Holmstrom Homotopy groups of spheres

Several good references, e.g. Hopkins ICM talk, the famous book.

A conjecture of Baues

Hatcher Spectral Seqs book draft

Hatcher AT pp 384

http://mathoverflow.net/questions/76541/what-do-the-stable-homotopy-groups-of-spheres-say-about-the-combinatorics-of-fini

http://mathoverflow.net/questions/22688/why-the-heck-are-the-homotopy-groups-of-the-sphere-so-damn-complicated

http://mathoverflow.net/questions/31004/computational-complexity-of-computing-homotopy-groups-of-spheres

http://mathoverflow.net/questions/22837/what-are-the-uses-of-the-homotopy-groups-of-spheres

http://mathoverflow.net/questions/24594/survey-articles-on-homotopy-groups-of-spheres

arXiv:0910.2817 Derived functors of non-additive functors and homotopy theory from arXiv Front: math.AT by Lawrence Breen, Roman Mikhailov We develop a functorial approach to the study of the homotopy groups of spheres and Moore spaces $M(A,n)$, based on the Curtis spectral sequence and the decomposition of Lie functors as iterates of simpler functors such as the symmetric or exterior algebra functors. The discussion takes place over the integers, and includes a functorial description of the derived functors of certain Lie algebra functors, as well as of all the main cubical functors (such as the degree 3 component $SP^3$ of the symmetric algebra functor). As an illustration of this method, we retrieve in a purely algebraic manner the 3-torsion component of the homotopy groups of the 2-sphere up to degree 14, and give a unified presentation of homotopy groups $\pi_i(M(A,n))$ for small values of both $n$ and $i$.

arXiv:1009.1125 The Goodwillie tower and the EHP sequence from arXiv Front: math.AT by Mark Behrens We study the interaction between the EHP sequence and the Goodwillie tower of the identity evaluated at spheres at the prime 2. Both give rise to spectral sequences (the EHP spectral sequence and the Goodwillie spectral sequence, respectively) which compute the unstable homotopy groups of spheres. We relate the Goodwillie filtration to the P map, and the Goodwillie differentials to the H map. Furthermore, we study an iterated Atiyah-Hirzebruch spectral sequence approach to the homotopy of the layers of the Goodwillie tower of the identity on spheres. We show that differentials in these spectral sequences give rise to differentials in the EHP spectral sequence. We use our theory to re-compute the 2-primary unstable stems through the Toda range (up to the 19-stem). We also study the homological behavior of the interaction between the EHP sequence and the Goodwillie tower of the identity. This homological analysis involves the introduction of Dyer-Lashof-like operations associated to M. Ching’s operad structure on the derivatives of the identity. These operations act on the mod 2 stable homology of the Goodwillie layers of any functor from spaces to spaces.

[arXiv:1108.3055] A combinatorial description of homotopy groups of spheres from arXiv Front: math.AT by Roman Mikhailov, Jie Wu We give a combinatorial description of general homotopy groups of $k$-dimensional spheres with $k\geq3$ as well as those of Moore spaces

For $n>k\geq 3,$ we construct a finitely generated group defined by explicit generators and relations, whose center is exactly $\pi_n(S^k)$.

nLab page on Homotopy groups of spheres