Exotic diffeomorphisms of reducible $4$-manifolds with odd $b_+$

A diffeomorphism of a $4$-manifold is said to be exotic if it is continuously isotopic to the identity but not smoothly isotopic to the identity.

Ruberman constructed the first examples of exotic diffeomorphisms on simply-connected closed $4$-manifolds.

His examples were reducible $4$-manifolds that necessarily have even $b_+$ in order that they can be detected by the families Seiberg--Witten or Donaldson invariants.

Later Konno and Baraglia produced exotic diffeomorphisms on irreducible $4$-manifolds with odd $b_+$.

In this paper, we will construct exotic diffeomorphisms on reducible $4$-manifolds with odd $b_+$.

Exoticness is detected using a families Bauer--Furuta invariant.

In proving our results we need to work with families moduli spaces which are not framed and so do not give rise to framed cobordism invariants.

We overcome this difficulty by considering a Bauer--Furuta type invariant valued in {\em pin-cobordism}.

In addition to constructing exotic diffeomorphisms, we also find new examples of simply-connected $4$-manifolds whose mapping class groups are not finitely generated.