PreprintCC BY 4.0arXiv
An abelian model for the Goodwillie tower of the circle
Abstract
We study the Goodwillie tower of the identity evaluated at the circle. Our main result is the p-local equivalence ΩPpkI(S1) ≃ ℤ × Ω∞+2kτ>0𝕊ppk obtained from the work of Behrens and Kuhn on the Whitehead conjecture. In particular, this identifies the Goodwillie approximations of the circle as infinite loop spaces, a phenomenon which does not hold in general. The equivalence also reduces the computation of their homotopy groups to a stable problem, making them accessible to tools such as the Adams spectral sequence and to computer calculations. We place these computations in a more general framework by developing a Goodwillie-calculus analogue of the Gray sequence, which relates the homotopy groups of successive stages of the tower.
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BibTeX
@misc{nervo2026abelian,
title = {{An abelian model for the Goodwillie tower of the circle}},
author = {Marco Nervo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31269},
eprint = {2609.31269},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31269},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An abelian model for the Goodwillie tower of the circle
AU - Marco Nervo
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31269
UR - https://arxiv.org/abs/2609.31269
AB - We study the Goodwillie tower of the identity evaluated at the circle. Our main result is the p-local equivalence ΩP_p^kI(S^1) ≃ ℤ × Ω^∞+2kτ_>0𝕊p^p^k obtained from the work of Behrens and Kuhn on the Whitehead conjecture. In particular, this identifies the Goodwillie approximations of the circle as infinite loop spaces, a phenomenon which does not hold in general. The equivalence also reduces the computation of their homotopy groups to a stable problem, making them accessible to tools such as the Adams spectral sequence and to computer calculations. We place these computations in a more general framework by developing a Goodwillie-calculus analogue of the Gray sequence, which relates the homotopy groups of successive stages of the tower.
ER -