Loop structures in Taylor towers
Loop structures in Taylor towers
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泰勒塔的环形结构
DOI：
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发表时间：
2008
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通讯作者：
Kathryn Lesh
中科院分区：
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作者：
G. Arone;W. Dwyer;Kathryn Lesh
我们研究了Goodwillie的同伦函子演算和韦斯的正交演算中的齐次函子之间的自然变换空间。我们用同伦不动点的构造来描述这种自然变换空间。我们的主要应用程序使用这种描述与Segal猜想相结合，以获得一个delooping定理连接地图的古德威利塔的身份和韦斯塔的BU.V/。对这种去环的兴趣源于第一和第三作者[4]所做的声明，这些塔为某些谱的射影链复形提供了收缩同伦的来源。55 P65; 55 P47，18 G55 1引言和记法本文研究了韦斯正交演算意义下的齐次函子[13]。更准确地说，我们计算的空间自然之间的转换，这样的函子，我们给出了一些例子和应用。我们的主要应用是一个delooping定理连接映射的古德威利塔的单位函子的点空间和韦斯塔的函子V7！BU.V/.这个去环定理的动机将在后面的介绍中讨论。为了说明我们的结果，我们总结了这种情况下的常用符号。设F是从点空间或从有限维真实的或复向量空间到点空间或谱的函子。古德威利和韦斯演算赋予这样一个函子F一个“泰勒”函子塔！Pn F！Pn 1 F！以及一个从F到塔的同伦逆极限的自然映射，这通常是一个弱同伦等价。映射Pn F的同伦纤维！Pn 1 F通常表示为Dn F，称为“F的第n个均匀层”，而Pn F称为“F的第n个泰勒多项式”。根据是否
We study spaces of natural transformations between homogeneous functors in Goodwillie’s calculus of homotopy functors and in Weiss’s orthogonal calculus. We give a description of such spaces of natural transformations in terms of the homotopy fixed point construction. Our main application uses this description in combination with the Segal Conjecture to obtain a delooping theorem for connecting maps in the Goodwillie tower of the identity and in the Weiss tower of BU.V/. The interest in such deloopings stems from conjectures made by the first and the third author [4] that these towers provide a source of contracting homotopies for certain projective chain complexes of spectra. 55P65; 55P47, 18G55 1 Introduction and notation In this paper, we study homogeneous functors in the sense of Weiss’s orthogonal calculus [13]. More precisely, we calculate the space of natural transformations between such functors, and we give a few examples and applications. Our main application is a delooping theorem for connecting maps in the Goodwillie tower of the identity functor for pointed spaces and in the Weiss tower of the functor V7!BU.V/. The motivation for this delooping theorem will be discussed later in this introduction. To state our results, we summarize the usual notation for this context. Let F be a functor from pointed spaces or from finite-dimensional real or complex vector spaces to pointed spaces or spectra. Goodwillie and Weiss calculus assign to such a functor F a “Taylor” tower of functors ! Pn F! Pn 1 F! ; together with a natural map from F to the homotopy inverse limit of the tower that is often a weak homotopy equivalence. The homotopy fiber of the map Pn F! Pn 1 F is customarily denoted Dn F and is referred to as “the nth homogeneous layer of F ,” while Pn F is referred to as “the nth Taylor polynomial of F .” Depending on whether