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**Monopole Floer homology for rational homology 3-spheres.**
*(English)*
Zbl 1237.57033

Adapting Floer’s original ideas to monopoles, the author gives a new construction of monopole Floer homology for \(\text{spin}^c\) rational homology 3-spheres. It seems likely that these are isomorphic to the groups of Kronheimer and Mrowka, when differences in grading conventions are taken into account.

The starting point is the “irreducible” Floer cohomology \(HF^*(Y,m)\), where the parameter \(m\) runs through a set \(\mathfrak{m}(Y)\) of the form \(m_0 + \mathbb{Z}\) with \(m_0 \in \mathbb{Q}\) and indicates which chamber is being used for the metric and perturbation. By taking suitable limits of \(HF^*(Y,m)\) as \(m \to \pm\infty\), the author obtains invariants \(\overline{HF}^*(Y)\) and \(\underline{HF}^*(Y)\) of the \(\text{spin}^c\) 3-manifold \(Y\). These “equivariant” Floer cohomology groups are modules over a polynomial ring and are related by a long exact sequence involving a third module which is essentially the module of Laurent polynomials. Exactness is established by geometric means, without recourse to homological algebra.

In the case of coefficients in a field \(\mathbb{F}\) of characteristic \(p\), the author gives a precise description of how \(HF^*(Y,m;\mathbb{F})\) depends on \(m\) in terms of an invariant \(h_p(Y) \in \mathfrak{m}(Y)\), which can be read off from the above exact sequence.

Let \(X\) be a \(\text{spin}^c\) smooth closed 4-manifold with \(b_1(X) = 1\) which contains a nonseparating smoothly embedded rational homology 3-sphere \(Y\). In the case \(b^+(X) > 1\), the author expresses the Seiberg-Witten invariant of \(X\) as the Lefshetz number of a certain endomorphism of the reduced Floer cohomology of \(Y\). This endomorphism is obtained by cutting \(X\) open along \(Y\), and considering the corresponding cobordism map. If \(b^+(X) = 0\), then, for certain \(\text{spin}^c\) structures on \(X\), this Lefshetz number yields an invariant of \(X\) (i.e., is independent of \(Y\)). If in addition \(Y\) is an integral homology sphere or \(b_2(X) = 0\), then \(h_p(Y)\) is an invariant of \((X,e)\), where \(e\) denotes the generator of \(H_3(X;\mathbb{Z})\) represented by \(Y\).

The starting point is the “irreducible” Floer cohomology \(HF^*(Y,m)\), where the parameter \(m\) runs through a set \(\mathfrak{m}(Y)\) of the form \(m_0 + \mathbb{Z}\) with \(m_0 \in \mathbb{Q}\) and indicates which chamber is being used for the metric and perturbation. By taking suitable limits of \(HF^*(Y,m)\) as \(m \to \pm\infty\), the author obtains invariants \(\overline{HF}^*(Y)\) and \(\underline{HF}^*(Y)\) of the \(\text{spin}^c\) 3-manifold \(Y\). These “equivariant” Floer cohomology groups are modules over a polynomial ring and are related by a long exact sequence involving a third module which is essentially the module of Laurent polynomials. Exactness is established by geometric means, without recourse to homological algebra.

In the case of coefficients in a field \(\mathbb{F}\) of characteristic \(p\), the author gives a precise description of how \(HF^*(Y,m;\mathbb{F})\) depends on \(m\) in terms of an invariant \(h_p(Y) \in \mathfrak{m}(Y)\), which can be read off from the above exact sequence.

Let \(X\) be a \(\text{spin}^c\) smooth closed 4-manifold with \(b_1(X) = 1\) which contains a nonseparating smoothly embedded rational homology 3-sphere \(Y\). In the case \(b^+(X) > 1\), the author expresses the Seiberg-Witten invariant of \(X\) as the Lefshetz number of a certain endomorphism of the reduced Floer cohomology of \(Y\). This endomorphism is obtained by cutting \(X\) open along \(Y\), and considering the corresponding cobordism map. If \(b^+(X) = 0\), then, for certain \(\text{spin}^c\) structures on \(X\), this Lefshetz number yields an invariant of \(X\) (i.e., is independent of \(Y\)). If in addition \(Y\) is an integral homology sphere or \(b_2(X) = 0\), then \(h_p(Y)\) is an invariant of \((X,e)\), where \(e\) denotes the generator of \(H_3(X;\mathbb{Z})\) represented by \(Y\).

Reviewer: A. Juhasz (Cambridge)

### MSC:

57R58 | Floer homology |

57R57 | Applications of global analysis to structures on manifolds |

57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |

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\textit{K. A. Frøyshov}, Duke Math. J. 155, No. 3, 519--576 (2010; Zbl 1237.57033)

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