Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 16 / 3

For an open subset $U\subseteq B$, let $E_U={\pi }^{-1}(U)$ and define the degree-preserving module mapUsing the prescribed fiber basis identifies its domain with the tensor product in the question. We prove that ${\Phi }_B$ is an isomorphism, giving the Leray-Hirsch theorem in this setting.

First suppose $U$ is a trivializing patch. The cohomological Künneth theorem over a field identifies $H^{\ast }(U\times F;\mathbb{Q})$ with the finite free graded $H^{\ast }(U;\mathbb{Q})$-module on a homogeneous basis of $H^{\ast }(F;\mathbb{Q})$. The required finiteness is supplied by the finite list $c_1,\dots ,c_v$ spanning the fiber's total cohomology. Express the restrictions of the $c_j$ in this product basis. A coefficient multiplying a fiber basis element of degree $k_i$ belongs to $H^{k_j-k_i}(U;\mathbb{Q})$; thus no coefficient can increase fiber degree. The degree-zero blocks are invertible on every path component, because restriction at every point gives a basis. Their inverses are again degree-zero cohomology classes.

After inverting those blocks, the remaining change-of-basis map is $I+N$, where $N$ strictly lowers fiber degree. There are only finitely many fiber degrees, so $N$ is nilpotent and $(I+N{)}^{-1}=I-N+N^2-\cdots$ is a finite sum. This proves the local basis criterion for Leray-Hirsch classes and therefore the isomorphism on every trivializing patch, without requiring the patch to be contractible.

Now use a finite trivializing open cover $U_1,\dots ,U_r$, supplied by compactness of $B$. Induct on $r$. Put $U=U_1\cup \cdots \cup U_{r-1}$ and $V=U_r$. The intersection $U\cap V$ is covered by the at most $r-1$ trivializing open sets $U_i\cap V$, so the induction hypothesis applies to $U$, $V$ and $U\cap V$.

Take the direct sum of shifted Mayer–Vietoris sequences for the base and compare it with the sequence for $E_U\cup E_V$. Restriction commutes with multiplication by each global cohomology class $c_j$. The connecting homomorphism also commutes: multiply the cochain representatives on the right by the restriction of a global cocycle representing $c_j$, and use $d(b⌣c_j)=db⌣c_j$. Consequently the maps $\Phi$ form a morphism of exact sequences. The Five lemma makes ${\Phi }_{U\cup V}$ an isomorphism, completing the finite-cover proof of the Leray-Hirsch theorem.

This is an additive graded module isomorphism. It does not assert that the fiber classes have the same multiplicative relations in the total space.