AboutHap

About HAP: Homotopy 2-types and third cohomology

Some classical theory

The third cohomology H³(G,A) of G with coefficients in a G-module A, together with the corresponding 3-cocycles, can be used to classify homotopy 2-types.

A 2-type is a CW-space whose homotopy groups are trivial in dimensions n=0 and n>2. There is an equivalence of categories

(Homotopy category of
connected CW-spaces X
with trivial homotopy
groups pi_n(X) for n>2)

<---------->

(Localisation of the category
of simplicial groups with
Moore complex of length
1, where the localisation is
with respect to homomorphisms
inducing isomorphisms on
homotopy groups)

which reduces the homotopy theory of 2-types to a "computable" algebraic theory. Furthermore, a simplicial group with Moore complex of length 1 can be represented by a group H with two endomorphisms s:H-->H and t:H-->H satisfying the axioms

ss = s, ts = s,
tt = t, st = t,
[Ker(s),Ker(t)] = 1.

Ths data {H,s,t} was termed a cat¹-group by J.-L. Loday since it can be regarded as a group H endowed with one compatible category structure.

The homotopy groups of a cat¹-group H are defined as

pi_1(H) = Image(s)/ t( Kernel(s) ),

pi_2(H) = Intersection( Kernel(s) , Kernel(t) ),

pi_n(H) = 0 for n>2 or n=0.

Note that pi_2(H) is a pi_1(H)-module where the action is induced by conjugation in H.

A number of standard group-theoretic constructions can be viewed naturally as a cat¹-group.

A ZG-module A can be viewed as a cat¹-group (H,s,t) where H is the semi-direct product Ax|G and s(a,g)=(1,g), t(a,g)=(1,g). Here pi_1(H)=G and p_2(H)=A.
A group G with normal subgroup N can be viewed as a cat¹-group (H,s,t) where H is the semi-direct product Nx|G and s(n,g)=(1,g), t(n,g)=(1,ng). Here pi_1(H)=G/N and pi_2(H)=0.
The homomorphism j:G--->Aut(G) which sends elements of G to the corresponding inner automorphism can be viewed as a cat¹-group (H,s,t) where H is the semi-direct product Gx|Aut(G) and s(g,a)=(1,a), t(g,a)=(1,j(g)a). Here pi_1(H)=Out(G) is the outer automorphism group of G and pi_2(H)=Z(G) is the centre of G.

These three examples are implemented in HAP.

For instance, the following commands begin by constructing the cat¹-group of the last example for the group G=SmallGroup(64,134). They then construct the fundamental group of H and then the second homotopy group of H as a pi_1-module. These have orders 8 and 2 respectively.

gap> G:=SmallGroup(64,134);;
gap> H:=AutomorphismGroupAsCatOneGroup(G);;

gap> pi_1:=HomotopyGroup(H,1);;

gap> pi_2:=HomotopyModule(H,2);;

gap> Order(pi_1)
8

gap> Order(ActedGroup(pi_2));
2

The third cohomology H³(pi_1,pi_2) classifies those cat¹-groups H with fundamental group equal to pi_1 and second homotopy module equal to pi_2. The classification is up to a Yoneda equivalence.

The following additional commands show that there are 1024 Yoneda equivalence classes of cat¹-groups with homotopy group pi_1 and homotopy module pi_2 equal to that in our example.

gap> R:=ResolutionFiniteGroup(pi_1,4);;
gap> C:=HomToGModule(R,pi_2);;
gap> CH:=CohomologyModule(C,3);;

gap> AbelianInvariants(ActedGroup(CH));
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]

A 3-cocycle f : pi_1 × pi_1 × pi_1 -----> pi_2 corresponding to the second cohomology class in H³(pi_1,pi_2) can be produced using the following command,

gap> x:=Elements(ActedGroup(CH))[2];;

gap> f:=CH!.representativeCocycle(x);
Standard 3-cocycle

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