Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory
Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory
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基于刚性无限循环空间理论的环谱、方向和托姆谱的单位
DOI：
10.1112/jtopol/jtu009
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发表时间：
2014
影响因子：
1.1
通讯作者：
Ando M
中科院分区：
文献类型：
--
作者：
Ando M
We extend the theory of Thom spectra and the associated obstruction theory for orientations in order to support the construction of the $E_{\infty }$ string orientation of $tmf$, the spectrum of topological modular forms. Specifically, we show that, for an $E_{\infty }$ ring spectrum $A$, the classical construction of $gl_{1} {A}$, the spectrum of units, is the right adjoint of the functor \[\Sigma ^{\infty }_{+ } \Omega ^{\infty } \colon \mathrm {ho} (\mbox {{connective spectra}}) \longrightarrow \mathrm {ho} ({\text {{$E_{\infty } $ ring spectra}}}).\] To a map of spectra \[f\colon b \longrightarrow bgl_{1} {A},\] we associate an $E_{\infty }$$A$-algebra Thom spectrum $Mf$, which admits an $E_{\infty }$$A$-algebra map to $R$ if and only if the composition \[b \longrightarrow bgl_{1} {A} \longrightarrow bgl_{1} {R}\] is null; the classical case developed by May, Quinn, Ray, and Tornehave arises when $A$ is the sphere spectrum. We develop the analogous theory for $A_{\infty }$ ring spectra: if $A$ is an $A_{\infty }$ ring spectrum, then to a map of spaces \[f\colon B \longrightarrow B{ GL}_{1} {A},\] we associate an $A$-module Thom spectrum $Mf,$ which admits an $R$-orientation if and only if \[B \longrightarrow B{ GL}_{1} {A} \longrightarrow B{ GL}_{1} {R}\] is null. Our work is based on a new model of the Thom spectrum as a derived smash product.
We extend the theory of Thom spectra and the associated obstruction theory for orientations in order to support the construction of the $E_{\infty }$ string orientation of $tmf$, the spectrum of topological modular forms. Specifically, we show that, for an $E_{\infty }$ ring spectrum $A$, the classical construction of $gl_{1} {A}$, the spectrum of units, is the right adjoint of the functor \[\Sigma ^{\infty }_{+ } \Omega ^{\infty } \colon \mathrm {ho} (\mbox {{connective spectra}}) \longrightarrow \mathrm {ho} ({\text {{$E_{\infty } $ ring spectra}}}).\] To a map of spectra \[f\colon b \longrightarrow bgl_{1} {A},\] we associate an $E_{\infty }$$A$-algebra Thom spectrum $Mf$, which admits an $E_{\infty }$$A$-algebra map to $R$ if and only if the composition \[b \longrightarrow bgl_{1} {A} \longrightarrow bgl_{1} {R}\] is null; the classical case developed by May, Quinn, Ray, and Tornehave arises when $A$ is the sphere spectrum. We develop the analogous theory for $A_{\infty }$ ring spectra: if $A$ is an $A_{\infty }$ ring spectrum, then to a map of spaces \[f\colon B \longrightarrow B{ GL}_{1} {A},\] we associate an $A$-module Thom spectrum $Mf,$ which admits an $R$-orientation if and only if \[B \longrightarrow B{ GL}_{1} {A} \longrightarrow B{ GL}_{1} {R}\] is null. Our work is based on a new model of the Thom spectrum as a derived smash product.
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几何拓扑：局域化、周期性和伽罗瓦对称性：1970 年 MIT 笔记
DOI：
--
发表时间：
2005
期刊：
影响因子：
--
作者：
D. Sullivan;A. Ranicki
通讯作者：
A. Ranicki
等变稳定同伦理论
影响因子：
1.4
作者：
L. Lewis;Jon P. May;M. Steinberger
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M. Steinberger
DOI：
10.1515/form.2002.016
发表时间：
2000-10
期刊：
arXiv: Algebraic Topology
影响因子：
--
作者：
V. Smirnov
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V. Smirnov
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影响因子：
2
作者：
A. Blumberg;R. Cohen;Christian Schlichtkrull
通讯作者：
Christian Schlichtkrull
影响因子：
0.7
作者：
John A. Lind
通讯作者：
John A. Lind