“Pontrjagin-Thom construction in non-linear Fredholm theories”
Mikio Furuta, University of Tokyo

Abstract:
For a single oriented closed 4-manifold X, we have gauge theoretic invariants of X defined by using the Anti-Self-Dual equation and the Seiberg-Witten equations. In both theories, under some compactness property, the invariants are defined as the fundamental classes of moduli spaces which are elements of homology groups of some configuration spaces C.

Atiyah and Singer introduced the notion of a manifold over T as a fiber bundle over T with fiber a manifold X and structure group Diff(X). When X is a closed oriented 4-manifold we would like to define gauge theoretical invariants for such a family. For a continuous map f:T' ——> T we naturally have a pullback family over T'. Therefore the groups in which the expected invariants take values behave like cohomology with respect to the base space T, and like homology with respect to the configuration space C.

Recall that we have a mixture of homology and cohomology for each generalized (co)homology theory. In this talk we explain and enhance such a mixture for the stable (co)homotopy theory, from which we are able to define new gauge theoretical invariants as long as the family of moduli spaces over T is compact. Our main tool is the Pontrjagin-Thom construction with the following two modifications.
(1) We introduce twisting in stable framing, which is defined by using a family of linear Fredholm operators.
(2) We also need a generalization of the above mentioned notion of a manifold over T to include the parametrized moduli spaces which might have some singularities.

For the Seiberg-Witten theory, our invariant is related to the Bauer-Furuta invariant by a version of S-duality.

So far the validity of our construction for the Donaldson theory is still limited because we need compactness of the moduli spaces. However, we do recover the 2 torsion invariants of Fintushel and Stern. In addition, if the noncompactness of the family of moduli spaces is caused only by bubbles with instanton number 1, then our construction applies to a refinement of Uhlenbeck compactification of the family of moduli spaces.

Formally our construction should also be valid for Gromov-Witten theory. In fact, we can also recover the bordism GW invariant of McDuff.

This is a joint work with Tian-Jun Li.