Seiberg-Witten Floer Spectra for $b_1 >0$

The Seiberg-Witten Floer spectrum is a stable homotopy refinement of the monopole Floer homology of Kronheimer and Mrowka. The Seiberg-Witten Floer spectrum was defined by Manolescu for closed, \operatorname{spin}^c 3-manifolds with b_1 = 0 in an S^1-equivariant stable homotopy category and has been producing interesting topological applications. Lidman and Manolescu showed that the S^1-equivariant homology of the spectrum is isomorphic to the monopole Floer homology.

For closed \operatorname{spin}^c 3-manifolds Y with b_1(Y) > 0, there are analytic and homotopy-theoretic difficulties in defining the Seiberg-Witten Floer spectrum. In this memoir, we address the difficulties and construct the Seiberg-Witten Floer spectrum for Y, provided that the first Chern class of the \operatorname{spin}^c structure is torsion and that the triple-cup product on H^1(Y;\mathbb{Z}) vanishes. We conjecture that its S^1-equivariant homology is isomorphic to the monopole Floer homology.

For a 4-dimensional \operatorname{spin}^c cobordism X between Y_0 and Y_1, we define the Bauer-Furuta map on these new spectra of Y_0 and Y_1, which is conjecturally a refinement of the relative Seiberg-Witten invariant of X. As an application, for a compact spin 4-manifold X with boundary Y, we prove a \frac{10}{8}-type inequality for X which is written in terms of the intersection form of X and an invariant \kappa(Y) of Y.

In addition, we compute the Seiberg-Witten Floer spectrum for some 3-manifolds.