arXiv math/03111732003-11-11math.GTmath.DG

The Integer Valued SU(3) Casson Invariant for Brieskorn spheres

Hans U. Boden, Christopher M. Herald, Paul A. Kirk

备注：: 50 pages, 3 figures
发表信息：: J. Diff. Geom. 71 (2005) 23-83
DOI：: 10.4310/jdg/1143644312

英文摘要：

We develop techniques for computing the integer valued SU(3) Casson invariant. Our method involves resolving the singularities in the flat moduli space using a twisting perturbation and analyzing its effect on the topology of the perturbed flat moduli space. These techniques, together with Bott-Morse theory and the splitting principle for spectral flow, are applied to calculate the invariant for all Brieskorn homology spheres.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

An integer valued SU(3) Casson invariant

We define an integer valued invariant of homology spheres using the methods of SU(3) gauge theory and study its behavior under orientation reversal and connected sum.

2000-08-11math.DGmath.GT

More Brieskorn spheres bounding rational balls

We call an integral homology sphere $\textit{non-trivially}$ bounds a rational homology ball if it is obstructed from bounding an integral homology ball. After Fintushel and Stern's well-known example $Σ(2,3,7)$, Akbulut and Larson recently provided the first infinite families of Brieskorn spheres non-trivially bounding rational homology balls: $Σ(2,4n+1,12n+5)$ and $Σ(3,3n+1,12n+5)$ for odd~$n$. Using their technique, we present new such families: $Σ(2,4n+3,12n+7)$ and $Σ(3,3n+2,12n+7)$ for even $n$. Also manipulating their main argument, we simply recover some classical results of Akbulut an…

2020-12-07math.GT

Defining an SU(3)-Casson/U(2)-Seiberg-Witten integer invariant for integral homology 3-spheres

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 obstruc…

2004-11-01math.DGmath.GT

A Casson-Lin type invariant for links

We define an integer valued invariant for two-component links in S^3 by counting projective SU(2) representations of the link group having non-trivial second Stiefel-Whitney class. We show that our invariant is, up to sign, the linking number of the link. Our construction generalizes that of X.-S. Lin who defined a similar invariant for knots in S^3; his invariant equals half the knot signature.

2009-11-23math.GT

Ribbon-moves of 2-knots: the Farber-Levine pairing and the Atiyah-Patodi-Singer-Casson-Gordon-Ruberman $\widetildeη$-invariants of 2-knots

Let K and K' be 2-knots. Suppose that K and K' are ribbon-move equivalent. Then the Farber-Levine pairing for K is equivalent to that for K' and the (Z-)torsion part of the first Alexander module of $K$ is isomorphic to that of K' as Z[Z] modules. Let K be a 2-knot which is ribbon-move equivalent to the trivial knot. Then the Atiyah-Patodi-Singer-Casson-Gordon-Ruberman Q/Z-valued \widetildeη-invariants of K for Z_d is zero. (d is a natural number. d>2.).

2000-04-02math-phmath.GTmath.MP

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.

1995-06-12math.GT

更多相关论文收起更多论文

Homological Casson type invariant of knotoids

We consider an analogue of well-known Casson knot invariant for knotoids. We start with a direct analogue of the classical construction which gives two different integer-valued knotoid invariants and then focus on its homology extension. Value of the extension is a formal sum of subgroups of the first homology group $H_1(Σ)$ where $Σ$ is an oriented surface with (maybe) non-empty boundary in which knotoid diagrams lie. To make the extension informative for spherical knotoids it is sufficient to transform an initial knotoid diagram in $S^2$ into a knotoid diagram in the annulus by removing smal…

2020-09-27math.GT