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.
These functions have been used to produce examples for Ronald Brown's paper
<E>Crossed modules, and the homotopy <M>2</M>-type of a free loop space</E>
<Cite Key="B18" />.
The relevant theorem in that paper is as follows.
<P/>
<B>Theorem 2.1</B>
<E>
Let <M>\calM = (\partial : M \to P)</M> be a crossed module of groups
and let <M>X = B\calM</M> be the classifying space of <M>\calM</M>.
Then the components of <M>LX</M>, the free loop space on <M>X</M>,
are determined by equivalence classes of elements <M>a \in P</M>
where <M>a,a' are equivalent if and only if there are elements
<M>m \in M,\, p \in P</M> such that <M>a'= p + a - \partial m - p.
</E>
<P/>
<E>
Further the homotopy <M>2</M>-type of a component of <M>LX</M>
given by <M>a \in P</M> is determined by the crossed module of groups
<M>L\calM[a] = (\partial_a : M \to P(a))</M> where:
</E>
<List>
<Item><E>
<M>P(a)</M> is the subgroup of the cat<M>^1</M>-group
<M>G = P \ltimes M</M> such that <M>\partial m = [p,a] = -p-a+p+a</M>;
</E></Item>
<Item><E>
<M>\partial_a(m) = (\partial m, m^{-1}m^a)</M> for <M>m \in M</M>;
</E></Item>
<Item><E>
the action of <M>P(a)</M> on <M>M</M> is given by
<M>n^{(p,m)} = n^p</M> for <M>n \in M,\, (p,m) \in P(a)</M>.
</E></Item>
</List>
<P/>
<E>
In particular <M>\pi_1(LX,a)</M> is isomorphic to
<M>\mathrm{cokernel}(\partial_a)</M>,
and <M>\pi_2(LX,a) \cong \pi_2(X,*)^{\bar{a}}</M>,
the elements of <M>\pi_2(X,*)</M> fixed under the action of <M>\bar{a}</M>,
the class of <M>a</M> in <M>\pi_1(X,*)</M>.
</E>
<P/>
<E>
There is an exact sequence
<M> \pi \stackrel{\,\phi\,}{\to} \pi \to \pi_1(LX,a)
\to C_{\bar{a}}(\pi_1(X,*)) \to 1</M>,
in which <M>\pi = \pi_2(X,*)</M>, and <M>\phi</M>
is the morphism <M>m \mapsto m^{-1}m^a</M>.
</E>
<P/>
<ManSection>
<Oper Name="LoopClasses"
Arg="M" />
<Oper Name="LoopsXMod"
Arg="M a" />
<Oper Name="AllLoopsXMod"
Arg="M" />
<Description>
The operation <C>LoopClasses</C> computes the equivalence
classes <M>[a]</M> described above.
These are all unions of conjugacy classes.
<P/>
The operation <C>LoopsXMod(M,a)</C> calculates the crossed module
<M>L\calM[a]</M> described in the theorem.
<P/>
The operation <C>AllLoopsXMod(M)</C> returns a list of crossed modules,
one for each equivalence class of elements <M>[a] \subseteq P</M>.
<P/>
In the example below the automorphism crossed module <C>X8</C> has
<M>M \cong C_2^3</M> and <M>P = PSL(3,2)</M>
is the automorphism group of <M>M</M>.
There are <M>6</M> equivalence classes which, in this case,
are identical with the conjugacy classes.
For each <M>LX</M> calculated, the <Ref Meth="IdGroup" Label="for 2d-groups"/> is printed out.
</Description>
</ManSection>
<P/>
<Example>
<![CDATA[
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
]]>
</Example>
</Section>
</Chapter>
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