On the inertia groups of fibre bundles

Abstract:

A subgroup $\tilde I(M \times {S^i})$ of the inertia group $I(M \times {S^i})$ is defined and shown to lie in $I(C)$ for every fibre bundle ${M^n} \to C \to {N^i}$. For certain $M$, examples of nontrivial elements in $\tilde I(M \times {S^i})$ are constructed using the $\tau$-pairing of Milnor-Munkres-Novikov. For compact mapping tori ${M_g}$ it is shown that $I({M_g}) = I(M \times {S^1})$ if ${\pi _1}M$ is finite and ${\text {Wh}}({\pi _1}M) = 0$.