An Infinite Genus Mapping Class Group and Stable Cohomology

Abstract

We exhibit a finitely generated group M whose rational homology is isomorphic to the rational stable homology of the mapping class group. It is defined as a mapping class group associated to a surface S∞ of infinite genus, and contains all the pure mapping class groups of compact surfaces of genus g with n boundary components, for any g ≥ 0 and n > 0. We construct a representation of M into the restricted symplectic group Spres(Hr) of the real Hilbert space generated by the homology classes of non-separating circles on S∞, which generalizes the classical symplectic representation of the mapping class groups. Moreover, we show that the first universal Chern class in H 2

The braided Ptolemy–Thompson group is asynchronously combable

“…An algebraic relation between T and the braid groups has been discovered in an article due to P. Greenberg and V. Sergiescu ([21]). Since then, several works ( [6], [7], [10], [11], [14], [15], [16], [28]) have contributed to improve our understanding of the links between Thompson groups and mapping class groups of surfaces -including braid groups.…”

End-periodic homeomorphisms and volumes of mapping tori

“…Remark. The behavior of end periodic homeomorphisms on the pants graph is reminicent of the work of Funar and Kapoudjian in [FK09]. Indeed, their asymptotic mapping class group of infinite genus is a subgroup of the mapping class group of the blooming Cantor tree that preserves a certain component of its pants graph.…”

Big mapping class groups: an overview

“…This was later generalized simultaneously by Brin [29] and Dehornoy [37] to the construction of an extension of V by B ∞ , the so-called braided Thompson groups. Funar-Kapoudjian [47,48], and later Funar and the first author [9], constructed finitely-generated (and often finitely-presented) extensions of V by a direct limit of mapping class groups of compact surfaces. Part of the motivation [47] is to construct a finitely-presented group whose homology agrees with the stable homology of pure mapping class groups, after a seminal result of Harer [60].…”

“…The case of the blooming Cantor tree. In [47], Funar and Kapoudjian constructed an asymptotic mapping class group B ∞ for the blooming Cantor tree, which we denote by Σ ∞ . In a similar fashion, the short exact sequence (5), when restricted to B ∞ , yields:…”