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

Mathematische Annalen (Impact Factor: 1.13). 01/2005; 337(1). DOI: 10.1007/s00208-006-0025-7
Source: arXiv


We construct a graph complex calculating the integral ho- mology of the
bordered mapping class groups. We compute the ho- mology of the bordered
mapping class groups of various surfaces. Using the circle action on this graph
complex, we build a double complex and a spectral sequence converging to the
homology of the unbordered mapping class groups. We compute the homology of the
punctured mapping class groups associated to certain surfaces. Finally, we use
Miller's operad to get the first Kudo-Araki and Browder operations on our graph
complex. We also consider an unstable version of the higher
Kudo-Araki-Dyer-Lashoff operations.

Full-text preview

Available from: ArXiv
  • Source
    • "The computational problem of determining the rank of a matrix has been extensively studied; it should be noted, however, that this step can actually be the most computationally burdening. It is worth mentioning that V. Godin [14] introduced a slightly different fatgraph complex, which computes the integral (co)homology of M g,n ; possible adaptation of the algorithms to this complex and an outlook on the expected problems is given in Section 7. An effective implementation (using the Python programming language [10]) of the algorithms presented here is available at "
    [Show abstract] [Hide abstract]
    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.
    Full-text · Article · Feb 2012
  • Source
    • "Fat graphs (without leaves) can be used to define a cell decomposition of Teichmuller space (see the work of Bowditch-Epstein [2], Harer [11], Penner [24] [25]), and the chain complex of isomorphism classes of graphs ( ¯ 0, 0)–Graphs is the corresponding cellular complex of the quotient of Teichmuller space by the action of the mapping class group, namely the (coarse) moduli space of Riemann surfaces. Similarly, graphs with leaves define a chain complex for the moduli space of surfaces with fixed boundaries, or with fixed intervals in its boundaries, (see Penner [26] [23], Godin [9], Costello [5, Sect. 6] and [6]). "
    [Show abstract] [Hide abstract]
    ABSTRACT: We give a general method for constructing explicit and natural operations on the Hochschild complex of algebras over any PROP with $A_\infty$--multiplication---we think of such algebras as $A_\infty$--algebras "with extra structure". As applications, we obtain an integral version of the Costello-Kontsevich-Soibelman moduli space action on the Hochschild complex of open TCFTs, the Tradler-Zeinalian action of Sullivan diagrams on the Hochschild complex of strict Frobenius algebras, and give applications to string topology in characteristic zero. Our main tool is a generalization of the Hochschild complex.
    Preview · Article · Oct 2011 · Advances in Mathematics
  • Source
    • "The category Fat b only depends on the topological type of the surface Σ, i.e., on its genus g. This category is denoted by Fat b g,1 in the paper [10], which applies to any compact connected oriented surface with non-empty boundary. "
    [Show abstract] [Hide abstract]
    ABSTRACT: Let S be a compact connected oriented surface with one boundary component. We extend each of Johnson's and Morita's homomorphisms to the Ptolemy groupoid of S. Our extensions are canonical and take values into finitely generated free abelian groups. The constructions are based on the 3-dimensional interpretation of the Ptolemy groupoid, and a chain map introduced by Suslin and Wodzicki to relate the homology of a nilpotent group to the homology of its Malcev Lie algebra.
    Preview · Article · Jun 2010 · Geometriae Dedicata
Show more