As the main result of the article the author shows that Carlsson’s deformation \(K\)-theory groups \(K^{\mathrm{def}}_*(\pi_{1}(\Sigma))\) for the fundamental group \(\pi_{1}(\Sigma)\) of an aspherical compact surface \(\Sigma\) coincide with the ordinary \(K\)-theory groups \(K^{-*}(\Sigma)\) for \(*>0\). The groups \(K^{\mathrm{def}}_*(\Gamma)\) for a discrete group \(\Gamma\) have been defined by G. Carlsson [“Derived representation theory and the algebraic \(K\)-theory of fields,” J. Topol. 4, No. 3, 543–572 (2011; Zbl 1230.19001)]. By definition they are the homotopy groups of the group completion of the topological monoid given by the direct sum over the natural numbers \(n \geq 0\) of the homotopy orbit spaces \(\mathrm{Hom}(\Gamma, U(n))_{hU(n)}\), where the action of the unitary group \(U(n)\) on \(\mathrm{Hom}(\Gamma, U(n))\) is given by conjugation. The groups \(K^{\mathrm{def}}_*(\Gamma)\) may be seen as a deformed version of the representation ring of \(\Gamma\), and since \(\Sigma\) can be viewed as a model for the classifying space \(B\pi_{1}(\Sigma)\) the established isomorphisms can be regarded as a variation of the Atiyah-Segal completion theorem: namely in the following formal sense that they give a way to determine the \(K\)-theory of the classifying space of \(\Gamma\) from a suitable modification of the representation ring of \(\Gamma\).
As a simple consequence of the established isomorphisms one can now compute the deformed \(K\)-theory of \(\pi_{1}(\Sigma)\) for surfaces \(\Sigma\) as above. In particular one obtains that the deformation \(K\)-theory groups in the considered cases satisfy certain excision properties in positive dimensions. This supports a more general conjecture of Carlsson, which claims that there is a Mayer-Vietoris sequence in deformation \(K\)-theory for amalgamated products of groups which is exact in high degrees. Moreover, for a surface \(\Sigma\) as above the established isomorphism yields (using additional results from T. Lawson [Math. Proc. Camb. Philos. Soc. 146, No. 2, 379–393 (2009; Zbl 1163.19002)]) a complete determination of the homotopy type of the space \(\mathrm{Hom}(\pi_{1}(\Sigma),U)/U\), which can be identified with the stable moduli space of unitary flat connections over \(\Sigma\).
In order to prove the main result the author uses elements of infinite dimensional Morse theory. More precisely, let \(\mathcal{A}_{\mathrm{flat}}(n)\) denote a suitable Sobolev completion of the space of unitary flat connections on \(\Sigma\times \mathbb{C}^{n}\) for an arbitrary choice of a Riemannian structure on \(\Sigma\), and let \(\mathcal{G}(n)\) denote the corresponding gauge group. As a first result the author shows that there is a \(\pi_{1}(\Sigma)\)-equivariant homeomorphism from \(\mathcal{A}_{\mathrm{flat}}(n)/\mathcal{G}(n)\) to \(\mathrm{Hom}(\pi_{1}(\Sigma), U(n))\), so that he can use the quotients \(\mathcal{A}_{\mathrm{flat}}(n)/\mathcal{G}(n)\) for the definition of the deformed \(K\)-theory of \(\pi_{1}(\Sigma)\). Next the author investigates the homotopy type of \(\mathcal{A}_{\mathrm{flat}}(n)\). Here he uses the fact that \(\mathcal{A}_{\mathrm{flat}}(n)\) is the critical set of the Yang-Mills functional, which is defined on the bigger space \(\mathcal{A}(n)\) of all connections on \(\Sigma\times \mathbb{C}^{n}\). Infinite dimensional Morse theory techniques applied to the Yang-Mills functional then yield a stratification for the space \(\mathcal{A}(n)\), and the author uses results from G. D. Daskalopoulos [J. Differ. Geom. 36, No. 3, 699–746 (1992; Zbl 0785.58014)] and J. Råde [J. Reine Angew. Math. 431, 123–163 (1992; Zbl 0760.58041)] to compare this stratification with the so-called Harder-Narasimhan stratification [established by G. Harder and M. S. Narasimhan, Math. Ann. 212, 215–248 (1975; Zbl 0324.14006)]. This allows him to deduce that the connectivity of the inclusion of \(\mathcal{A}_{\mathrm{flat}}(n)\) into \(\mathcal{A}(n)\) increases when \(n\) gets larger. If follows that the colimit over \(n\) of the spaces \(\mathcal{A}_{\mathrm{flat}}(n)/\mathcal{G}(n)\) becomes a classifying space for the stable gauge group \(\mathcal{G}\). Using this and the result mentioned at the first place the author now derives that the homotopy groups \(\pi_{*}B\mathcal{G}\) can be identified with \(K^{\mathrm{def}}_*(\pi_{1}(\Sigma))\) in positive degrees. However, by a well-known result of M. F. Atiyah and R. Bott [Philos. Trans. R. Soc. Lond., A 308, 523–615 (1983; Zbl 0509.14014)] the homotopy groups \(\pi_i B\mathcal{G}\) also coincide with \(K^{-i}(\Sigma)\) for \(i>0\).