Due to the deep work of Tillmann, Madsen, Weiss and Galatius, the cohomology of the stable mapping class group Γ∞ 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 BΓ∞, the Quillen plus construction applied to BΓ∞. Our method ...

For the last thirty years the EHP sequence has been a major conceptual tool in the attempt to understand the homotopy groups of spheres. It is a collection of long exact sequences of homotopy groups induced by certain fibrations in which all three spaces are loop spaces of spheres. These fibrations are due originally to James, G. W. Whitehead, and Toda. The Freudenthal suspension theorem and th...

The Adams spectral sequence was invented by J.F.Adams [1] almost fifty years ago for calculations of stable homotopy groups of topological spaces and in particular of spheres. The calculation of differentials of this spectral sequence is one of the most difficult problem of Algebraic Topology. Here we consider an approach to solve this problem in the case of Z/2 coefficients and find inductive ...

The Adams spectral sequence was invented by J.F.Adams [1] almost fifty years ago for calculations of stable homotopy groups of topological spaces and in particular of spheres. The calculation of differentials of this spectral sequence is one of the most difficult problem of Algebraic Topology. Here we consider an approach to solve this problem in the case of Z/2 coefficients and find inductive ...