Lefschetz Theory on Fibre bundles via Gysin homomorphism

Abstract

For a pair of fibre preserving continuous functions $f,g:E_{1}\rightarrow E_{2}$ between two compact smooth fibre bundles over $B$, we construct a transfer map $T(f,g):H^{*}(B)\rightarrow H^{*}(B)$ that generalizes Lefschetz number $\lambda_{f,g}$ of the pair of maps. If the pair $(f,g)$ is smooth satisfying a transversality condition and $T(f,g)$ is non-zero, then there is a surjective submersion from any connected component of $\{x\mid f(x)=g(x)\}$ to $B$. This yields a necessary and sufficient condition for a principal $G$-bundle over a simply connected compact manifold to be trivial and we also get a necessary condition for every smooth map from $S^{2n+1}$ to $S^{1}$ for all $n\geq1$.