</tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">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 ] 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 255); vertical-align: top;">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 <a
href="http://hamilton.nuigalway.ie/software.html">here</a>.) 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
abelian group 
 
 <table
style="width: 80%; text-align: left; margin-left: auto; margin-right: auto;"
border="0" cellpadding="20" cellspacing="2">
 <tbody>
 <tr>
 <td
style="text-align: center; vertical-align: top; background-color: rgb(204, 255, 255);">J2(G)
= Ker ( µ : G(×)G → G
)</td>
 </tr>
 </tbody>
 </table>
 
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 
 
 <table
style="width: 80%; text-align: left; margin-left: auto; margin-right: auto;"
border="0" cellpadding="20" cellspacing="0">
 <tbody>
 <tr>
 <td
style="text-align: center; vertical-align: top; background-color: rgb(204, 255, 255);">→
H3(G,Z) → Gamma(Gab) → J2(G)
→ H2(G,Z) → 0 
 
 <div style="text-align: right;">(1) 
 </div>
 </td>
 </tr>
 </tbody>
 </table>
 
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. 
 
 <table
style="margin-left: auto; margin-right: auto; width: 80%; text-align: left;"
border="0" cellpadding="20" cellspacing="0">
 <tbody>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(204, 255, 255);">For any Sylow p-subgroup P of a finite
group G, the p-primary part of J2(G) is a quotient of J2(P).</td>
 </tr>
 </tbody>
 </table>
 
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. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
ThirdHomotopyGroupOfSuspensionB_alt(SymmetricGroup(12)); 
[ [ 2 ], [ 2 ] ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">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, 
 <ul>
 <li>114 pairs have Gab = Qab and H2(G,Z)
= H2(Q,Z).</li>
 <li>47 pairs have Gab = Qab and J2(G)
= J2(Q).</li>
 <li>45 pairs have Gab = Qab and H2(G,Z)
= H2(Q,Z) and J2(G) = J2(Q).</li>
 <li>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). 
 </li>
 </ul>
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">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 ] ] ] ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">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). 
 
 <table
style="margin-left: auto; margin-right: auto; width: 80%; text-align: left;"
border="0" cellpadding="20" cellspacing="0">
 <tbody>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(204, 255, 255);">For
any group G with normal subgroup N there is a natural exact sequence 
 
 <div style="text-align: center;">J2(G)
→ J2(G/N) → N/[N,G] → Gab
→ (G/N)ab → 0 
 
 <div style="text-align: right;">(2) 
 </div>
 </div>
 </td>
 </tr>
 </tbody>
 </table>
 
 
Sequences (1) and (2) yield the following result. 
 
 <table
style="margin-left: auto; margin-right: auto; width: 80%; text-align: left;"
border="0" cellpadding="20" cellspacing="0">
 <tbody>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(204, 255, 255);">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. 
 <ul>
 <li>If H2(G,Z) is finite then there is a
finite abelian group A
such that </li>
 </ul>
 <div style="text-align: center;">J2(S/Tn)
= A + (Tn/Tn+1) 
 </div>
 
 for all
sufficiently large n. 
 
 <ul>
 <li>If H2(G,Z) is infinite then the order of J2(S/Tn)
is unbounded.</li>
 </ul>
 </td>
 </tr>
 </tbody>
 </table>
 
 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. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">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 ] ] 
 </td>
 </tr>
 <tr>
 <td style="vertical-align: top;">
 <table
style="margin-left: auto; margin-right: auto; width: 100%; text-align: left;"
border="0" cellpadding="2" cellspacing="2">
 <tbody>
 <tr>
 <td style="vertical-align: top;"><a
style="color: rgb(0, 0, 102);" href="aboutAspherical.html">Previous
Page</a> 
 </td>
 <td style="text-align: center; vertical-align: top;"><a
href="aboutContents.html">Contents</a> 
 </td>
 <td style="text-align: right; vertical-align: top;"><a
href="aboutNonabelian.html">Next
page 
 </a> </td>
 </tr>
 </tbody>
 </table>
 <a href="aboutTopology.html"> 
 </a> </td>
 </tr>
 </tbody>
</table>
 
 
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 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: Third Homotopy Groups Of
Suspensions Of Classifying Spaces 
 </big></td>
 <td style="text-align: right; vertical-align: top;"><a
href="aboutNonabelian.html">next</a> 
 </td>
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 <big></big> 
 </th>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">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 
 
 <table
style="margin-left: auto; margin-right: auto; width: 80%; text-align: left;"
border="0" cellpadding="2" cellspacing="2">
 <tbody>
 <tr>
 <td
style="background-color: rgb(204, 255, 255); text-align: center; vertical-align: top;"> 
 Pi3(SK(G,1))
= Ker ( µ : G(×)G → G ) 
 
 </td>
 </tr>
 </tbody>
 </table>
 
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",
style="font-style: italic;">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. 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">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 ] 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 255); vertical-align: top;">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.

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