The SU(3) Casson Invariant for Integral Homology 3-Spheres

Topological field theories and formulae of Casson and Meng-Taubes

The goal of this paper is to give a new proof of a theorem of Meng and Taubes that identifies the Seiberg-Witten invariants of 3-manifolds with Milnor torsion. The point of view here will be that of topological quantum field theory. In particular, we relate the Seiberg-Witten equations on a 3-manifold with the Abelian vortex equations on a Riemann surface. These techniques also give a new proof of the surgery formula for the Casson invariant, interpreted as an invariant of a homology S^2 x S^1.

等变塞弗理论与扰动的SU(n)卡索恩不变量

Equivariant Cerf theory and perturbative $SU(n)$ Casson invariants

本文通过等变塞弗理论研究扰动SU(n)连接模空间，证明了整数同调球上扰动SU(n)卡索恩不变量的存在性，并在n=4时给出显式公式。

Rohlin's invariant and gauge theory II. Mapping tori

This is the second in a series of papers studying the relationship between Rohlin's theorem and gauge theory. We discuss an invariant of a homology S^1 cross S^3 defined by Furuta and Ohta as an analogue of Casson's invariant for homology 3-spheres. Our main result is a calculation of the Furuta-Ohta invariant for the mapping torus of a finite-order diffeomorphism of a homology sphere. The answer is the equivariant Casson invariant (Collin-Saveliev 2001) if the action has fixed points, and a version of the Boyer-Nicas (1990) invariant if the action is free. We deduce, for finite-order mapping …

Gauge Theoretic Invariants of, Dehn Surgeries on Knots

New methods for computing a variety of gauge theoretic invariants for homology 3-spheres are developed. These invariants include the Chern-Simons invariants, the spectral flow of the odd signature operator, and the rho invariants of irreducible SU(2) representations. These quantities are calculated for flat SU(2) connections on homology 3-spheres obtained by 1/k Dehn surgery on (2,q) torus knots. The methods are then applied to compute the SU(3) gauge theoretic Casson invariant (introduced in [H U Boden and C M Herald, The SU(3) Casson invariant for integral homology 3--spheres, J. Diff. Geom.…

同调配边群不由图同调$3$-球面生成

The homology cobordism group is not generated by graph homology $3$-spheres

本文证明同调配边群不由图同调3-球面生成，方法结合滤过瞬子Floer理论与Gromov单纯体积。

Seiberg-Witten-Floer Theory for Homology 3-Spheres

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.

Monopole Floer homology for rational homology 3-spheres

We give a new construction of monopole Floer homology for spin-c rational homology 3-spheres. As applications we define two invariants of certain smooth compact 4-manifolds with b_1=1 and b^+=0.

A monopole homology of integral homology 3-spheres

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 in studying smooth 4-manifolds. We show that the monopole homology $MH_*(Y, I_{\e}(\T; \e_0))$ is invariant among Riema…

Dual submanifolds in rational homology spheres

Let $Σ$ be a simply connected rational homology sphere. A pair of disjoint closed submanifolds $M_+, M_-$ in $Σ$ are called dual to each other if the complement $Σ- M_+$ strongly homotopy retracts onto $M_-$ or vice-versa. In this paper we will give a complete answer of which integral triples $(n; m_+, m_-)$ can appear, where $n=dim Σ-1$, $m_+={codim}M_+ -1$ and $m_-={codim}M_- -1$.

Tight Lagrangian homology spheres in compact homogeneous Kähler manifolds

For any irreducible compact homogeneous Kähler manifold, we classify the compact tight Lagrangian submanifolds which have the Z_2-homology of a sphere.

具有二阶挠的有理同调六维球面上的复结构

A Complex Structure on A Rational Homology Six-Sphere with Two-Torsion

本文构造了一个紧致光滑单连通复三维流形，其第二、第三同调群为二阶循环群，其余同调群与六维球面相同。

A non-abelian Seiberg-Witten invariant for integral homology 3-spheres

A new diffeomorphism invariant of integral homology 3-spheres is defined using a non-abelian 'quaternionic' version of the Seiberg-Witten equations.

Simply connected Sasaki-Einstein rational homology 5-spheres

We completely determine which simply connected rational homology 5-spheres admit Sasaki-Einstein metrics.

Einstein Metrics on Rational Homology 7-Spheres

In this paper we demonstrate the existence of Sasakian-Einstein structures on certain 2-connected rational homology 7-spheres. These appear to be the first non-regular examples of Sasakian-Einstein metrics on simply connected rational homology spheres. We also briefly describe the rational homology 7-spheres that admit regular positive Sasakian structures.

Berglund-Hübsch转置与Sasaki-Einstein有理同调7维球面

Beglund-Hübsch transpose and Sasaki-Einstein rational homology 7-spheres

本研究探讨源自Johnson与Kollár的Kähler-Einstein三维簇列表中可逆多项式的链环，在Berglund-Hübsch转置规则下的性质，证明其仍为有理同调7维球面，给出保持Sasaki-Einstein性质的条件，发现75个新例子。

Einstein Metrics on Rational Homology Spheres

We prove the existence of Sasakian-Einstein metrics on infinitely many rational homology spheres in all odd dimensions greater than 3. In dimension 5 we obain somewhat sharper results. There are examples where the number of effective parameters in the Einstein metric grows exponentially with dimension.