The SU(3)-Casson invariant for integral homology 3-spheres as studied by Boden-Herald possesses a 'spectral flow obstruction' to being an integer valued invariant which depends only on the non-degenerate (perturbed) moduli space of flat SU(3)-connections. This obstruction is the non-trivial spectral flow of a family of twisted signature operators in 3-dimensions. The parallel U(2)-Seiberg-Witten construction also has an obstruction but from the non-trivial spectral flow of a family of twisted Dirac operators. By taking the SU(3)-flat and U(2)-Seiberg-Witten equations simultaneously the obstructions can be made to cancel and an integer invariant is obtained.
Topological field theories and formulae of Casson and Meng-Taubes
本文的目的是给出Meng和Taubes的一个定理的新证明，该定理确定了具有Milnor挠的3-流形的Seiberg-Witten不变量。这里的观点将是拓扑量子场论的观点。特别地，我们将三维流形上的Seiberg-Witten方程与Riemann曲面上的Abel涡旋方程联系起来。这些技巧也给出了卡森不变量的外科手术公式的新证明，卡森不变量被解释为同调S^2 x S^1的不变量。
Casson-type invariants emerging from Donaldson theory over certain negative definite 4-manifolds were recently suggested by Andrei Teleman. These are defined by a count of a zero-dimensional moduli space of flat instantons. Motivated by the cobordism program of proving Witten's conjecture, we use a moduli space of PU(2) Seiberg-Witten monopoles to exhibit an oriented one-dimensional cobordism of the instanton moduli space to the empty space. The Casson-type invariant must therefore vanish.
We introduce a new class of perturbations of the Seiberg-Witten equations. Our perturbations offer flexibility in the way the Seiberg-Witten invariants are constructed and also shed a new light to LeBrun's curvature inequalities.
We prove a gluing formula for the families Seiberg-Witten invariants of families of $4$-manifolds obtained by fibrewise connected sum. Our formula expresses the families Seiberg-Witten invariants of such a connected sum family in terms of the ordinary Seiberg-Witten invariants of one of the summands, under certain assumptions on the families. We construct some variants of the families Seiberg-Witten invariants and prove the gluing formula also for these variants. One variant incorporates a twist of the families moduli space using the charge conjugation symmetry of the Seiberg-Witten equations.…
We show how the families Seiberg-Witten invariants of a family of smooth $4$-manifolds can be recovered from the families Bauer-Furuta invariant via a cohomological formula. We use this formula to deduce several properties of the families Seiberg-Witten invariants. We give a formula for the Steenrod squares of the families Seiberg-Witten invariants leading to a series of mod $2$ relations between these invariants and the Chern classes of the spin$^c$ index bundle of the family. As a result we discover a new aspect of the ordinary Seiberg-Witten invariants of a $4$-manifold $X$: they obstruct t…
We prove that, under a simple condition on the cohomology ring, every closed 4-manifold has mod 2 Seiberg-Witten simple type. This result shows that there exists a large class of topological 4-manifolds such that all smooth structures have mod 2 simple type, and yet some have non-vanishing (mod 2) Seiberg-Witten invariants. As corollaries, we obtain adjunction inequalities and show that, under a mild topological condition, every geometrically simply connected closed 4-manifold has the vanishing mod 2 Seiberg-Witten invariant for at least one orientation.
We introduce an invariant of tuples of commutative diffeomorphisms on a 4-manifold using families of Seiberg-Witten equations. This is a generalization of Ruberman's invariant of diffeomorphisms defined using 1-parameter families of Seiberg-Witten equations. Our invariant yields an application to the homotopy groups of the space of positive scalar curvature metrics on a 4-manifold. We also study the extension problem for families of 4-manifolds using our invariant.
We construct infinitely many examples of finite volume 4-manifolds with $T^3$ ends that do not admit any cusped asymptotically hyperbolic Einstein metrics yet satisfy a strict logarithmic version of the Hitchin-Thorpe inequality due to Dai-Wei. This is done by using estimates from Seiberg-Witten theory due to LeBrun as well as a method for constructing solutions to the Seiberg-Witten equations on noncompact manifolds due to Biquard. We also use constructions coming from the $Pin^-(2)$ monopole equations to obtain a larger class of manifolds where these techniques apply.
This is an expansion on my talk at the Geometry and Topology conference at McMaster University, May 2004. We outline a program to relate the Heegaard Floer homologies of Ozsvath-Szabo, and Seiberg-Witten-Floer homologies as defined by Kronheimer-Mrowka. The center-piece of this program is the construction of an intermediate version of Floer theory, which exhibits characteristics of both theories.
Let $M$ be a closed oriented $4$-manifold admitting a rank-$2$ oriented foliation with a metric of leafwise positive scalar curvature. If $b^+>1$, then we will show that the Seiberg-Witten invariant vanishes for all \spinc structures.
I show that flat PSL(2;R)-connections on three-manifolds satisfying certain 'stability condition' can be interpreted as solutions of the Seiberg-Witten equations with two spinors. This is used to construct explicit examples of the Seiberg-Witten moduli spaces. Also, I show that in this setting blow up sets satisfy certain non-trivial topological restrictions.
We prove an abstract compactness theorem for a family of generalized Seiberg-Witten equations in dimension three. This result recovers Taubes' compactness theorem for stable flat $\mathbf{P}\mathrm{SL}_2(\mathbf{C})$-connections as well as the compactness theorem for Seiberg-Witten equations with multiple spinors. Furthermore, this result implies a compactness theorem for the ADHM$_{1,2}$ Seiberg-Witten equation, which partially verifies a conjecture by Doan and Walpuski.
In this paper, we focus on the moduli space of Seiberg-Witten equation on non-compact manifold with periodic end. Suppose that the scalar curvature on the periodic end is identically zero and the topological conditions: the first de-Rham cohomology and the self-dual cohomology restricting on the periodic end vanish. Then, we will show that the moduli space of the perturbed Seiberg-Witten equation is compact.
I describe a relation (mostly conjectural) between the Seiberg-Witten monopoles, Fueter sections, and G2 instantons. In the last part of this article I gathered some open questions connected with this relation.
