AboutHap

About HAP: Third Homotopy Groups Of Suspensions Of Classifying Spaces

The homology groups H_n(X,Z) of a space are reasonably straighforward invariants to compute. Its homotopy groups Pi_n(X) are, by contrast, extremely difficult to compute and the difficulty increases with n. For instance, the homotopy groups of one of the most basic spaces, the 2-dimensional sphere S², are still only known for relatively small n.

When homotopy groups were introduced back in the 1930s they were at first believed to be isomorphic to the homology groups. Heinz Hopf's surprising discovery that Pi₃(S²) = Z soon put pay to that belief.

The sphere can be regarded as a suspension S²=SK(Z,1) of an Eilenberg-Mac Lane space K(Z,1) for the infinite cyclic group Z. There has recently been some interest in computing the third homotopy group Pi₃(SK(G,1)) of the suspension of Eilenberg-Mac Lane spaces for other groups G. (The Hurewicz Theorem shows that Pi₂(G)=G_ab and Pi₁(G)=0.)

A purely group theoretic description of Pi₃(SK(G,1)) was found in [R. Brown & J.-L. Loday, "van Kampen theorems diagrams for diagrams of spaces", Topology 1987]. They described it as a kernel of a group homomorphism

Pi₃(SK(G,1)) = Ker ( µ : G(×)G → G )

where G(×)G denotes a certain "nonabelian tensor square" of the group G. The following command uses this isomorphism to compute Pi₃(SK(G,1)) = Z³⁰for Gthe free nilpotent group of class 2 on four generators. The command implements a method for calculating the nonabelian tensor which is described in [G. Ellis & F. Leonard, "Computations of nonabelian tensor products and Schur multipliers of finite groups", Proc. Royal Irish Acad. 1997]. A result in [A. McDermott, "Nonabelian tensor products", PhD thesis, Galway 1997] enables the implementation to run on certain infinite groups. The computational significance of McDermott's result was pointed out by R.F. Morse.

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 ]

The Blakers-Massey Theorem can be used to show that, for any abelian group A, the third homotopy group Pi₃(SK(A,1)) is isomorphic to the usual tensor square of A. The homotopy groups Pi₃(SK(G,1)) were calculated for all nonabelian groups G of order |G|<31 by R. Brown, D.L. Johnson and E. Robertson in ["The nonabelian tensor square of groups", J. Algebra 1987]. These calculations are recovered by the following commands which use GAP's small groups library.

gap> for i in [2..30] do for G in AllSmallGroups(i) do

> if not IsAbelian(G) then

> name:=IdSmallGroup(G); pi:=ThirdHomotopyGroupOfSuspensionB(G);

> Print("Small group G = ", name," has Pi3(SK(G,1))= ", pi,"\n");

