×

On the mapping class group action on the cohomology of the representation space of a surface. (English) Zbl 0863.58010

Let \(X\) be a compact oriented surface and \({\mathcal R}\) the space of gauge equivalence classes of flat unitary connections on \(X\) with some finite number of punctures. Certain conditions are imposed on \({\mathcal R}\). Then the mapping class group \({\mathcal M}\) gives a representation \(\rho: {\mathcal M} \to H^* ({\mathcal R}; \mathbb{Q})\). \({\mathcal M}\) also acts on \(H_1 (X; \mathbb{Z})\), giving an exact sequence \(O\to {\mathcal T} \to {\mathcal M} \to G\to O\). Then \(\rho|{\mathcal T}\) is trivial, and there is constructed a surjective algebra homomorphism \(A\to H^* ({\mathcal R}; \mathbb{Q})\) which is a homomorphism of \(G\)-modules.

MSC:

58D19 Group actions and symmetry properties
14D20 Algebraic moduli problems, moduli of vector bundles
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Biswas, I., Raghavendra, N. : Canonical generators of the cohomology of moduli of parabolic bundles on curves, Math. Ann. (to appear). · Zbl 0853.14005
[2] Dostoglou, S., Salamon, D. : Self-dual instantons and holomorphic curves. Ann. Math. 139 (1994) 581–640. CMP 94:15 · Zbl 0812.58031
[3] Franz W. Kamber and Philippe Tondeur, Foliated bundles and characteristic classes, Lecture Notes in Mathematics, Vol. 493, Springer-Verlag, Berlin-New York, 1975. · Zbl 0308.57011
[4] V. B. Mehta and C. S. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980), no. 3, 205 – 239. · Zbl 0454.14006 · doi:10.1007/BF01420526
[5] Carlos T. Simpson, Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990), no. 3, 713 – 770. · Zbl 0713.58012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.