25 research outputs found
Yang-Mills theory over surfaces and the Atiyah-Segal theorem
In this paper we explain how Morse theory for the Yang-Mills functional can
be used to prove an analogue, for surface groups, of the Atiyah-Segal theorem.
Classically, the Atiyah-Segal theorem relates the representation ring R(\Gamma)
of a compact Lie group to the complex K-theory of the classifying
space . For infinite discrete groups, it is necessary to take into
account deformations of representations, and with this in mind we replace the
representation ring by Carlsson's deformation --theory spectrum \K
(\Gamma) (the homotopy-theoretical analogue of ). Our main theorem
provides an isomorphism in homotopy \K_*(\pi_1 \Sigma)\isom K^{-*}(\Sigma)
for all compact, aspherical surfaces and all . Combining this
result with work of Tyler Lawson, we obtain homotopy theoretical information
about the stable moduli space of flat unitary connections over surfaces.Comment: 43 pages. Changes in v4: improved results in Section 7, simplified
arguments in the Appendix, various minor revision