GAP (HAP commands) - Chapter 5: Group theoretic computations

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5 Group theoretic computations
5.1 Third homotopy group of a supsension of an Eilenberg-MacLane space
5.2 Representations of knot quandles
5.3 Aspherical 2-complexes
5.4 Bogomolov multiplier

### 5 Group theoretic computations

#### 5.1 Third homotopy group of a supsension of an Eilenberg-MacLane space

The following example uses the nonabelian tensor square of groups to compute the third homotopy group

π_3(S(K(G,1))) = Z^30

of the suspension of the Eigenberg-MacLane space K(G,1) for G the free nilpotent group of class 2 on four generators.

    gap> F:=FreeGroup(4);;G:=NilpotentQuotient(F,2);;
    gap> ThirdHomotopyGroupOfSuspensionB(G);
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0 ]

#### 5.2 Representations of knot quandles

The following example constructs the finitely presented quandles associated to the granny knot and square knot, and then computes the number of quandle homomorphisms from these two finitely prresented quandles to the 17-th quandle in HAP's library of connected quandles of order 24. The number of homomorphisms differs between the two cases. The computation therefore establishes that the complement in R^3 of the granny knot is not homeomorphic to the complement of the square knot.

    gap> Q:=ConnectedQuandle(24,17,"import");;
    gap> K:=PureCubicalKnot(3,1);;
    gap> L:=ReflectedCubicalKnot(K);;
    gap> square:=KnotSum(K,L);;
    gap> granny:=KnotSum(K,K);;
    gap> gcsquare:=GaussCodeOfPureCubicalKnot(square);;
    gap> gcgranny:=GaussCodeOfPureCubicalKnot(granny);;
    gap> Qsquare:=PresentationKnotQuandle(gcsquare);;
    gap> Qgranny:=PresentationKnotQuandle(gcgranny);;
    gap> NumberOfHomomorphisms(Qsquare,Q);
    408
    gap> NumberOfHomomorphisms(Qgranny,Q);
    24

#### 5.3 Aspherical 2-complexes

The following example uses Polymake's linear programming routines to establish that the 2-complex associated to the group presentation <x,y,z : xyx=yxy, yzy=zyz, xzx=zxz> is aspherical (that is, has contractible universal cover). The presentation is Tietze equivalent to the presentation used in the computer code, and the associated 2-complexes are thus homotopy equivalent.

    gap> F:=FreeGroup(6);;
    gap> x:=F.1;;y:=F.2;;z:=F.3;;a:=F.4;;b:=F.5;;c:=F.6;;
    gap> rels:=[a^-1*x*y, b^-1*y*z, c^-1*z*x, a*x*(y*a)^-1,
    >    b*y*(z*b)^-1, c*z*(x*c)^-1];;
    gap> Print(IsAspherical(F,rels),"\n");
    Presentation is aspherical.

    true

#### 5.4 Bogomolov multiplier

The Bogomolov multiplier of a group is an isoclinism invariant. Using this property, the following example shows that there are precisely three groups of order 243 with non-trivial Bogomolov multiplier. The groups in question are numbered 28, 29 and 30 in GAP's library of small groups of order 243.

    gap> L:=AllSmallGroups(3^5);;
    gap> C:=IsoclinismClasses(L);;
    gap> for c in C do
    > if Length(BogomolovMultiplier(c[1]))>0 then
    > Print(List(c,g->IdGroup(g)),"\n\n\n"); fi;
    > od;
    [ [ 243, 28 ], [ 243, 29 ], [ 243, 30 ] ]

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