Cell decomposition and odd cycles on compactified Riemann's moduli space

Details Cell decomposition and odd cycles on compactified Riemann's moduli space Cell decomposition and odd cycles on compactified Riemann's moduli space

2011-12-16

arxiv.org/abs/1112.3915v1

Mathematics

Geometric Topology

60 pages, 16 figures

Scientific paper

We introduce and investigate a bundle over the augmented Teichmueller space of a multiply punctured surface F by suitably decorating both punctures and double points of nodal surfaces; the total space of this bundle is shown to be equivariantly homotopy equivalent to the augmented Teichmueller space of F itself. We furthermore study a new bordification of the decorated Teichmueller space of F, which is shown to be equivalently homeomorphic to a real blow-up of its arc complex, by a space of filtered screens on the surface that arises from a natural elaboration of earlier work of McShane-Penner. An appropriate quotient of this space of filtered screens on F is shown to be equivariantly homotopy equivalent to its augmented Teichmueller space and to admit a natural cell decomposition, where cells are indexed by a suitable generalization of fatgraphs. The further quotient by the mapping class group gives the long-sought after cell decomposition of a space homotopy equivalent to the Deligne-Mumford compactification of the Riemann moduli space of F. As a first application of this technology, a plethora of odd-degree cycles is readily constructed, and we conjecture that these classes are homologically non-trivial.

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