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The Segal conjecture for large objects and semiadditive lifts
Abstract
We prove a projection formula for homotopy fixed points of trivial actions of finite p-groups. More precisely, for every bounded-below, p-complete spectrum X whose mod-p homology is I-complete, the canonical map (𝕊p)BG⊗ X overset≃→ XBG is an equivalence for every finite p-group G, where I is the augmentation ideal of the Steenrod algebra. Building on earlier joint work with Burklund, our proof combines the Adams--Gunawardena--Miller calculation with an adaptation of Carlsson's induction. As an application, using the work of Yuan, we construct height-one semiadditive lifts of such spectra and show that their cohomology of p-finite spaces depends only on the $1$-type.
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BibTeX
@misc{subramanian2026segal,
title = {{The Segal conjecture for large objects and semiadditive lifts}},
author = {Vignesh Subramanian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40209},
eprint = {2610.40209},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40209},
}
RIS
TY - UNPB
TI - The Segal conjecture for large objects and semiadditive lifts
AU - Vignesh Subramanian
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40209
UR - https://arxiv.org/abs/2610.40209
AB - We prove a projection formula for homotopy fixed points of trivial actions of finite p-groups. More precisely, for every bounded-below, p-complete spectrum X whose mod-p homology is I-complete, the canonical map (𝕊_p)^BG⊗ X overset≃→ X^BG is an equivalence for every finite p-group G, where I is the augmentation ideal of the Steenrod algebra. Building on earlier joint work with Burklund, our proof combines the Adams--Gunawardena--Miller calculation with an adaptation of Carlsson's induction. As an application, using the work of Yuan, we construct height-one semiadditive lifts of such spectra and show that their cohomology of p-finite spaces depends only on the $1$-type.
ER -