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

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

Funding Information:
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 © 2022, Prasit Bhattacharya, Bertrand Guillou and Ang Li. Permission to copy for private use granted.

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

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

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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) ^ 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",

note = "Funding Information: 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 {\textcopyright} 2022, Prasit Bhattacharya, Bertrand Guillou and Ang Li. Permission to copy for private use granted. Publisher Copyright: {\textcopyright} 2022. Prasit Bhattacharya, Bertrand Guillou and Ang Li. All Rights Reserved.",

year = "2022",

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

language = "English",

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AU - Bhattacharya, Prasit

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N1 - Funding Information: 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 © 2022, Prasit Bhattacharya, Bertrand Guillou and Ang Li. Permission to copy for private use granted. Publisher Copyright: © 2022. Prasit Bhattacharya, Bertrand Guillou and Ang Li. All Rights Reserved.

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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.

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