KonovalovN042023D.pdf (922.24 kB)
Algebraic Goodwillie Spectral Sequence
thesis
posted on 2023-04-11, 00:00 authored by Nikolai KonovalovLet sL be the infinity-category of simplicial restricted Lie algebras over F, the algebraic closure of a finite field F_p. By the work of A. K. Bousfield et al. on the unstable Adams spectral sequence, the category sL can be viewed as an algebraic approximation of the infinity-category of pointed p-complete spaces. We study the functor calculus in the category sL. More specifically, we consider the Taylor tower for the functor L^r of a free simplicial restricted Lie algebra together with the associated Goodwillie spectral sequence. We show that this spectral sequence evaluated at Sigma^l F, l>=0 degenerates on the third page after a suitable re-indexing, which proves an algebraic version of the Whitehead conjecture.
In our proof we compute explicitly the differentials of the Goodwillie spectral sequence in terms of the Lambda-algebra of A. K. Bousfield et al. and the Dyer-Lashof-Lie power operations, which naturally act on the homology groups of a spectral Lie algebra. As an essential ingredient of our calculations, we establish a general Leibniz rule in functor calculus associated to the composition of mapping spaces, which conceptualizes certain formulas of W. H. Lin. Also, as a byproduct, we identify previously unknown Adem relations for the Dyer-Lashof-Lie operations in the odd-primary case.
In our proof we compute explicitly the differentials of the Goodwillie spectral sequence in terms of the Lambda-algebra of A. K. Bousfield et al. and the Dyer-Lashof-Lie power operations, which naturally act on the homology groups of a spectral Lie algebra. As an essential ingredient of our calculations, we establish a general Leibniz rule in functor calculus associated to the composition of mapping spaces, which conceptualizes certain formulas of W. H. Lin. Also, as a byproduct, we identify previously unknown Adem relations for the Dyer-Lashof-Lie operations in the odd-primary case.
History
Date Modified
2023-04-17Defense Date
2023-03-31CIP Code
- 27.0101
Research Director(s)
Mark J. BehrensDegree
- Doctor of Philosophy
Degree Level
- Doctoral Dissertation
Alternate Identifier
1376262168OCLC Number
1376262168Program Name
- Mathematics
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