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### 12 Applications

This chapter was added in April 2018 for version 2.66 of XMod. Initially it describes crossed modules for free loop spaces. Further applications may arise in due course.

#### 12.1 Free Loop Spaces

These functions have been used to produce examples for Ronald Brown's paper Crossed modules, and the homotopy $$2$$-type of a free loop space [Bro18]. The relevant theorem in that paper is as follows.

Let $$\calM = (\partial : M \to P)$$ be a crossed module of groups and let $$X = B\calM$$ be the classifying space of $$\calM$$. Then the components of $$LX$$, the free loop space on $$X$$, are determined by equivalence classes of elements $$a \in P$$ where $$a,a'$$ are equivalent if and only if there are elements $$m \in M,\, p \in P$$ such that $$a'= p + a - \partial m - p$$.

Further the homotopy $$2$$-type of a component of $$LX$$ given by $$a \in P$$ is determined by the crossed module of groups $$L\calM[a] = (\partial_a : M \to P(a))$$ where:

• $$P(a)$$ is the subgroup of the cat$$^1$$-group $$G = P \ltimes M$$ such that $$\partial m = [p,a] = -p-a+p+a$$;

• $$\partial_a(m) = (\partial m, m^{-1}m^a)$$ for $$m \in M$$;

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

In particular $$\pi_1(LX,a)$$ is isomorphic to $$\mathrm{cokernel}(\partial_a)$$, 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 $$\pi_1(X,*)$$.

There is an exact sequence $$\pi \stackrel{\,\phi\,}{\to} \pi \to \pi_1(LX,a) \to C_{\bar{a}}(\pi_1(X,*)) \to 1$$, in which $$\pi = \pi_2(X,*)$$, and $$\phi$$ is the morphism $$m \mapsto m^{-1}m^a$$.

##### 12.1-1 LoopClasses
 ‣ LoopClasses( M ) ( operation )
 ‣ LoopsXMod( M, a ) ( operation )
 ‣ AllLoopsXMod( M ) ( operation )

The operation LoopClasses computes the equivalence classes $$[a]$$ described above. These are all unions of conjugacy classes.

The operation LoopsXMod(M,a) calculates the crossed module $$L\calM[a]$$ described in the theorem.

The operation AllLoopsXMod(M) returns a list of crossed modules, one for each equivalence class of elements $$[a] \subseteq P$$.

In the example below the automorphism crossed module X8 has $$M \cong C_2^3$$ and $$P = PSL(3,2)$$ is the automorphism group of $$M$$. There are $$6$$ equivalence classes which, in this case, are identical with the conjugacy classes. For each $$LX$$ calculated, the IdGroup (2.8-1) is printed out.


gap> SetName( k8, "k8" );
gap> Y8 := XModByAutomorphismGroup( k8 );;
gap> X8 := Image( IsomorphismPerm2DimensionalGroup( Y8 ) );;
gap> SetName( X8, "X8" );
gap> Print( "X8: ", Size( X8 ), " : ", StructureDescription( X8 ), "\n" );
X8: [ 8, 168 ] : [ "C2 x C2 x C2", "PSL(3,2)" ]
gap> classes := LoopClasses( X8 );;
gap> List( classes, c -> Length(c) );
[ 1, 21, 56, 42, 24, 24 ]
gap> LX := LoopsXMod( X8, (1,2)(5,6) );;
gap> Size2d( LX );
[ 8, 64 ]
gap> IdGroup( LX );
[ [ 8, 5 ], [ 64, 138 ] ]
gap> SetInfoLevel( InfoXMod, 1 );
gap> LX8 := AllLoopsXMod( X8 );;
#I  LoopsXMod with a = (),  IdGroup = [ [ 8, 5 ], [ 1344, 11686 ] ]
#I  LoopsXMod with a = (4,5)(6,7),  IdGroup = [ [ 8, 5 ], [ 64, 138 ] ]
#I  LoopsXMod with a = (2,3)(4,6,5,7),  IdGroup = [ [ 8, 5 ], [ 32, 6 ] ]
#I  LoopsXMod with a = (2,4,6)(3,5,7),  IdGroup = [ [ 8, 5 ], [ 24, 13 ] ]
#I  LoopsXMod with a = (1,2,4,3,6,7,5),  IdGroup = [ [ 8, 5 ], [ 56, 11 ] ]
#I  LoopsXMod with a = (1,2,4,5,7,3,6),  IdGroup = [ [ 8, 5 ], [ 56, 11 ] ]
gap> iso := IsomorphismGroups( Range( LX ), Range( LX8[2] ) );;
gap> iso = fail;
false


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