Cohomotopy invariants and the universal cohomotopy invariant jump formula

The first stable-homotopy Seiberg-Witten invariants have been introduced independently by M. Furuta and S. Bauer. Furuta first used “finite dimensional approximations” of the monopole map in his work on the 11/8 conjecture [Fu1], and then introduced a class of refined Seiberg-Witten invariants (called “stable homotopy version of the Seiberg-Witten invariants”) in a geometric, non-formalized way in [Fu2]. In this preprint Furuta acknowledges independent work by Bauer [B3]. According to Furuta, the new invariants belong to a certain inductive limit of sets of homotopy classes of maps associated with “finite dimensional approximations” of the Seiberg-Witten map. The structure and the functorial properties of this inductive limit (with respect to diffeomorphisms between 4-manifolds) have not been worked out in this article. A precise version of the new invariants has been introduced later by Bauer-Furuta in [BF]: the Bauer-Furuta classes belong to certain stable cohomotopy groups associated with a presentation (E, F ) of the K-theory element ind( D) defined by a fixed Spin c -structure. This element ind( D) belongs to the K-theory group K (B), where B = H 1 (X; R)/H 1 (X; Z) is the Picard group of the base manifold X.

In this article we propose a different construction of cohomotopy invariants which has the following advantages: Our construction yields a larger class of invariants, which are well defined in all cases, are always finer than the classical invariants, and have clear functorial properties. In order to explain the advantages of the new formalism in a non-technical way, we consider again the Seiberg-Witten case.

It is well known that the Seiberg-Witten map µ can be regarded as an S 1equivariant bundle map between Hilbert bundles over the torus B (see [BF] and section 3.4 of this article). We first choose the perturbing form in the second Seiberg-Witten equation in the “bad way”, i.e. such that the equations have reducible solutions (solutions with trivial spinor component); we make this “bad choice” even in the case b + (X) > 1! In “classical” Seiberg-Witten theory one perturbs the second Seiberg-Witten equation using a nontrivial self-dual harmonic form κ ∈ iH + \ {0}, and gets a new map µ κ which defines a moduli space which does not contain reductions. Instead of a constant perturbation κ, we consider a map κ : B → iH + {0}, and perturb the Seiberg-Witten map µ (regarded as bundle map over B) using this map. The associated invariant will depend on the homotopy class [κ] ∈ [B, S(iH + )]. This leads to the following questions:

(1) Does one obtain new invariants in this way?

(2) If so, does one have a universal cohomotopy invariant jump formula, i.e. a formula which describes the jump of the cohomotopy invariant when one passes from one homotopy class to another? (3) Use again constant perturbation forms κ, but let κ vary in the sphere S(iH + ) and regard the obtained map μ as an S 1 -equivariant bundle map over the larger basis B × S(iH + ). Does this universal perturbation μ yield more differential topological information than the individual perturbations µ κ ? If not, express the cohomotopy invariant associated with μ in terms of the invariant associated with µ κ and topological invariants of X.

These questions are interesting as soon as b 1 ≥ b + -1 (even for b + > 1!) and they are also interesting for the classical invariant, because for non-constant perturbations κ one gets new Seiberg-Witten type moduli spaces. The universal wall-crossing formula [LL], [OO], [OT] for the full Seiberg-Witten invariant 1 should be a formal consequence of a universal cohomotopy invariant jump formula. These questions will be completely answered in this article.

Another motivation for proposing a new formalism was the need to have well defined invariants, with clear functorial properties. Recall that the classical full Seiberg-Witten invariant can be regarded as an element of [∧ * H 1 (X, Z)] Spin c (X) , where Spin c (X) denotes the torsor of equivalence classes of Spin c -structures. Therefore this invariant belongs to a group which is obviously functorial with respect to pairs (h, θ) consisting of an orientation preserving homotopy equivalence h : X → X ′ , and a bijection θ : Spin c (X ′ ) → Spin c (X) which is compatible with the Chern class maps Spin c (X) → H 2 (X, Z), Spin c (X ′ ) → H 2 (X ′ , Z) and the H 2 (X, Z), H 2 (X ′ , Z)-actions on the two sets. Such a pair will be called a Spin c -homotopy equivalence. We will say that an assignement X → G(X) ∈ Ab is topologically functorial on the category of smooth 4-manifold

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