On the inertia groups of fibre bundles

Abstract

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