Thesis Proposal: Periodic Homotopy Theory of Unstable Spheres

The unstable homotopy groups of spheres can be approached by the EHP spectral sequence. There are computations of the low dimensional portion of the EHP sequence by Toda [1, 2] for the 2,3-primary part, and Behrens [3], Harper [4] for the 5-primary part. There are certain stable phenomenon in the EHP sequence. In fact, there is one portion in the E1-term which are in the stable range, which means the input of the EHP spectral sequence is the stable homotopy groups. This portion is called the metastable range. There is the James-Hopf map mapping the (certain iterated loop space of) spheres to the infinite loop spaces of the infinite suspension of the real projective space (for the prime 2, and classifying space of the symmetric group Σp for odd prime p), respecting the unstable filtration on the sphere side and skeletal filtration on the projective space side. So we have a comparison map of the EHP spectral sequence and the AHSS of the projective space. Moreover, in the metastable range, this induce an isomorphism of the E1-term. So the differentials of the EHP spectral sequence in the metastable range are in fact stable, and we can apply stable techniques to compute them. For example, the behavior of the image of J is determined by Mahowald [5] and Gray [6]. Above the metastable range in the EHP spectral sequence, there is the portion which has input the metastable homotopy groups – the meta-metastable range, and we also have the meta-meta-metastable range whose input are the meta-metastable groups, and so on. In fact, these pieces can be spitted into separated spectral sequences, each of which is a stable AHSS. So the differentials inside each piece are all stable ones, and only the differentials between different pieces are actually unstable. This splitting is achieved by the technique of the Goodwillie calculus developed in [7]. Using this, any space is decomposed into the homogeneous parts, which are infinite loop spaces. The interrelation of the EHP sequence and the Goodwillie tower of the identity is studied by Behrens [8]. In particular, the EHP spectral sequence in each homogeneous part of the Goodwillie tower is a stable AHSS for certain spectra Ln(k). To study these sequences, and in particular to do actual computations, we will take the chromatic point of view. The rational part is the classical computations by Serre. The v1-periodic part is essentially the Image of J. So these are