Surfaces in a background space and the homology of mapping class groups

Abstract

Abstract. In this paper we study the topology of the space of Riemann surfaces in a simply connected space X, Sg,n(X, γ). This is the space consisting of triples, (Fg,n, φ, f ), where Fg,n is a Riemann surface of genus g and n-boundary components, φ is a parameterization of the boundary, ∂Fg,n, and f : Fg,n → X is a continuous map that satisfies a boundary condition γ. We prove three theorems about these spaces. Our main theorem is the identification of the stable homology type of the space S∞,n(X; γ), defined to be the limit as the genus g gets large, of the spaces Sg,n(X; γ). Our result about this stable topology is a parameterized version of the theorem of Madsen and Weiss proving a generalization of the Mumford conjecture on the stable cohomology of mapping class groups. Our second result describes a stable range in which the homology of Sg,n(X; γ) is isomorphic to the stable homology. Finally we prove a stability theorem about the homology of mapping class groups with certain families of twisted coefficients. The second and third theorems are generalizations of stability theorems of Harer and Ivanov.

Monoids of moduli spaces of manifolds

“…Thus we can cut open the surface W along L, so the spaces in (1)(2)(3) are effectively moduli spaces of surfaces with one boundary component.…”

“…In many cases of interest, 0 M is a finitely generated monoid, so the localisation (1)(2)(3)(4) can be formed as a sequential direct limit…”

“…For an oriented surface F , Cohen and Madsen [2] studied spaces S g;n .Y / D Map @ .F g;n ; Y /= =Diff C .F g;n /…”

Homology of the moduli spaces and mapping class groups of framed, r -Spin and Pin surfaces

“…This theory has been built upon by several workers to-amongst other things-deal with tangential structures other than orientations. Thus much is known about the homology of moduli spaces of unoriented surfaces [24], moduli spaces of Spin surfaces [13,1,9], and moduli spaces of oriented surfaces with maps to a simply-connected background space [3,4].…”

“…Now q ξ (a i ) = [1] or [3] in Z/4, so 2 − q ξ (a 1 ) = q ξ (a 1 ) and 2 · q ξ (a 1 ) + 2 = 0, so c acts trivially on Pin − (S n,b+1 ; δ) and in particular preserves A. …”