An application of transversality to the topology of the moduli space of stable bundles
Let \(M\) be a Riemann surface of genus \(g \geq 2\) and let \(\overline {\mathcal M} = \overline {\mathcal M} (n,k)\) be the normal projective variety of equivalence classes of semistable holomorphic vector bundles \(E\) over \(M\) of rank \(n\) and first Chern class \(k\). If \(n\) and \(k\) are coprime then the variety \(\overline {\mathcal M}\) is smooth and its cohomology has been determined by \textit{G. Harder} and \textit{M. S. Narasimhan} [Math. Ann. 212, 215-248 (1975; Zbl 0324.14006)] and by \textit{M. F. Atiyah} and \textit{R. Bott} [Philos. Trans. R. Soc. Lond. A 308, 523-615 (1983; Zbl 0509.14014)]. If \(n\) and \(k\) are not coprime, \textit{F. C. Kirwan} [Proc. Lond. Math. Soc., III. Ser. 53, 237-266 (1986; Zbl 0607.14017); Corrigendum ibid. 65, No. 3, 474 (1992; Zbl 0778.14012)] described a method of computing the intersection Betti numbers of \(\overline {\mathcal M}\). In the case \(n = 2\) she wrote explicitly the intersection Poincaré series for \(\overline {\mathcal M}\) and deduced from general results about intersection homology that, up to dimension \(2g - 4\), the intersection Poincaré series of \(\overline {\mathcal M}\) coincides with the ordinary Poincaré series of the nonsingular part \(\mathcal M\). In [the author, J. Differ. Geom. 36, No. 3, 699-746 (1992; Zbl 0785.58014)] the homotopy groups of \({\mathcal M} (n,k)\) were computed up to dimension \(2g - 4\) by using Morse theory. In this paper some low dimensional homotopy and cohomology groups of \(\mathcal M\) are computed by direct infinite dimensional transversality arguments. The results are as follows: For \((n,k) \neq (2,2)\) one has \(\pi_1 ({\mathcal M} (n,k)) \simeq H_1 (M,\mathbb{Z})\), \(\pi_2 ({\mathcal M} (n,k)) \simeq \mathbb{Z} \oplus (\mathbb{Z}/\text{gcd} (n,k) \mathbb{Z})\), and for \(2 < i \leq 2(g - 1) (n - 1) - 2\) the homotopy group \(\pi_i ({\mathcal M} (n,k))\) equals \(\pi_{i - 1}\) of the gauge group. Also the \(i\)-th Betti number of \({\mathcal M} (n,k)\) is given for \(i \leq 2 (g - 1) (n - 1) - 2\). An appendix derives bundle versions of Thom's transversality theorem in infinite dimensions from Smale's infinite dimensional version of Sard's theorem. These are used in the proofs.
scientific article; zbMATH DE number 722026