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, ${\mathrm{spin}}^c$ $3$-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 ${\mathrm{spin}}^c$ $3$-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 ${\mathrm{spin}}^c$ 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 ${\mathrm{spin}}^c$ 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.