> fi;od;od;
Small group G = [ 6, 1 ] has Pi3(SK(G,1))= [ 2 ]
Small group G = [ 8, 3 ] has Pi3(SK(G,1))= [ 2, 2, 2, 2 ]
Small group G = [ 8, 4 ] has Pi3(SK(G,1))= [ 2, 4, 4 ]
Small group G = [ 10, 1 ] has Pi3(SK(G,1))= [ 2 ]
Small group G = [ 12, 1 ] has Pi3(SK(G,1))= [ 4 ]
Small group G = [ 12, 3 ] has Pi3(SK(G,1))= [ 2, 3 ]
Small group G = [ 12, 4 ] has Pi3(SK(G,1))= [ 2, 2, 2, 2 ]
Small group G = [ 14, 1 ] has Pi3(SK(G,1))= [ 2 ]
Small group G = [ 16, 3 ] has Pi3(SK(G,1))= [ 2, 2, 2, 2, 4 ]
Small group G = [ 16, 4 ] has Pi3(SK(G,1))= [ 2, 2, 4, 4 ]
Small group G = [ 16, 6 ] has Pi3(SK(G,1))= [ 2, 2, 8 ]
Small group G = [ 16, 7 ] has Pi3(SK(G,1))= [ 2, 2, 2, 2 ]
Small group G = [ 16, 8 ] has Pi3(SK(G,1))= [ 2, 2, 4 ]
Small group G = [ 16, 9 ] has Pi3(SK(G,1))= [ 2, 2, 4 ]
Small group G = [ 16, 11 ] has Pi3(SK(G,1))= [ 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
Small group G = [ 16, 12 ] has Pi3(SK(G,1))= [ 2, 2, 2, 2, 2, 2, 4, 4 ]
Small group G = [ 16, 13 ] has Pi3(SK(G,1))= [ 2, 2, 2, 2, 2, 2, 2, 2 ]
Small group G = [ 18, 1 ] has Pi3(SK(G,1))= [ 2 ]
Small group G = [ 18, 3 ] has Pi3(SK(G,1))= [ 2, 3 ]
Small group G = [ 18, 4 ] has Pi3(SK(G,1))= [ 2, 3 ]
Small group G = [ 20, 1 ] has Pi3(SK(G,1))= [ 4 ]
Small group G = [ 20, 3 ] has Pi3(SK(G,1))= [ 4 ]
Small group G = [ 20, 4 ] has Pi3(SK(G,1))= [ 2, 2, 2, 2 ]
Small group G = [ 21, 1 ] has Pi3(SK(G,1))= [ 3 ]
Small group G = [ 22, 1 ] has Pi3(SK(G,1))= [ 2 ]
Small group G = [ 24, 1 ] has Pi3(SK(G,1))= [ 8 ]
Small group G = [ 24, 3 ] has Pi3(SK(G,1))= [ 3 ]
Small group G = [ 24, 4 ] has Pi3(SK(G,1))= [ 2, 4, 4 ]
Small group G = [ 24, 5 ] has Pi3(SK(G,1))= [ 2, 2, 2, 4 ]
Small group G = [ 24, 6 ] has Pi3(SK(G,1))= [ 2, 2, 2, 2 ]
Small group G = [ 24, 7 ] has Pi3(SK(G,1))= [ 2, 2, 2, 4 ]
Small group G = [ 24, 8 ] has Pi3(SK(G,1))= [ 2, 2, 2, 2 ]
Small group G = [ 24, 10 ] has Pi3(SK(G,1))= [ 2, 2, 2, 2, 3 ]
Small group G = [ 24, 11 ] has Pi3(SK(G,1))= [ 2, 3, 4, 4 ]
Small group G = [ 24, 12 ] has Pi3(SK(G,1))= [ 2, 2 ]
Small group G = [ 24, 13 ] has Pi3(SK(G,1))= [ 2, 2, 3 ]
Small group G = [ 24, 14 ] has Pi3(SK(G,1))= [ 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
Small group G = [ 24, 15 ] has Pi3(SK(G,1))= [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 3 ]
Small group G = [ 26, 1 ] has Pi3(SK(G,1))= [ 2 ]
Small group G = [ 27, 3 ] has Pi3(SK(G,1))= [ 3, 3, 3, 3, 3 ]
Small group G = [ 27, 4 ] has Pi3(SK(G,1))= [ 3, 3, 3 ]
Small group G = [ 28, 1 ] has Pi3(SK(G,1))= [ 4 ]
Small group G = [ 28, 3 ] has Pi3(SK(G,1))= [ 2, 2, 2, 2 ]
Small group G = [ 30, 1 ] has Pi3(SK(G,1))= [ 2, 5 ]
Small group G = [ 30, 2 ] has Pi3(SK(G,1))= [ 2, 3 ]
Small group G = [ 30, 3 ] has Pi3(SK(G,1))= [ 2 ]

The functions NonableianTensorSquare(G) and ThirdHomotopyGroupOfSuspensionB(G) first decide if G is nilpotent or solvable. If it is then the nilpotent or solvable quotient algorithm is used on a certain finitely presented group H of order |H|=|G|²|G(×)G|. Otherwise coset enumeration is used on H.

(A more elaborate set of functions for computing the nonabelian tesnor product of distinct groups, and functorial homomorphisms, have been implemented in Magma and can be downloaded from here.)

