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Hans U. Boden, An SU(n) Casson-Lin invariant for links

Date: 23. 5. 2012
Source: Topology seminar
Ponedeljek 28.5. 2012, od 12h do 14h, soba 3.06, Jadranska 21
 
Around 1980 Casson defined an invariant for homology 3-spheres giving an integral lift of the Rochlin invariant, and heapplied these ideas to the Hauptvermutung in dimension four. This talkwill give a gentle introduction to Casson's invariant, its definitionin terms of SU(2) representations of the fundamental group, and wewill discuss the gauge-theoretic interpretation of Casson's invariant,which is due to Taubes and Floer, as well as methods to generalizeCasson's invariant in different directions. Of particular interest isthe Casson-Lin invariant, which is a closely related invariant definedfor knots by X.-S. Lin in 1992 and which surprisingly turns out to bea multiple of the knot signature. We will describe more current workof Harper-Saveliev and also our own joint work with Harper on thegeneralized Casson-Lin type invariants, which are a family of linkinvariants. We will outline the construction of the invariants anddescribe some of their basic properties.