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The homology of moduli spaces on a Riemann surface as a representation of the mapping class group. (English) Zbl 1026.14004

Let \(\Sigma\) be a compact Riemann surface of genus \(g\geq 2\), \(p\) a fixed point of \(\Sigma\) and \(M_{g,k}\) the moduli space of semistable rank 2 vector bundles on \(\Sigma\) with fixed determinant \({\mathcal O}_\Sigma(kp)\). It is a compact complex variety of real dimension \(6g-6\). In this paper, the author gives an explicit decomposition of the rational (intersection) homology of \(M_{g,k}\) into irreducible representations of the oriented mapping class group \(\text{Map}^+(\Sigma)\) of oriented diffeomorphisms of \(\Sigma\) (which acts on \(M_{g,k}\), too, by diffeomorphisms).
In the non-singular case, \(k\) odd, the author uses a result of P. E. Newstead [Trans. Am. Math. Soc. 169, 337-345 (1972; Zbl 0256.14008)] which gives a system of generators of the cohomology ring \(H^*(M_{g,1})\) and a lemma of M. Thaddeus [J. Differ. Geom. 35, 131-149 (1992; Zbl 0772.53013)] which computes the cohomology class of \(M_{g-1,1}\) embedded in \(M_{g,1}\) such that \(H^*(M_{g-1,1})\) is a subring of \(H^*(M_{g,1})\).
In the more complicated singular case, \(k\) even, the author considers the moduli space \(N_{g,0}\) of semistable rank 2 vector bundles on \(\Sigma\) of degree 0, its realization as a quotient by geometric invariant theory, and uses the “cohomology of quotients” method of F. C. Kirwan [Ann. Math. (2) 122, 41-85 (1985; Zbl 0592.14011)]. It turns out, in both cases, that the intersection homology of \(M_{g,k}\) is a direct sum of (several) copies of the standard representations \(\bigwedge^{g-p}H_1(\Sigma,\mathbb{Q})\) of \(\text{Map}^+ (\Sigma)\). Their multiplicities are independent of the genus \(g\). As a corollary, the author observes that \(\text{IH}^*(M_{g,0};\mathbb{Q})\) is a \(\text{Map}^+ (\Sigma)\)-equivariant quotient of \(H^*(M_{g,1};\mathbb{Q})\).
Using the fact that a 0-surgery on a fibred knot \(K\) in \(S^3\) results in a mapping torus \(\Sigma \times[0,1]\) with ends identified together by the monodromy map \(f:\Sigma \to \Sigma\), the author gives some applications to knot theory.

MSC:

14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14H60 Vector bundles on curves and their moduli
14D20 Algebraic moduli problems, moduli of vector bundles
55N33 Intersection homology and cohomology in algebraic topology
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