The abelian group

J₂(G) = Ker ( µ : G(×)G → G )

was first studied algebraically in [R.K. Dennis, `In search of new "Homology" functors having a close relationship to K-theory', preprint, Cornell, 1976] as an alternative to the second homology functor H₂(G,Z). The isomorphism J₂(G) = Pi₃(SK(G,1)) of Brown and Loday can be inserted into the famous Certain Exact Sequence of J.H.C. Whitehead to obtain the following exact sequence

→ H₃(G,Z) → Gamma(G_ab) → J₂(G) → H₂(G,Z) → 0

(1)

in which the abelian group Gamma(G_ab) is Whitehead's universal quadratic functor. We immediately get that J₂(G) coincides with H₂(G,Z) for perfect groups G. This sequence, together with the fact that the homology of a Sylow p-subgroup of G maps onto the p-part of the homology of G, also yields the following.

For any Sylow p-subgroup P of a finite group G, the p-primary part of J₂(G) is a quotient of J₂(P).

This last result is used in the function NonabelianTensorSquare(G) to bound the order of J₂(G) for a non-nilpotent solvable group G.

The kernel of the quotient homomorphism J₂(P) → J₂(G) can be described by a Cartan-Eilenberg double coset type formula like the one for group homology. So one should be able to compute the functor J₂(G) for an arbitrary finite group G from its values on the Sylow subgroups. This approach has, as yet, only been partially implemented in HAP. The command ThirdHomotopyGroupOfSuspensionB_alt(G) returns the abelian invariants of groups A and B related by a short exact sequence 0 → B → J₂(G) → A → 0. So, for instance, the following commands show that J₂(S₁₂) has order 4 for the symmetric group on 12 symbols.

gap> ThirdHomotopyGroupOfSuspensionB_alt(SymmetricGroup(12));
[ [ 2 ], [ 2 ] ]

HAP commands can be used to compare the two functors J₂(G) and H₂(G,Z) empirically. For example, there are 92 prime-power groups of order at most 32. There are thus 4140 unordered pairs (G,Q) of distinct prime-power groups of order at most 32. The following commands show that, of these pairs,

114 pairs have G_ab = Q_ab and H₂(G,Z) = H₂(Q,Z).
47 pairs have G_ab = Q_ab and J₂(G) = J₂(Q).
45 pairs have G_ab = Q_ab and H₂(G,Z) = H₂(Q,Z) and J₂(G) = J₂(Q).
G_ab = Q_ab and J₂(G) = J₂(Q) but H₂(G,Z) is different to H₂(Q,Z) in just two cases: either G=SmallGroup(16,12) and Q=SmallGroup(32,31) or G=SmallGroup(32,29) and Q=SmallGroup(32,31).

L:=[];; H2:=[];; J2:=[];; H2J2:=[];;

for i in [1..32] do
if IsPrimePowerInt(i) then
for G in AllSmallGroups(i) do
x:=[IdSmallGroup(G),AbelianInvariants(G),GroupHomology(G,2),
ThirdHomotopyGroupOfSuspensionB(G)];
Append(L,[x]);
od;
od;fi;od;

gap> for x in L do for y in L do
> if Position(L,x)<Position(L,y) then
> if x[2]=y[2] and x[3]=y[3] then Append(H2,[[x,y]]); fi;
> fi;
> od;od;
gap> Length(H2);
114

gap> for x in L do for y in L do
> if Position(L,x)<Position(L,y) then
> if x[2]=y[2] and x[4]=y[4] then Append(J2,[[x,y]]); fi;
> fi;
> od;od;
gap> Length(J2);
47

gap> for x in L do for y in L do
> if Position(L,x)<Position(L,y) then
> if x[2]=y[2] and x[3]=y[3] and x[4]=y[4] then Append(H2J2,[[x,y]]); fi;
> fi;
> od;od;
gap> Length(H2J2);
45

gap> Difference(J2,H2J2);
[ [ [ [ 16, 12 ], [ 2, 2, 2 ], [ 2, 2 ], [ 2, 2, 2, 2, 2, 2, 4, 4 ] ],
[ [ 32, 31 ], [ 2, 2, 2 ], [ 2, 4 ], [ 2, 2, 2, 2, 2, 2, 4, 4 ] ] ],
[ [ [ 32, 29 ], [ 2, 2, 2 ], [ 2, 2 ], [ 2, 2, 2, 2, 2, 2, 4, 4 ] ],
[ [ 32, 31 ], [ 2, 2, 2 ], [ 2, 4 ], [ 2, 2, 2, 2, 2, 2, 4, 4 ] ] ] ]

Recall that a prime-power group G of order pⁿ and nilpotency class c is said to have coclass r=n-c . In a recent preprint Bettina Eick has shown that the second integral homology functor H₂(G,Z) sometimes has only finitely many values for the infinitely many 2-groups G of a given coclass. This result extends to the functor J₂(G) thanks to the exact sequence (1) . Her proof relies on the five term exact sequence in integral group homology. There is an analogous sequence for the functor J₂(G).

