Unstable chromatic homotopy theory

In this thesis, I study unstable homotopy theory with chromatic methods. Using the v, self maps provided by the Hopkins-Smith periodicity theorem, we can decompose the unstable homotopy groups of a space into its periodic parts, except some lower stems. For fixed n, using the Bousfield-Kuhn functor [Phi]n, we can associate to any space a spectrum, which captures the vo-periodic part of its homotopy groups. I study the homotopy type of the spectra LK(n)[Phi]nfSk, which would tell us much about the vn-periodic part of the homotopy groups of spheres provided we have a good understanding of the telescope conjecture. I make use the Goodwillie tower of the identity functor, which resolves the unstable spheres into spectra which are the Steinberg summands of classifying spaces of the additive groups of vector spaces over F,. By understanding the attaching maps of the Goodwillie tower after applying the Bousfield-Kuhn functor, we would be able to determine the homotopy type of LK(n)[Phi]nSk. As an example of how this works in concrete computations, I will compute the homotopy groups of LK(2)[Phi]nS3 at primes p >/= 5. The computations show that the unstable homotopy groups not only have finite p-torsion, their K(2)-local parts also have finite vo-torsion, which indicates there might be a more general finite v-torsion phenomena in the unstable world.

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2015

Author dc:creator

• Wang, Guozhen, Ph. D. Massachusetts Institute of Technology

Advisor dc:contributor.advisor

• Mark Behrens and Haynes Miller.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/99321

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/99321