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, the authors address the difficulties and construct the Seiberg–Witten Floer spectrum for Y, provided that the first Chern class of the spinc structure is torsion and that the triple-cup product on \(H^{1}(Y; \mathbb{Z})\) vanishes. The authors 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 \(Y_{1}\), the authors 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 \(Y\), the authors 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, the authors compute the Seiberg–Witten Floer spectrum for some 3-manifolds.