and a compact p‑dimensional submanifold Y⊂X, assuming that Y is h*‑orientable (i.e., its normal bundle admits a Thom class in h^{n‑p}).

The AHSS associated to the cellular filtration of X has E₂‑term
E₂^{s,t}=H^{s}(X;h^{t}(pt)) ⇒ h^{s+t}(X).
The Poincaré dual of Y in X, denoted PD_X