Einstein Metrics on Rational Homology Spheres

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.

Seiberg-Witten invariants of rational homology spheres

We prove that the Seiberg-Witten invariants of a rational homology sphere are determined in a very explicit fashion by the Casson-Walker invariant and the Reidemeister torsion

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.

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…

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

We derive a gauge theoretic invariant of integral homology 3-spheres which counts gauge orbits of irreducible, perturbed flat SU(3) connections with sign given by spectral flow. To compensate for the dependence of this sum on perturbations, the invariant includes contributions from the reducible, perturbed flat orbits. Our formula for the correction term generalizes that given by Walker in his extension of Casson's SU(2) invariant to rational homology 3-spheres.

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。

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 $m_-={codim}M_- -1$.

有理同调三维球的谱不变量与等变单极Floer同调

Spectral invariants and equivariant monopole Floer homology for rational homology three-spheres

本文利用$\mathcal{S}$-complex研究有理同调三维球的$S^1$-等变单极Floer同调，通过Chern-Simons-Dirac泛函定义$\mathbf{R}$-过滤与谱不变量$ρ$，并证明$ρ$可阻碍带状同调配边上正标量曲率度量的存在。

Rational Homology 5-Spheres with Positive Ricci Curvature

We prove that for every integer k>1 there is a simply connected rational homology 5-sphere $\scriptstyle{M^5_k}$ with spin such that $\scriptstyle{H_2(M^5_k,\bbz)}$ has order $\scriptstyle{k^2},$ and $\scriptstyle{M^5_k}$ admits a Riemannian metric of positive Ricci curvature. Moreover, if the prime number decomposition of $\scriptstyle{k}$ has the form $\scriptstyle{k=p_1... p_r}$ for distinct primes $\scriptstyle{p_i}$ then $\scriptstyle{M^5_k}$ is uniquely determined.

Anosov diffeomorphisms of products I. Negative curvature and rational homology spheres

We show that various classes of products of manifolds do not support transitive Anosov diffeomorphisms. Exploiting the Ruelle-Sullivan cohomology class, we prove that the product of a negatively curved manifold with a rational homology sphere does not support transitive Anosov diffeomorphisms. We extend this result to products of finitely many negatively curved manifolds of dimensions at least three with a rational homology sphere that has vanishing simplicial volume. As an application of this study, we obtain new examples of manifolds that do not support transitive Anosov diffeomorphisms, inc…