2 omitted one of double `the'

You might like to look at my preprint http://arxiv.org/abs/1003.5617 on the homotopy 2-type of a free loop space $LX$. It assumes that $X$ is a 2-type, i.e. the classifying space of a crossed module, and then gives precise formulae for crossed modules representing the the 2-types of the components.

I am aware that the main interest in free loop spaces seems to be their homology, and I can't see how these results help on that.

The paper is planned to be revised with Chris Wensley and to include specific computer calculations, hence the delay.

I might as well quote the theorem.

Let $\mathcal M$ be the crossed module of groups $\delta: M \to P$ and let $X=B\mathcal M$ be the classifying space of $\mathcal M$. Then the components of $LX$, the free loop space on $X$, are determined by equivalence classes of elements $a \in P$ where $a,b$ are equivalent if and only if there are elements $m \in M, p \in P$ such that $$b= p + a + \delta m -p.$$ Further the homotopy $2$-type of a component of $LX$ given by $a \in P$ is determined by the crossed module of groups $L\mathcal M [a]=(\delta_a: M \to P(a))$ where

(i) $P(a)$ is the group of elements $(m,p)\in M \times P$ such that $\delta m= [a,p]$, with composition $(n,q)+(m,p)= (m+n^p,q+p)$;

(ii) $\delta_a(m)= ( -m^a + m,\delta m)$, for $m \in M$;

(iii) the action of $P(a)$ on $M$ is given by $n^{(m,p)}= n^p$ for $n \in M, (m,p) \in P(a)$.

In particular $\pi_1(LX,a)$ is isomorphic to Cok $\delta_a$, and $\pi_2(LX,a) \cong \pi_2(X,*)^{\bar{a}}$, the elements of $\pi_2(X,*)$ fixed under the action of $\bar{a}$, the class of $a$ in $G=\pi_1(X,*)$.

1

You might like to look at my preprint http://arxiv.org/abs/1003.5617 on the homotopy 2-type of a free loop space $LX$. It assumes that $X$ is a 2-type, i.e. the classifying space of a crossed module, and then gives precise formulae for crossed modules representing the the 2-types of the components.

I am aware that the main interest in free loop spaces seems to be their homology, and I can't see how these results help on that.

The paper is planned to be revised with Chris Wensley and to include specific computer calculations, hence the delay.

I might as well quote the theorem.

Let $\mathcal M$ be the crossed module of groups $\delta: M \to P$ and let $X=B\mathcal M$ be the classifying space of $\mathcal M$. Then the components of $LX$, the free loop space on $X$, are determined by equivalence classes of elements $a \in P$ where $a,b$ are equivalent if and only if there are elements $m \in M, p \in P$ such that $$b= p + a + \delta m -p.$$ Further the homotopy $2$-type of a component of $LX$ given by $a \in P$ is determined by the crossed module of groups $L\mathcal M [a]=(\delta_a: M \to P(a))$ where

(i) $P(a)$ is the group of elements $(m,p)\in M \times P$ such that $\delta m= [a,p]$, with composition $(n,q)+(m,p)= (m+n^p,q+p)$;

(ii) $\delta_a(m)= ( -m^a + m,\delta m)$, for $m \in M$;

(iii) the action of $P(a)$ on $M$ is given by $n^{(m,p)}= n^p$ for $n \in M, (m,p) \in P(a)$.

In particular $\pi_1(LX,a)$ is isomorphic to Cok $\delta_a$, and $\pi_2(LX,a) \cong \pi_2(X,*)^{\bar{a}}$, the elements of $\pi_2(X,*)$ fixed under the action of $\bar{a}$, the class of $a$ in $G=\pi_1(X,*)$.