On Casson-type instanton moduli spaces over negative definite four-manifolds

A vanishing result for a Casson-type instanton invariant

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.

Floer homologies for Lagrangian intersections and instantons

In 1985 lectures at MSRI, A. Casson introduced an interesting integer valued invariant for any oriented integral homology 3-sphere Y via beautiful constructions on representation spaces (see [1] for an exposition). The Casson invariant λ(Y) is roughly defined by measuring the oriented number of irreducible representations of the fundamental group π_1(Y) in SU(2). Such an invariant generalized the Rohlin invariant and gives surprising corollaries in low dimensional topology.

Lescop's Invariant and Gauge Theory

Taubes proved that the Casson invariant of an integral homology 3-sphere equals half the Euler characteristic of its instanton Floer homology. We extend this result to all closed oriented 3-manifolds with positive first Betti number by establishing a similar relationship between the Lescop invariant of the manifold and its instanton Floer homology. The proof uses surgery techniques.

Harmonic sections in sphere bundles, normal neighborhoods of reduction loci, and instanton moduli spaces on definite 4-manifolds

We prove an existence theorem for gauge invariant $L^2$-normal neighborhoods of the reduction loci in the space ${\cal A}_a(E)$ of oriented connections on a fixed Hermitian 2-bundle $E$. We use this to obtain results on the topology of the moduli space ${\cal B}_a(E)$ of (non-necessarily irreducible) oriented connections, and to study the Donaldson $μ$-classes globally around the reduction loci. In this part of the article we use essentially the concept of harmonic section in a sphere bundle with respect to an Euclidean connection. Second, we concentrate on moduli spaces of instantons on defin…

关于以同调三维球面为边界的定号格与 Yang-Mills 瞬子 Floer 理论

On definite lattices bounded by a homology 3-sphere and Yang-Mills instanton Floer theory

本文利用瞬子 Floer 理论推广 Froyshov 方法，分类了以特定同调三维球面为边界的定号格，并证明 (2,5) 环面结 +1 手术情形下非对角格仅为 E8 和秩 12 的不可分解幺模定号格。

Niemeier lattices, smooth 4-manifolds and instantons

We show that the set of even positive definite lattices that arise from smooth, simply-connected 4-manifolds bounded by a fixed homology 3-sphere can depend on more than the ranks of the lattices. We provide two homology 3-spheres with distinct sets of such lattices, each containing a distinct nonempty subset of the rank 24 Niemeier lattices.