A new gauge slice for the relative Bauer–Furuta invariants

Abstract

Abstract. In this paper, we study Manolescu's construction of the relative Bauer-Furuta invariants arising from the Seiberg-Witten equations on 4-manifolds with boundary. The main goal of this paper is to introduce a new gauge fixing condition in order to apply the finite dimensional approximation technique. We also hope to provide a framework to extend Manolescu's construction to general 4-manifolds.

Pin(2)-Equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture

“…The irreducibles come in k pairs related by the action of j ∈ G. Each irreducible is connected to the reducible by a single flow line. From (29), (30) we deduce that there is a long exact sequences on Borel homology: [1] denotes the respective degree. The same discussion as in [37, Section 7.2] shows that the connecting map from (F k 2 ) [1] to H 0 (BG; F 2 ) ∼ = F 2 must be nontrivial.…”

“…Let W be a compact four-manifold with boundary Y , such that b 1 (Y ) = 0. Suppose we have a spin structure t on W whose restriction to Y is s. Following [36, Section 9] (as corrected by Khandhawit in [29]), we can do finite dimensional approximation for the Seiberg-Witten equations on W with suitable boundary conditions.…”

“…Pick a Riemannian metric g on W so that the boundary has a neighborhood isometric to [0, 1] × Y . Let S + and S − denote the two spinor bundles on W , and Ω 1 g (W ) denote the space of one-forms on W in double Coulomb gauge, as in [29,Definition 1]. For a fixed cut-off ν ≫ 0, we can define a Seiberg-Witten map for W :…”

“…(They also used some nonstandard connections instead of the Levi-Civita connections, but finite-dimensional approximation still works in this setting, and yields the same answers.) One then uses the attractor-repeller sequences (29), (30) to obtain information about the Floer spectrum. The same methods as in [37,Section 7.2] can be employed to compute the G-equivariant Seiberg-Witten Floer homology; we just have to keep track of the additional symmetry when describing the Seiberg-Witten flow.…”

Low-regularity Seiberg-Witten moduli spaces on manifolds with boundary

“…Firstly, we choose a gauge twisting τ (Definition 3.24) and define the (S 1 -equivariant) twisted restriction map R τ : M • s (X, ŝ) → C cC (∂X, s), taking values in the configurations on ∂X in the Coulomb slice (Definition 3.2). The gauge splittings and gauge twistings we introduce generalize the double Coulomb slice used in [Lip08,Kha] and twistings utilized in [KLS18]. We prove regularity, denseness and "semi-infinite-dimensionality" of the Seiberg-Witten moduli spaces:…”