Heegaard Splittings and Seiberg-Witten monopoles

This paper presents the construction of the Seiberg-Witten-Floer homology of three-manifolds with non-trivial rational homology, and some properties of the invariant of three-manifolds obtained by computing the Euler characteristic. This new version corrects a mistake and some misprints that appear in the published version (Internat. J. Math., Vol.7, N.5 (1996) 671-696). Various sections are rewritten in more detail.

Let M be a closed, connected and oriented 3-manifold. This article is the first of a five part series that constructs an isomorphism between the Heegaard Floer homology groups of M and the corresponding Seiberg-Witten Floer homology groups of M.

We show that monopole Floer homology (as defined by Kronheimer and Mrowka) is isomorphic to the S^1-equivariant homology of the Seiberg-Witten Floer spectrum constructed by the second author.

This is the author's PhD thesis, as submitted to the Princeton University. The results of this paper have already appeared in arXiv:math/0607777v4, arXiv:math/0607691 and arXiv:0901.2156.

These are lecture notes from a series of lectures at the SMF summer school on "Geometric and Quantum Topology in Dimension 3", June 2014. The focus is on Heegaard Floer homology from the perspective of sutured Floer homology.

This is the fourth of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold. The isomorphism is given as a composition of three isomorphisms; the first of these relates a version of embedded contact homology on an an auxillary manifold to the Heegaard Floer homology on the original. The second isomorphism relates the relevant version of the embedded contact homology on the au...

We study the Seiberg-Witten invariant $\lambda_{\rm{SW}} (X)$ of smooth spin $4$-manifolds $X$ with integral homology of $S^1\times S^3$ defined by Mrowka, Ruberman, and Saveliev as a signed count of irreducible monopoles amended by an index-theoretic correction term. We prove a splitting formula for this invariant in terms of the Fr{\o}yshov invariant $h(X)$ and a certain Lefschetz number in the reduced monopole Floer homology of Kronheimer and Mrowka. We apply this formula ...

This is the last of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold.

To an integral homology 3-sphere $Y$, we assign a well-defined $\Z$-graded (monopole) homology $MH_*(Y, I_{\e}(\T; \e_0))$ whose construction in principle follows from the instanton Floer theory with the dependence of the spectral flow $I_{\e}(\T; \e_0)$, where $\T$ is the unique U(1)-reducible monopole of the Seiberg-Witten equation on $Y$ and $\e_0$ is a reference perturbation datum. The definition uses the moduli space of monopoles on $Y \x \R$ introduced by Seiberg-Witten...

The earlier article tried to construct an algorithm to compute the Heegaard Floer homology \hat{HF}(Y) for a 3-manifold Y. However there is an error in a proof which the author, as of now, is unable to fix.