The low-dimensional homotopy of the stable mapping class group

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 $\pi_3 (B \gaminf)$.