Gysin Sequence and Thom Isomorphism

These results relate the cohomology of a sphere bundle and a disk bundle to the cohomology of the base, providing key tools for computing characteristic classes and understanding the topology of vector bundles.

The Thom Isomorphism

Theorem10.8Thom Isomorphism

Let $\xi : E \to B$ be an oriented rank- $n$ real vector bundle with disk bundle $D(\xi)$ and sphere bundle $S(\xi)$ . Then there exists a Thom class $u \in H^n(D(\xi), S(\xi); \mathbb{Z})$ such that the map $\Phi : H^k(B; \mathbb{Z}) \xrightarrow{\;\cong\;} H^{k+n}(D(\xi), S(\xi); \mathbb{Z}), \quad \alpha \mapsto \pi^*(\alpha) \smile u$ is an isomorphism for all $k$ , where $\pi : D(\xi) \to B$ is the projection. The group $H^*(D(\xi), S(\xi))$ is called the Thom space cohomology.

The Thom class restricts to the generator of $H^n(D^n, S^{n-1}) \cong \mathbb{Z}$ on each fiber. Its existence is equivalent to the orientability of the bundle.

The Euler Class and Gysin Sequence

Definition

The Euler class of an oriented rank- $n$ vector bundle $\xi : E \to B$ is $e(\xi) = s^*(u) \in H^n(B; \mathbb{Z})$ where $s : B \to D(\xi)$ is the zero section and $u$ is the Thom class (pulled back via the natural map $H^n(D(\xi), S(\xi)) \to H^n(D(\xi)) \xrightarrow{s^*} H^n(B)$ ). The Euler class vanishes if and only if $\xi$ admits a nowhere-zero section (when the base is a CW complex of dimension $\leq n$ ).

Theorem10.9Gysin Exact Sequence

Let $S^{n-1} \to E \xrightarrow{p} B$ be an oriented sphere bundle (the sphere bundle of an oriented rank- $n$ vector bundle). Then there is a long exact sequence $\cdots \to H^k(B) \xrightarrow{\smile e} H^{k+n}(B) \xrightarrow{p^*} H^{k+n}(E) \xrightarrow{\tau} H^{k+1}(B) \to \cdots$ where $e$ is the Euler class of the associated vector bundle and $\tau$ is the transfer (or integration along the fiber).

ExampleCohomology of $\mathbb{CP}^n$ via Gysin sequence

The unit circle bundle $S^1 \to S^{2n+1} \to \mathbb{CP}^n$ is the sphere bundle of the tautological line bundle $\gamma$ over $\mathbb{CP}^n$ . The Gysin sequence with the Euler class $e(\gamma) = x \in H^2(\mathbb{CP}^n)$ gives: $0 \to H^{2k}(\mathbb{CP}^n) \xrightarrow{\smile x} H^{2k+2}(\mathbb{CP}^n) \to H^{2k+2}(S^{2n+1}) \to \cdots$ Since $H^*(S^{2n+1})$ is known, induction shows $H^{2k}(\mathbb{CP}^n) \cong \mathbb{Z}$ generated by $x^k$ for $0 \leq k \leq n$ .

RemarkConnection to the Gauss-Bonnet theorem

For the tangent bundle of a closed oriented even-dimensional manifold $M^{2n}$ , the Euler class satisfies $\langle e(TM), [M] \rangle = \chi(M)$ , the Euler characteristic. When combined with Chern-Weil theory, this yields the generalized Gauss-Bonnet theorem: $\chi(M) = \int_M \operatorname{Pf}(\Omega / 2\pi)$ , where $\operatorname{Pf}$ is the Pfaffian of the curvature form.