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

Frøyshov, Kim A.

Duke Math. J. 155, No. 3, 519-576 (2010).

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 \(Y\).

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)

Keywords:

monopole; Floer homology; 3-manifold; gauge theory; Seiberg-Witten

References:

[1]	D. M. Austin and P. J. Braam, “Morse-Bott theory and equivariant cohomology” in The Floer Memorial Volume , Progr. Math. 133 , Birkhäuser, Basel, 1995, 123–183. · Zbl 0834.57017
[2]	A. Banyaga and D. Hurtubise, Lectures on Morse Homology , Kluwer Texts Math. Sci. 29 , Kluwer Acad. Publ., Dordrecht, 2004. · Zbl 1080.57001
[3]	Ch. Bär, Harmonic spinors for twisted Dirac operators, Math. Ann. 309 (1997), 225–246. · Zbl 0889.53025
[4]	S. K. Donaldson, Floer Homology Groups in Yang-Mills Theory , Cambridge Tracts in Math. 147 , Cambridge Univ. Press, Cambridge, 2002. · Zbl 0998.53057
[5]	N. D. Elkies, A characterization of the \(\z^n\) lattice , Math. Res. Lett. 2 (1995), 321–326. · Zbl 0855.11032
[6]	A. Floer, An instanton-invariant for 3-manifolds , Comm. Math. Phys. 118 (1988), 215–240. · Zbl 0684.53027
[7]	K. A. Frøyshov, The Seiberg-Witten equations and four-manifolds with boundary , Math. Res. Lett. 3 (1996), 373–390. · Zbl 0872.57024
[8]	-, Equivariant aspects of Yang-Mills Floer theory , Topology 41 (2002), 525–552. · Zbl 0999.57032
[9]	-, Compactness and Gluing Theory for Monopoles, Geom. Topol. Monogr. 15 , Geom. Topol. Publ., Coventry, 2008. · Zbl 1207.57044
[10]	M. Furuta and H. Ohta, Differentiable structures on punctured \(4\)-manifolds , Topology Appl. 51 (1993), 291–301. · Zbl 0799.57009
[11]	N. Hitchin, Harmonic spinors , Adv. Math. 14 (1974), 1–55. · Zbl 0284.58016
[12]	J. Kazdan, Unique continuation in geometry , Comm. Pure Appl. Math. 41 (1988), 667–681. · Zbl 0632.35015
[13]	P. Kronheimer and T. Mrowka, Monopoles and Three-Manifolds , New Math. Monogr. 10 , Cambridge Univ. Press, Cambridge, 2007. · Zbl 1158.57002
[14]	Y. Lim, The equivalence of Seiberg-Witten and Casson invariants for homology \(3\)-spheres , Math. Res. Lett. 6 (1999), 631–643. · Zbl 0948.57007
[15]	M. Marcolli and B.-L. Wang, Equivariant Seiberg-Witten Floer homology , Comm. Anal. Geom. 9 (2001), 451–639. · Zbl 1026.57025
[16]	Ch. Okonek and A. Teleman, “Seiberg-Witten invariants for \(4\)-manifolds with \(b_+=0\)” in Complex Analysis and Algebraic Geometry , Walter de Gruyter, Berlin, 2000, 347–357. · Zbl 1034.53091
[17]	P. Ozsváth and Z. Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary , Adv. Math. 173 (2003), 179–261. · Zbl 1025.57016
[18]	-, Holomorphic disks and topological invariants for closed three-manifolds , Ann. of Math. (2) 159 (2004), 1027–1158. JSTOR: · Zbl 1073.57009
[19]	J. Robbin and D. Salamon, The spectral flow and the Maslov index , Bull. Lond. Math. Soc. 27 (1995), 1–33. · Zbl 0859.58025
[20]	D. Ruberman and N. Saveliev, Rohlin’s invariant and gauge theory, II: Mapping tori , Geom. Topol. 8 (2004), 35–76. · Zbl 1063.57025
[21]	A. Teleman, Donaldson theory on non-Kählerian surfaces and class VII surfaces with \(b_2=1\) , Invent. Math. 162 (2005), 493–521. · Zbl 1093.32006