Rigidity of the mod 2 families Seiberg-Witten invariants and topology of families of spin 4-manifolds

Abstract

We show a rigidity theorem for the Seiberg-Witten invariants mod 2 for families of spin 4-manifolds. We also give a family version of 10/8-type inequality using this rigidity theorem. As applications, we shall give a new series of non-smoothable topological actions on some 4-manifolds, and also prove the existence of a non-smoothable topological family of 4-manifolds whose fiber, base space, and total space are smoothable as manifolds. As a consequence, it follows that the inclusion map Diff(M ) ֒→ Homeo(M ) is not a weak homotopy equivalence for a smooth 4-manifold M which is homeomorphic to K3#nS 2 × S 2 with 0 ≤ n ≤ 3.

On the Bauer-Furuta and Seiberg-Witten invariants of families of $4$-manifolds

“…(2) For comparison, see Theorem 1.4 of [10], in which other non-smoothable families have been detected by a different technique based on 10/8-type inequalities.…”

“…(3) The proofs that E satisfies the first and second conditions are based on [10] and [17] respectively. To prove that E satisfies the third condition, we shall use Corollary 1.3.…”

“…(1) By the long exact sequence of homotopy groups it follows that π 0 (Dif f (K3)) → π 0 (Homeo(K3)) is not injective or π 1 (Dif f (K3)) → π 1 (Homeo(K3)) is not surjective, although it remains an open problem to determine which (if not both) of these statements hold. (2) It is interesting to compare Corollary 1.6 with Theorem 1.9 of [10], which states that…”

“…This helped the authors to consider its families version. After the version 1 of this paper had appeared on arXiv, the second author uploaded the paper [10] with Tsuyoshi Kato and Nobuhiro Nakamura on arXiv. The work on [10] and a comment by Nakamura on the version 1 of this paper prompted the authors to conceive the idea of Theorem 1.4 and Corollary 1.6, which are added after [10]…”

A note on exotic families of 4-manifolds

“…In the previous paper [3] we showed that there exists a smooth manifold which is realized as the total space of a non-smoothable topological fiber bundle (i.e. a bundle with structure group the homeomorphism group which does not admit a reduction to the diffeomorphism group) with a 4-dimensional fiber.…”