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Register a new userUnfolded Seiberg-Witten Floer spectra, II: Relative invariants and the gluing theorem
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We use the construction of unfolded Seiberg-Witten Floer spectra of general 3-manifolds defined in our previous paper to extend the notion of relative Bauer-Furuta invariants to general 4-manifolds with boundary. One of the main purposes of this paper is to give a detailed proof of the gluing theorem for the relative invariants.
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Let $Y$ be a closed and oriented $3$-manifold. We define differe
We construct a generalization of the Seiberg-Witten Floer spectrum for suitable three-manifolds $Y$ with $b_1(Y)>0$. For a cobordism between three-manifolds we define Bauer-Furuta maps on these new spectra, and additionally compute some examples.
The moduli space of stable pairs on a local surface $X=K_S$ is in general non-compact. The action of $mathbb{C}^*$ on the fibres of $X$ induces an action on the moduli space and the stable pair invariants of $X$ are defined by the virtual localization formula. We study the contribution to these invariants of stable pairs (scheme theoretically) supported in the zero section $S subset X$. Sometimes there are no other contributions, e.g. when the curve class $beta$ is irreducible. We relate these surface stable pair invariants to the Poincare invariants of Durr-Kabanov-Okonek. The latter are equal to the Seiberg-Witten invariants of $S$ by work of Durr-Kabanov-Okonek and Chang-Kiem. We give two applications of our result. (1) For irreducible curve classes the GW/PT correspondence for $X = K_S$ implies Taubes GW/SW correspondence for $S$. (2) When $p_g(S) = 0$, the difference of surface stable pair invariants in class $beta$ and $K_S - beta$ is a universal topological expression.
We give the definition of the Seiberg-Witten-Floer homology group for a homology 3-sphere. Its Euler characteristic number is a Casson-type invariant. For a four-manifold with boundary a homology sphere, a relative Seiberg-Witten invariant is defined taking values in the Seiberg-Witten-Floer homology group, these relative Seiberg-Witten invariants are applied to certain homology spheres bounding Stein surfaces.
We study the Seiberg-Witten invariant $lambda_{rm{SW}} (X)$ of smooth spin $4$-manifolds $X$ with integral homology of $S^1times S^3$ 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 to obstruct existence of metrics of positive scalar curvature on certain 4-manifolds, and to exhibit new classes of integral homology $3$-spheres of Rohlin invariant one which have infinite order in the homology cobordism group.
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