We obtain constraints on the topology of families of smooth
–manifolds
arising from a finite-dimensional approximation of the families Seiberg–Witten
monopole map. Amongst other results these constraints include a families
generalisation of Donaldson’s diagonalisation theorem and Furuta’s
theorem. As an application we construct examples of continuous
–actions, for
any odd prime
,
which cannot be realised smoothly. As a second application we show that the
inclusion of the group of diffeomorphisms into the group of homeomorphisms is not a
weak homotopy equivalence for any compact, smooth, simply connected, indefinite
–manifold with signature of
absolute value greater than
.
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