Idempotent symmetries of spaces

Wojciech Chacholski
Monday, July 22, 2013 - from 14:50 to 15:30
This is a joint work with E.\ Dror Farjoun, R.\ Flores, and J.\ Scherer. Bousfield showed that if $X$Ê is a connected space for which $\text{Map}_{\ast}(X,X)$ Êis discrete and $\pi_1X$ acts trivially on $\pi_0\text{Map}_{\ast}(X,X)$, then $X$ is weakly equivalent to a product of Eilenberg-Mac Lane spaces $\prod_{i\geq 1} K(\pi_i,i)$ with $\pi_1$ abelian. This statemant is what Bousfield called the key lemma as all his results related to understanding the failure of preservation of fibrations by localizations depended on it. The key lemma implies for example that if $X$ Êis simply connected, then the map $\pi_n\colon\text{Map}_{\ast}(X,X)\to \text{Hom}(\pi_nX,\pi_nX)$ is a weak equivalence if and only if $X$Ê is weakly equivalent to the Eilenberg-Mac Lane space $K(\pi_nX,n)$. If $X$ Êis not simply connected, then the situation is much more complicated. Let $G$ be a group. Define $\calb G$ to be the collection of these connected spaces $X$ for which $\pi_1X=G$ and the map $\pi_1\colon\text{Map}_{\ast}(X,X)\to\text{Hom}(G,G)$ is a weak equivalence. Thus a space $X$ belongs to $\calb G$ if the space of self maps $\text{Map}_{\ast}(X,X)$ is discrete and its components are in bijection with the set $\text{Hom}(G,G)$ via the $\pi_1$ map. The Eilenberg-Mac Lane space $K(G,1)$ clearly belongs to $\calb G$. The question is if there are other spaces in $\calb G$. Flores and Scherer shown that the collection $\calb \Sigma_3$ contains a space whose universal cover is the homotopy fiber of the degree $3$ map on the sphere $S^3$, and hence its homotopy groups are non trivial in infinitely many degrees. This is in contrast with what I present in my talk for $G$ nilpotent: \begin{thmab} If $G$ is nilpotent, then any space in $\calb G$ is weakly equivalent to $K(G,1)$. \end{thmab} This theorem is an illustration of a more general phenomena which is reflected by certain symmetries of the category of pointed spaces. One way to understand symmetry of $\text{Spaces}_{\ast}$ is to look at what acts on this category and study functors $\phi\colon\text{Spaces}_{\ast}\to \text{Spaces}_{\ast}$. To understand how such an operation deforms spaces we consider natural transformations $c_X\colon \phi(X)\to X$. A choice of a homotopy invariant operation $\phi\colon\text{Spaces}_{\ast}\to \text{Spaces}_{\ast}$ and a comparison $c_X\colon \phi(X)\to X$ is called a co--augmented functor. Among all co--augmented functors there are the idempotent ones for which the map $\text{Map}_{\ast}(\phi(X),c_X)\colon \text{Map}_{\ast}(\phi(X),\phi(X))\to\text{Map}_{\ast}(\phi(X),X)$ is a weak equivalence for any $X$. The $A$--cellularization $c_{A,X}\colon\text{CW}_AX\to X$ is a typical example of an idempotnent functor. During the talk I will describe the action of idempotent operations on spaces, in particular how such functors deform classifying spaces of nilpotent groups, or more generally nilpotent finite Postnikov sections.