Example s.t. the unbased loop-space is not $\Omega X \times X$

Question

For a connected pointed CW-complex $X$, let us write (as usual) $\Omega X$ for the space of based loops at $X$. I am looking for an example where the space $\Omega' X$ of all (unbased) loops in $X$ is not weakly equivalent to $X \times \Omega X$.

Answer 1

Let $G$ be a discrete group. Let $P$ denote the groupoid whose objects are the elements of $G$, and whose morphisms from $a$ to $b$ are the elements $g\in G$ such that $gag^{-1}=b$. It is then a standard fact that $\Omega'BG=BP$. Less naturally but more concretely, if we choose a set $C$ of representatives for the conjugacy classes in $G$, we get an equivalence $\Omega'BG=\coprod_{c\in C}BZ_G(c)$. If $G$ is nonabelian then this will be different from $BG\times \Omega BG=BG\times G$. For example, you could take $G$ to be the free group on two generators, so $BG$ is a figure eight.

UPDATE: I can't immediately remember a convenient reference for the above "standard fact". However, if we regard $\mathbb{Z}$ and $G$ as groupoids with one object, then there is a functor $e\colon\mathbb{Z}\times P\to G$ given by $e((n,g)\colon a \to b)=g^n$. This gives a map $$ Be \colon S^1\times BP = B(\mathbb{Z}\times P) \to BG, $$ and adjointly a map $BP\to\text{Map}(S^1,BG)=\Omega'BG$, which is the one you want. Another way to analyse this is to consider $S^1$ as the pushout of two semicircles along a copy of $S^0$. After applying the functor $\text{Map}(-,BG)$ (and recalling that the semicircles are contractible) we get a homotopy pullback diagram $$ \begin{array}{ccc} \Omega'BG & \to & BG \\ \downarrow && \downarrow \\ BG & \to & BG\times BG \end{array} $$ After thinking a bit about how the maps work, one can recover the previous description of $\Omega'BG$.

Answer 2

The unbased loop space is also known as the free loop space and is often also denoted by $LX$ or $\Lambda X$. If you compute the homology of $LX$ (which might be difficult in general) you will often see that it does not agree with that of $X\times \Omega X$. In particular, one discovers that for $X = S^{2n}$ and $X = \mathbb{CP}^n$ there is torsion in the homology, but no torsion in the homology of $X\times \Omega X$. (The original computation is in Ziller, The free loop space of globally symmetric spaces using Morse theory and it was recomputed by Nora Seeliger using the Serre spectral sequence.)

Often, as in these examples, you already see the difference in rational cohomology, which can usually be computed more easily using rational homotopy theory; a Sullivan minimal model of $X$ gives a minimal model for $LX$. You can find the details about that either in Rational Homotopy Theory: A Brief Introduction or the original article by Sullivan and Vigué-Poirrier .