Cosimplicial invariants and Calculus of homotopy functors

Eldred, Rosona

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https://hdl.handle.net/2142/29665 Copy

Description

Title

Cosimplicial invariants and Calculus of homotopy functors

Author(s)

Eldred, Rosona

Issue Date

2012-02-06T20:09:50Z

Director of Research (if dissertation) or Advisor (if thesis)

McCarthy, Randy

Committee Member(s)

• McCarthy, Randy
• Rezk, Charles
• Guillou, Bertrand

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Goodwillie Calculus
• Homotopy Theory
• Algebraic Topology
• homotopy functor
• homotopy limit

Abstract

The origin of these investigations was the successful attempt by myself and coauthors to generalize rational equivalences of two constructions which suggest possible definitions of deRham cohomology of “brave new” rings, one of Rezk (using a cosimplicial resolution) and the other by Waldhausen (using a variant of Goodwillie’s Taylor tower). The proof of agreement of these constructions relies heavily on the fact that the functors involved take values in Spectra. Goodwillie conjectured the extension of these results to include the case of functors taking value in Spaces. The main result of my thesis is a proof of this conjecture (Theorem 7.1.2), using significantly different methods than in the stable setting of the joint work. This makes strong use of the intermediate constructions T_n F in Goodwillie’s Calculus of homotopy functors. I give a new model which naturally gives rise to a new family of towers filtering the Taylor Tower of a functor. I also establish a surprising equivalence between the homotopy inverse limits of these towers and the homotopy inverse limits of certain cosimplicial resolutions. This equivalence gives a greatly simplified construction for the homotopy inverse limit of the Taylor tower of a functor F under general assumptions.

Graduation Semester

2011-12

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http://hdl.handle.net/2142/29665

Copyright and License Information

Copyright 2011 Rosona Eldred

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