The second homology groups of mapping class groups of orientable surfaces

Let $\Sigma_{g,r}^n$ be a connected orientable surface of genus $g$ with $r$ boundary components and $n$ punctures and let $\Gamma_{g,r}^n$ denote the mapping class group of $\Sigma_{g,r}^n$, namely the group of isotopy classes of orientation-preserving diffeomorphisms of $\Sigma_{g,r}^n$ which are the identity on the boundary and on the punctures. Here, we see the punctures on the surface as distinguished points. The isotopies are required to be the identity on the boundary and on the punctures. If $r$ and/or $n$ is zero, then we omit it from the notation.


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The class of almost completely decomposable groups with a critical typeset of type (2,2) and a homocyclic regulator quotient of exponent p(3) is shown to be of bounded representation type. There are only 16 isomorphism at p types of indecomposables, all of rank 8 or lower.
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We investigate the structure of the components of the moduli space Of Surfaces of general type, which parametrize surfaces admitting nonsmooth genus 2 fibrations of nonalbanese type, over curves of genus g(b) greater than or equal to 2.
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In this paper we construct a family of variational families for a Legendrian embedding, into the 1-jet bundle of a closed manifold, that can be obtained from the zero section through Legendrian embeddings, by discretising the action functional. We compute the second variation of a generating function obtained as above at a nondegenerate critical point and prove a formula relating the signature of the second variation to the Maslov index as the mesh goes to zero. We use this to prove a generalisation of the ...
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Citation Formats
M. Korkmaz, “The second homology groups of mapping class groups of orientable surfaces,” MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, pp. 479–489, 2003, Accessed: 00, 2020. [Online]. Available: