Seiberg-Witten Floer spectra for $b_1>0$

Abstract

We construct a generalization of the Seiberg-Witten Floer spectrum for suitable three-manifolds Y with b1pY q ą 0. For a cobordism between three-manifolds we define Bauer-Furuta maps on these new spectra, and additionally compute some examples. HIROFUMI SASAHIRA AND MATTHEW STOFFREGEN 7.1. The Unstable parameterized homotopy category 100 7.2. The Parameterized Conley Index 104 7.3. Spectra 107 8. Afterword: Finite-dimensional Approximation in other settings 108 References 109Theorem 1.2. Let E be a (possibly S 1 -equivariant) complex-oriented (resp. S 1 -equivariantly complex oriented) cohomology theory. Then E ˚´npY,s,Pq pSWF u pY, s, Pqq is (canonically) independent of P.In particular, the complex-cobordism theories FMU ˚pY, sq " Ą M U ˚´npY,s,g,Pq pSWF u pY, s, Pqq,pSWF u pY, s, Pqq, are invariants of the pair pY, sq, which we call the Floer (equivariant) complex cobordism of pY, sq.c j e j `ÿ µnăη j ďµn`δ c j e j `ÿ µn`δăη j c j e j .Then π n φ " ÿ η j ďµn c j e j `ÿ µnăη j ďµn`δ µnăηpďµn`δ c j α jp e p .

Positive scalar curvature and homology cobordism invariants

“…by [5, Proposition 3.10], which is shown considering a certain ordinary equation [5,Lemma 3.11] corresponding to the cylinder (−∞, ∞) × 𝑌 appearing the neck stretching of 𝑀 1 ∪ 𝑌 𝑀 2 . Now we have checked (36) and this completes the proof. □…”

“…In this paper, we developed Seiberg-Witten theory for 4-manifolds with periodic ends to prove Theorem 1.1. But we expect that an alternative proof of Theorem 1.1 without using Seiberg-Witten theory for 4-manifolds with periodic ends could be given by using Schoen-Yau's argument [39] combined with a kind of gluing theorems for relative Bauer-Furuta invariants [16,17,36].…”

Diffeomorphisms of 4-manifolds with boundary and exotic embeddings

“…• What kind of gauge-theoretic invariants of diffeomorphisms in the case of 4-manifold with boundary should be considered? One possibility might be to use families Floer theoretic relative invariants such as those in [56], but they are hard to compute in general.…”

A knot Floer stable homotopy type

“…In [6], Cohen, Jones, and Segal proposed the problem of lifting Floer homology to a Floer spectrum or pro-spectrum, in the sense of stable homotopy theory. Since then, stable homotopy refinements of Floer homology have been constructed in Seiberg-Witten theory [19,12,36] and symplectic geometry [7,15,1]. In a similar vein, there is a lift of Khovanov homology to a stable homotopy type [18,17].…”