Homology of the Derivatives of the Identity Functor of Spaces

Doctoral Dissertation

Abstract

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.

Attributes

Attribute NameValues
Author Jens Jakob Kjaer
Contributor Mark J. Behrens, Research Director
Degree Level Doctoral Dissertation
Degree Discipline Mathematics
Degree Name Doctor of Philosophy
Banner Code
  • PHD-MATH

Defense Date
  • 2019-06-17

Submission Date 2019-06-19
Record Visibility and Access Public
Content License
Departments and Units
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