Bauer-Furuta Invariants and Galois Symmetries

Abstract

The Bauer-Furuta invariants of smooth 4-manifolds are investigated from a functorial point of view. This leads to a definition of equivariant Bauer-Furuta invariants for compact Lie group actions. These are studied in Galois covering situations. We show that the ordinary invariants of all quotients are determined by the equivariant invariants of the covering manifold. In the case where the Bauer-Furuta invariants can be identified with the Seiberg-Witten invariants, this implies relations between the invariants in Galois covering situations, and these can be illustrated through elliptic surfaces. It is also explained that the equivariant Bauer-Furuta invariants potentially contain more information than the ordinary invariants.

Characteristic cohomotopy classes for families of 4-manifolds

“…No further ideas are needed for that. In this sense the present paper conceptionally complements [21].…”

“…If a compact Lie group G acts on X preserving a complex spin structure σ X , there is an extension G of G by T such that the homomorphism G → Diff(X) lifts to a homomorphism G → Spiff c (X, σ X ). In [21] there has been constructed an equivariant invariant which lives in π 0 G (S λ ) and maps to the Bauer-Furuta invariant under the forgetful map π 0 G (S λ ) → π 0 T (S λ ), and I will now explain how this relates to the present construction.…”

“…Section 6 illustrates how the characteristic cohomotopy classes for mapping tori may be used to define an invariant of isotopy classes of diffeomorphisms of 4-manifolds. The final Section 7 comments on the relationship of the family invariants with the equivariant invariants of [21].…”

Constraints on families of smooth 4-manifolds from Bauer-Furuta invariants

“…We assume that G preserves the isomorphism class of a spin c -structure s but does not necessarily lift to a G-action on the spinor bundles. The G-equivariant Bauer-Furuta invariant was constructed in [14] and we refer the reader to [14] for the details of the construction.…”

Mod $p$ Equality Theorem for Seiberg-Witten Invariants under ${\mathbb Z}_p$-actions

“…In several cases, one can relate the Seiberg-Witten invariants of a 4-manifold X with an action of a group G to those of its quotient (V -)manifold X/G. In fact, in the case of free actions of prime order cyclic groups G = Z p , it is proved that the Seiberg-Witten invariant of X is equal modulo p to a sum of invariants of X/G, by Ruan-Wang [11], Szymik [12] and the author [7]. This mod p equality theorem is extended to the case of double branched coverings by Ruan-Wang [11], B. D. Park [9] and Cho-Hong [2].…”