AN (Formula presented)-MOTIVIC v1-SELF-MAP OF PERIODICITY 1

Prasit Bhattacharya; Bertrand Guillou; Ang Li

doi:10.4310/HHA.2022.v24.n1.a15

AN (Formula presented)-MOTIVIC v₁-SELF-MAP OF PERIODICITY 1

Prasit Bhattacharya
, Bertrand Guillou
, Ang Li

Mathematics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

We consider a nontrivial action of C2 on the type 1 spectrum Y: = M₂(1) ^ C(n), which is well-known for admitting a 1-periodic v₁-self-map. The resultant finite C2-equivariant spectrum y^C2 can also be viewed as the complex points of a finite (Formula presented)-motivic spectrum (Formula presented). In this paper, we show that one of the 1-periodic v₁-self-maps of y can be lifted to a self-map of y^C2 as well as (Formula presented). Further, the cofiber of the self-map of (Formula presented) is a realization of the subalgebra (Formula presented) (1) of the R-motivic Steenrod algebra. We also show that the C2-equivariant self-map is nilpotent on the geometric fixed-points of y^C2.

Original language	English
Pages (from-to)	299-324
Number of pages	26
Journal	Homology, Homotopy and Applications
Volume	24
Issue number	1
DOIs	https://doi.org/10.4310/HHA.2022.v24.n1.a15
State	Published - 2022

Bibliographical note

Publisher Copyright:
© 2022. Prasit Bhattacharya, Bertrand Guillou and Ang Li. All Rights Reserved.

Funding

B. Guillou and A. Li were supported by NSF grants DMS-1710379 and DMS-2003204. Received October 30, 2020, revised February 22, 2021; published on April 13, 2022. 2010 Mathematics Subject Classification: 14F42, 55Q51, 55Q91. Key words and phrases: self-map, motivic homotopy, equivariant homotopy. Article available at http://dx.doi.org/10.4310/HHA.2022.v24.n1.a15 Copyright \u00A9 2022, Prasit Bhattacharya, Bertrand Guillou and Ang Li. Permission to copy for private use granted.

Funders	Funder number
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	DMS-2003204, 2003204, DMS-1710379

Keywords

equivariant homotopy
motivic homotopy
self-map

ASJC Scopus subject areas

Mathematics (miscellaneous)

Access to Document

10.4310/HHA.2022.v24.n1.a15

Cite this

@article{57b06691a3584d1c8c42ce7dcaba3125,

title = "AN (Formula presented)-MOTIVIC v1-SELF-MAP OF PERIODICITY 1",

abstract = "We consider a nontrivial action of C2 on the type 1 spectrum Y: = M2(1) \textasciicircum{} C(n), which is well-known for admitting a 1-periodic v1-self-map. The resultant finite C2-equivariant spectrum yC2 can also be viewed as the complex points of a finite (Formula presented)-motivic spectrum (Formula presented). In this paper, we show that one of the 1-periodic v1-self-maps of y can be lifted to a self-map of yC2 as well as (Formula presented). Further, the cofiber of the self-map of (Formula presented) is a realization of the subalgebra (Formula presented) (1) of the R-motivic Steenrod algebra. We also show that the C2-equivariant self-map is nilpotent on the geometric fixed-points of yC2.",

keywords = "equivariant homotopy, motivic homotopy, self-map",

author = "Prasit Bhattacharya and Bertrand Guillou and Ang Li",

year = "2022",

doi = "10.4310/HHA.2022.v24.n1.a15",

language = "English",

volume = "24",

pages = "299--324",

number = "1",

}

TY - JOUR

T1 - AN (Formula presented)-MOTIVIC v1-SELF-MAP OF PERIODICITY 1

AU - Bhattacharya, Prasit

AU - Guillou, Bertrand

AU - Li, Ang

PY - 2022

Y1 - 2022

N2 - We consider a nontrivial action of C2 on the type 1 spectrum Y: = M2(1) ^ C(n), which is well-known for admitting a 1-periodic v1-self-map. The resultant finite C2-equivariant spectrum yC2 can also be viewed as the complex points of a finite (Formula presented)-motivic spectrum (Formula presented). In this paper, we show that one of the 1-periodic v1-self-maps of y can be lifted to a self-map of yC2 as well as (Formula presented). Further, the cofiber of the self-map of (Formula presented) is a realization of the subalgebra (Formula presented) (1) of the R-motivic Steenrod algebra. We also show that the C2-equivariant self-map is nilpotent on the geometric fixed-points of yC2.

AB - We consider a nontrivial action of C2 on the type 1 spectrum Y: = M2(1) ^ C(n), which is well-known for admitting a 1-periodic v1-self-map. The resultant finite C2-equivariant spectrum yC2 can also be viewed as the complex points of a finite (Formula presented)-motivic spectrum (Formula presented). In this paper, we show that one of the 1-periodic v1-self-maps of y can be lifted to a self-map of yC2 as well as (Formula presented). Further, the cofiber of the self-map of (Formula presented) is a realization of the subalgebra (Formula presented) (1) of the R-motivic Steenrod algebra. We also show that the C2-equivariant self-map is nilpotent on the geometric fixed-points of yC2.

KW - equivariant homotopy

KW - motivic homotopy

KW - self-map

UR - https://www.scopus.com/pages/publications/85129718408

UR - https://www.scopus.com/pages/publications/85129718408#tab=citedBy

U2 - 10.4310/HHA.2022.v24.n1.a15

DO - 10.4310/HHA.2022.v24.n1.a15

M3 - Article

AN - SCOPUS:85129718408

SN - 1532-0073

VL - 24

SP - 299

EP - 324

JO - Homology, Homotopy and Applications

JF - Homology, Homotopy and Applications

IS - 1

ER -