Seiberg-Witten-Floer homology and the geometric structure \(\mathbb R\times H^{2}\) (Q1017354)

Let \(M\) be a closed oriented \(3\)-manifold carrying the geometric structure \(\mathbb R\times H^{2}\) with the (anti-) canonical line bundle \(L=K_{M}^{\pm1}\). When \(b_{1}(M)>1\), the author and M. Itoh showed in a previous paper [\textit{M. Itoh} and \textit{T. Yamase}, Hokkaido Math. J. 38, No.~1, 67--81 (2009; Zbl 1177.57026)] that the first Chern class \(\alpha=c_{1}(K_{M}^{\pm1})\) is a monopole class and the associated Seiberg-Witten invariant equals \(\pm 1\). In the paper under review, T. Yamase computes the Seiberg-Witten-Floer homology \(HF_{k}(M,L)\), under the same hypothesis. The latter was introduced by \textit{A. Floer} in [Commun. Math. Phys. 118, No.~2, 215--240 (1988; Zbl 0684.53027)]. It can be viewed as a generalization of the Seiberg-Witten invariant since this invariant equals the Euler characteristic of the Seiberg-Witten-Floer homology.