Computations in Equivariant and Chromatic Homotopy Theory

Adams spectral sequences have been an efficient tool in computing stable homotopy groups, classically and equivariantly. In Chapter 2 of this thesis, we explore the relationship between the classical Adams spectral sequences for stunted real projective spectra, the Borel C₂-equivariant Adams spectral sequence for the 2-completed sphere, and the genuine C₂-equivariant Adams spectral sequence for the 2-completed sphere. This allows us to understand the genuine C₂-equivariant Adams spectral sequence from the Borel Adams spectral sequence. We show that the Borel Adams spectral sequence is computable as a classical Adams spectral sequence.

Given a fiber sequence with good properties, the geometric boundary theorem relates the connecting map in the long exact sequence of homotopy groups to the connecting homomorphism in the map between the E₂-page of the Adams spectral sequences. This theorem was generalized by Behrens. In Chapter 3 of this thesis, we reformulate the generalized geometric boundary theorem to fix a mistake made by Behrens, and give a new proof using the language of filtered spectra.

A finite spectrum can be reassembled from its K(n)-localizations. While the homotopy groups of the K(2)-local sphere have been computed completely at odd primes, they are not fully understood at the prime 2. On the other hand, the K(2)-local homotopy groups of Z, a finite spectrum of type 2 with 32 cells, have been determined up to a possible extension. In Chapter 4 of this thesis, we compute an approximation of the K(2)-local homotopy groups of $Y_{\mathbb H}$ at the prime 2 using the algebraic duality spectral sequence and the topological duality spectral sequence, where $Y_{\mathbb H}\simeq\Sigma^{-7}\Sigma^\infty\mathbb{RP}^2\wedge\mathbb{CP}^2\wedge\mathbb{HP}^2$ is a finite spectrum of type 1 with 8 cells. We fully compute its algebraic duality spectral sequence. Furthermore, we construct a filtered duality spectral sequence, using which we compute the topological duality spectral sequence associated to $Y_{\mathbb H}$ up to some possible $d_3$-differentials.