Instantons mod 2 and indefinite 4-manifolds

This talk is on work in preparation with Ali Daemi.

I will explain why Kim Froyshov’s mod 2 instanton invariant q_3 gives information about indefinite 4-manifolds: if H_1(W;Z/2) = 0 and W has boundary Y, then -b^+(W) <= q_3(Y) <= b^-(W). This is the first invariant known to enjoy comparable bounds for indefinite manifolds W.

The key observation is that even when b^+(W) > 0, one can define a Donaldson invariant in the (tilde) instanton homology of boundary(W) — not in the usual instanton tilde complex, but rather a “suspension”. This suspension process accounts for the role of obstructed gluing theory, and does not destroy information about q_3 (but does destroy information about all other types of h-invariant).

As a corollary, we show that there exist integer homology spheres with arbitrary integral surgery number S(Y_n) = n. This answers a question of Dave Auckly. Previously, the state of the art was n=2, and further progress was obstructed by the possibility that the linking matrix be indefinite.