Reference request for Homology Gysin sequence.

Question

I am trying to study the Homology Gysin sequence (not cohomology). I am interested in finding references that either use, or explain the Homology Gysin sequence, especially if it gives descriptions for the maps in the sequence.

Answer 1

Let $\xi\colon E\to B$ be an orientable $n$-bundle. You can always consider the l.e.s. for the pair $(E,E_0)$ (where $E_0:= E\setminus B$ via the zero section). In order to avoid pathologies, assume $B$ being a CW-complex.

Notice that $H_k(E,E_0;\Bbb Z)\cong H_{k-n}(E)\cong H_{k-n}(B)$ via the homological Thom Iso plus the fact $\xi_*$ is an isomorphism in homology. If you use these identification you will end up with your Gysin sequence where the only non-obvious map will be cap-product with the Euler class.