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
of a compact
Lie group
to the
complex
–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
(the homotopy-theoretical
analogue of
).
Our main theorem provides an isomorphism in homotopy
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.
Keywords
Atiyah–Segal theorem, deformation $K$–theory, flat
connection, Yang–Mills theory
