Gysin sequence

Remarks

The space $E_0$ deformation retracts onto the unit sphere bundle $S(E)$ of any metric on $E$, so the sequence is often written with $S(E)$, a bundle with fibre $S^{n-1}$. The proof writes the long exact sequence of the pair $(E, E_0)$, replaces $H^\bullet(E)$ by $H^\bullet(B)$ since $E$ deformation retracts onto $B$, and replaces $H^{\bullet}(E, E_0)$ by $H^{\bullet - n}(B)$ with the Thom isomorphism. The composite $H^{i}(B) \to H^{i+n}(E, E_0) \to H^{i+n}(E) \cong H^{i+n}(B)$ is then the cup product with $e$. With $\Z/2$ coefficients the sequence holds without the orientation hypothesis, with the top Stiefel–Whitney class in place of $e$.

Example: the tautological complex line bundle over complex projective space $\C P^m$ has rank $2$ as a real bundle, and its unit sphere bundle is the Hopf map $S^{2m+1} \to \C P^m$. Since $H^j(S^{2m+1}) = 0$ for $0 < j < 2m+1$, the sequence shows that $\smile e \colon H^i(\C P^m) \to H^{i+2}(\C P^m)$ is an isomorphism for $0 \le i \le 2m - 2$. With $H^0(\C P^m) = \Z$, this gives $H^\bullet(\C P^m; \Z) \cong \Z[e]/(e^{m+1})$.

Going deeper

For a student

The Gysin sequence is the special case of the Serre spectral sequence for a sphere bundle, where only two rows are nonzero. It is a standard way to compute cohomology rings of projective spaces, of unit tangent bundles and of lens spaces. It also shows that an oriented bundle with a nowhere zero section has Euler class zero, because then $\pi_0^*$ is injective in the relevant degree.