Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.1014

The low-dimensional homotopy of the stable mapping class group

Johannes Ebert
View original source ↗Read source document ↗Export metadata ↓

Abstract

Due to the deep work of Tillmann, Madsen, Weiss and Galatius, the cohomology of the stable mapping class group $\gaminf$ is known with rational or finite field coefficients. Little is known about the integral cohomology. In this paper, we study the first four cohomology groups. Also, we compute the first few steps of the Postnikov tower of $B \gaminf^+$, the Quillen plus construction applied to $B \gaminf$. Our method relies on the Madsen-Weiss theorem, a few known computations of stable homotopy groups of spheres and projective spaces and on a certain action of the binary icosahedral group on a surface. Using the latter, we can also describe an explicit geometric generator of the third homotopy group $π_3 (B \gaminf)$.

Explore related subjects

math.AT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Johannes Ebert. 2007-07-06. The low-dimensional homotopy of the stable mapping class group. https://arxiv.org/abs/0707.1014

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On The Rational Realization of Even-dimensional Spheres and Products of Eilenberg--MacLane Spaces as Classifying Spaces

In this paper, we study the rational realization problem for the classifying space $B\mathrm{aut}_1(X)$. We prove that if $S^{2n}$ is realized as $B\mathrm{aut}_1(X)_{\mathbb{Q}}$ for a simply-connected space $X$, then $X$ is $π$-infinite and has vanishing rational Gottlieb elements above degree $2n-1$. In particular, even-dimensional spheres cannot be realized as $B\mathrm{aut}_1(X)_{\mathbb{Q}}$ for any simply-connected $π$-finite space $X$. We also prove that, for all $n\geq 2$ and $s,t\geq 1$, the product of Eilenberg--MacLane spaces $K(\mathbb{Q}^s,n)\times K(\mathbb{Q}^t,n+1)$ cannot be realized as $B\mathrm{aut}_1(X)_{\mathbb{Q}}$ for any simply-connected $π$-finite space $X$. Moreover, we prove that if $r\geq 2$, $n\geq 3$ and $K(\mathbb{Q}^r,n)$ is realized as $B\mathrm{aut}_1(X)_{\mathbb{Q}}$ for a simply-connected $π$-finite space $X$, then $X$ has the rational homotopy type of $K(\mathbb{Q}^r,n-1)$. The proofs are based on two structural results for Gottlieb elements in the derivation Lie algebra of a Sullivan minimal model, which provide a uniform method for these realization problems.

math.AT↗
arXiv2026

A complete classification of the rational realization of gap-two products of Eilenberg--MacLane spaces as classifying spaces

In this paper, we study the rational realization of the gap-two product $K(\mathbb{Q},n)\times K(\mathbb{Q},n+2)$ as the classifying space $B\mathrm{aut}_1(X)$ for a simply-connected $π$-finite space $X$ with rational homology of finite type. We prove that $K(\mathbb{Q},n)\times K(\mathbb{Q},n+2)$ can be realized as $B\mathrm{aut}_1(X)$ up to rational homotopy equivalence if and only if $n=3,4$ or $4m+2$ for $m\geq 1$. Moreover, we determine all their minimal Sullivan models up to isomorphism. The rational homotopy type of such $X$ is unique for $n=3$ or $4m+2$ for $m\geq 1$; there exist infinitely many rational homotopy types of such $X$ for $n=4$.

math.AT↗