Homology of the Derivatives of the Identity Functor of Spaces

The present thesis consists of an introduction and two chapters of mathematics. The latter two are as follows.

Chapter 2 The derivatives of the identity functor on spaces in Goodwillie calculus forms an operad in spectra. Antolin-Camarena computed the mod 2 homology of free algebras over this operad for 1-connected spectra. In this chapter we carry out similar computations for mod p homology for odd primes p, and all spectra.

Chapter 3 In this chapter we recover Bousfield's computation of v_1-periodic homotopy groups of simply connected, finite H-spaces from Bousfield's work using the techniques of Goodwillie calculus. This is done through first computing André-Quillen cohomology over the monad T that encodes the power operations of complex K-theory. We then lift this computation to computing K-theory of topological André-Quillen cohomology. We can then use results of Behrens and Rezk relating topological André-Quillen cohomology back to the Bousfield-Kuhn functor. The fact that we recover the result of Bousfield allows us to conclude that the v_1-periodic Goodwillie tower for simply connected, finite H-spaces converges.

The two chapters can be read almost completely independently, only relying on the definitions from Section 1.1.