Realising automorphisms of the extended intersection form by diffeomorphisms

Abstract

Suppose that $f : M \rightarrow B$ is a normal $(q-1)$-smoothing of a $2q$-manifold $M$ over some $(B,\xi)$, where $q$ is even and $B$ is simply-connected. A diffeomorphism of $M$ over $B$ induces an automorphism of the extended intersection form (or ``Q-form") of $f$, which consists of the intersection form of $M$ and the induced homomorphism $H_q(f)$. Assuming that $H_q(B)$ is free, we determine precisely which automorphisms can be realised by such diffeomorphisms. In particular, if $H_{q-1}(B)$ is also free, then we show that every automorphism can be realised. These results are obtained by studying a ``two-sided" version of the extended surgery obstruction, which was introduced in earlier work of the author. As an application, we show that for a complex $q$-dimensional complete intersection, with $q > 2$ even, every automorphism of the cohomology ring is realised by a diffeomorphism.