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The homology groups Hn(X,Z) of a space are reasonably straighforward invariants to compute. Its homotopy groups Pin(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)=Gab 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)) = Z30 for G the 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(Gab) → J2(G) → H2(G,Z) → 0 (1)
in which the abelian group Gamma(Gab) 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 Gab = Qab and H2(G,Z) = H2(Q,Z). 47 pairs have Gab = Qab and J2(G) = J2(Q). 45 pairs have Gab = Qab and H2(G,Z) = H2(Q,Z) and J2(G) = J2(Q). Gab = Qab 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] → Gab → (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 Tn+1=[Tn,S] . Suppose that S/T is a finite p-group and that Tn/Tn+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/Tn) = A + (Tn/Tn+1) for all sufficiently large n. If H2(G,Z) is infinite then the order of J2(S/Tn) is unbounded.
To prove the proposition we consider the canonical homomorphisms fn:J2(S) → J2(S/Tn) and hn:J2(S/Tn) → J2(S/Tn-1) . Since fn-1 = hnfn we have that Ker(fn+1) lies in Ker(fn) . Since S/T and T/T2 are finite, then so too is S/T2. Hence Sab is finite. If H2(S,Z) is finite, then so is J2(S) by sequence (1) above. It follows that Ker(fn+1) = Ker(fn) for all sufficiently large n. Let A denote the cokernel of fn (for n sufficiently large). The exact sequence (2) implies that J2(S/Tn) is an extension of A by (Tn/Tn+1). Since (Tn/Tn+1) is elementary abelian we get J2(S/Tn) = A + (Tn/Tn+1) as required. The statement for infinite H2(G,Z) is got by showing that Ker(fn+1) is in this case a proper subgroup of Ker(fn) 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 Tn/Tn+1 is cyclic of order p and S/Tn has the same coclass as G for all sufficiently large n. Furthermore, for some n the quotient S/Tn is isomorphic to a lower central quotient of G. By the proposition, nearly all the values of J(S/Tn) will be equal if H2(S,Z) is finite. Also, nearly all the values of H2(S/Tn,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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