For any group G with normal subgroup N there is a natural exact sequence

J₂(G) → J₂(G/N) → N/[N,G] → G_ab → (G/N)_ab → 0

(2)

Sequences (1) and (2) yield the following result.

Proposition

Let S be a group with normal subgroup T. Define a relative lower central series by setting T₁=T and T_n+1=[T_n,S] . Suppose that S/T is a finite p-group and that T_n/T_n+1 is elementary abelian for all n.

If H₂(G,Z) is finite then there is a finite abelian group A such that

J₂(S/T_n) = A + (T_n/T_n+1)

for all sufficiently large n.

If H₂(G,Z) is infinite then the order of J₂(S/T_n) is unbounded.

To prove the proposition we consider the canonical homomorphisms f_n:J₂(S) → J₂(S/T_n) and h_n:J₂(S/T_n) → J₂(S/T_n-1) . Since f_n-1 = h_nf_n we have that Ker(f_n+1) lies in Ker(f_n) . Since S/T and T/T₂are finite, then so too is S/T₂. Hence S_ab is finite. If H₂(S,Z) is finite, then so is J₂(S) by sequence (1) above. It follows that Ker(f_n+1) = Ker(f_n) for all sufficiently large n. Let A denote the cokernel of f_n (for n sufficiently large). The exact sequence (2) implies that J₂(S/T_n) is an extension of A by (T_n/T_n+1). Since (T_n/T_n+1) is elementary abelian we get J₂(S/T_n) = A + (T_n/T_n+1) as required. The statement for infinite H₂(G,Z) is got by showing that Ker(f_n+1) is in this case a proper subgroup of Ker(f_n) for all n.

The interest in the proposition is that, associated to any p-group G, there is an infinite space group S with normal translation subgroup T such that T_n/T_n+1 is cyclic of order p and S/T_n has the same coclass as G for all sufficiently large n. Furthermore, for some n the quotient S/T_n is isomorphic to a lower central quotient of G. By the proposition, nearly all the values of J(S/T_n) will be equal if H₂(S,Z) is finite. Also, nearly all the values of H₂(S/T_n,Z) will be equal. The following commands illustrate this phenomenon.

gap> S:=Image(IsomorphismPcpGroup(SpaceGroup(2,11)));;
gap> IsAlmostCrystallographic(S);;
gap> T:=Kernel(NaturalHomomorphismOnHolonomyGroup(S));;
gap> RCS:=[];; RCS[1]:=T;;
gap> for i in [2..7] do
> RCS[i]:=CommutatorSubgroup(RCS[i-1],S);
> od;
gap> Quotients:=List([1..7],i->RefinedPcGroup(Range(IsomorphismPcGroup(S/RCS[i]))));;

gap> #RCS is the relative lower central series. Quotients is the list of quotient groups.

gap> List([2..7],i->Order(RCS[i-1]/RCS[i]));
[ 2, 2, 2, 2, 2, 2 ]

gap> List([1..7],i->Coclass(Quotients[i]));
[ 1, 2, 3, 3, 3, 3, 3 ]

gap> List([1..7],i->ThirdHomotopyGroupOfSuspensionB(Quotients[i]));
[ [ 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2, 2, 2, 2, 2 ],
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ],
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ],
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ] ]

gap> List([1..7],i->GroupHomology(Quotients[i],2));
[ [ 2 ], [ 2, 2, 2 ], [ 2, 2, 2, 2 ], [ 2, 2, 2, 2 ], [ 2, 2, 2, 2 ],
[ 2, 2, 2, 2 ], [ 2, 2, 2, 2 ] ]

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