An application of transversality to the topology of the moduli space of stable bundles. (English) Zbl 0835.58005

Daskalopoulos, Georgios D.; Uhlenbeck, Karen K.

Topology 34, No. 1, 203-315 (1995).

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 G. Harder and M. S. Narasimhan [Math. Ann. 212, 215-248 (1975; Zbl 0324.14006)] and by M. F. Atiyah and R. Bott [Philos. Trans. R. Soc. Lond. A 308, 523-615 (1983; Zbl 0509.14014)].
If \(n\) and \(k\) are not coprime, 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.

Reviewer: P.Michor (Wien)

MSC:

58D27	Moduli problems for differential geometric structures
14D20	Algebraic moduli problems, moduli of vector bundles
53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
55Q52	Homotopy groups of special spaces
32G13	Complex-analytic moduli problems

Keywords:

modulo space; stable bundles; homotopy groups; transversality

Citations:

Zbl 0324.14006; Zbl 0509.14014; Zbl 0607.14017; Zbl 0778.14012; Zbl 0785.58014