Article

# The unstable integral homology of the mapping class groups of a surface with boundary

Mathematische Annalen (Impact Factor: 1.38). 01/2005; 337(1). DOI: 10.1007/s00208-006-0025-7

Source: arXiv

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**ABSTRACT:**We provide a general method for finding all natural operations on the Hochschild complex of E-algebras, where E is any algebraic structure encoded in a PROP with multiplication, as for example the PROP of Frobenius, commutative or A_infty-algebras. We show that the chain complex of all such natural operations is approximated by a certain chain complex of formal operations, for which we provide an explicit model that we can calculate in a number of cases. When E encodes the structure of open topological conformal field theories, we identify this last chain complex, up quasi-isomorphism, with the moduli space of Riemann surfaces with boundaries, thus establishing that the operations constructed by Costello and Kontsevich-Soibelman via different methods identify with all formal operations. When E encodes open topological quantum field theories (or symmetric Frobenius algebras) our chain complex identifies with Sullivan diagrams, thus showing that operations constructed by Tradler-Zeinalian, again by different methods, account for all formal operations. As an illustration of the last result we exhibit two infinite families of non-trivial operations and use these to produce non-trivial higher string topology operations, which had so far been elusive.Journal für die reine und angewandte Mathematik (Crelles Journal) 12/2012; · 1.08 Impact Factor -
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**ABSTRACT:**Fatgraphs are multigraphs enriched with a cyclic order of the edges incident to a vertex. This paper presents algorithms to: (1) generate the set of all fatgraphs having a given genus and number of boundary cycles; (2) compute automorphisms of any given fatgraph; (3) compute the homology of the fatgraph complex. The algorithms are suitable for effective computer implementation. In particular, this allows us to compute the rational homology of the moduli space of Riemann surfaces with marked points. We thus compute the Betti numbers of $M_{g,n}$ with $(2g + n) \leq 6$, corroborating known results.02/2012;

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