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            <td style="vertical-align: top;"><a
href="aboutBogomolov.html"><small style="color: rgb(0, 0, 102);">Previous</small></a><br>
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            <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big><span
style="font-weight: bold;">About HAP: Homotopy 2-types and third
cohomology<br>
            </span></big></td>
            <td style="text-align: right; vertical-align: top;"><a
href="aboutSimplicialGroups.html"><small style="color: rgb(0, 0, 102);">next</small></a><br>
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      <big><span style="font-weight: bold;"></span></big><br>
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      <div style="text-align: center;"></div>
      <div style="text-align: center;">
      <h3>Some classical theory</h3>
      </div>
      <p>The third <a href="aboutTwistedCoefficients.html">cohomology</a>
H<sup>3</sup>(G,A) of G with coefficients in a G-module A, together
with the corresponding 3-<a href="aboutCocycles.html">cocycles</a>, can
be used to classify homotopy 2-types. <br>
      </p>
A <span style="font-style: italic;">2-type</span> is a CW-space whose
homotopy groups are trivial in dimensions n=0 and n>2. There is an
equivalence of categories<br>
      <br>
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            <td style="vertical-align: top; color: rgb(0, 0, 102);">(Homotopy
category
of<br>
connected CW-spaces X <br>
with trivial homotopy <br>
groups pi_n(X) for n>2)</td>
            <td style="vertical-align: top; color: rgb(0, 0, 102);"><br>
            <br>
<----------><br>
            </td>
            <td style="vertical-align: top; color: rgb(0, 0, 102);">(Localisation
of
the
category <br>
of simplicial groups with <br>
Moore complex of length<br>
1, where the localisation is<br>
with respect to homomorphisms<br>
inducing isomorphisms on<br>
homotopy groups)<br>
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      <br>
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<br>
      <ul style="text-align: left;">
        <li>ss = s,  ts = s,</li>
        <li>tt = t, st = t,</li>
        <li>[Ker(s),Ker(t)] = 1.</li>
      </ul>
Ths data {H,s,t} was termed a <span style="font-style: italic;">cat</span><sup
style="font-style: italic;">1</sup><span style="font-style: italic;">-group</span>
by J.-L. Loday since it can be regarded as a group H endowed with one
compatible category structure.<br>
      <br>
The homotopy groups of a cat<sup>1</sup>-group H are defined as<br>
      <ul>
        <li>pi_1(H) = Image(s)/ t( Kernel(s) ),</li>
      </ul>
      <ul>
        <li>pi_2(H) = Intersection( Kernel(s) , Kernel(t) ),</li>
      </ul>
      <ul>
        <li>pi_n(H) = 0 for n>2 or n=0.</li>
      </ul>
Note that pi_2(H) is a pi_1(H)-module where the action is induced by
conjugation in H. <br>
      </td>
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      <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">A
number of standard group-theoretic constructions can be viewed
naturally as a cat<sup>1</sup>-group.<br>
      <ol>
        <li>A ZG-module A can be viewed as a cat<sup>1</sup>-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.</li>
        <li>A group G with normal subgroup N can be viewed as a cat<sup>1</sup>-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.</li>
        <li>The homomorphism j:G--->Aut(G) which sends elements of G
to the corresponding inner automorphism can be viewed as a cat<sup>1</sup>-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.</li>
      </ol>
These three examples are implemented in HAP. <br>
      <br>
For instance, the following commands begin by constructing the cat<sup>1</sup>-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.<br>
      </td>
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      <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
G:=SmallGroup(64,134);;<br>
gap> H:=AutomorphismGroupAsCatOneGroup(G);;<br>
      <br>
gap> pi_1:=HomotopyGroup(H,1);;<br>
      <br>
gap> pi_2:=HomotopyModule(H,2);;<br>
      <br>
gap> Order(pi_1)<br>
8<br>
      <br>
gap> Order(ActedGroup(pi_2));<br>
2<br>
      </td>
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      <td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">The
third
cohomology H<sup>3</sup>(pi_1,pi_2) classifies those cat<sup>1</sup>-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. <br>
      <br>
The following additional commands show that there are 1024 Yoneda
equivalence classes of cat<sup>1</sup>-groups with homotopy group pi_1
and homotopy module pi_2 equal to that in our example.<br>
      </td>
    </tr>
    <tr>
      <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
R:=ResolutionFiniteGroup(pi_1,4);;<br>
gap> C:=HomToGModule(R,pi_2);;<br>
gap> CH:=CohomologyModule(C,3);;<br>
      <br>
gap> AbelianInvariants(ActedGroup(CH));<br>
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]<br>
      </td>
    </tr>
    <tr>
      <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">A
3-cocycle  f : pi_1 × pi_1 × pi_1 -----> pi_2
corresponding to the second cohomology class in H<sup>3</sup>(pi_1,pi_2)
can
be produced using the following command, </td>
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    <tr>
      <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
x:=Elements(ActedGroup(CH))[2];;<br>
      <br>
gap> f:=CH!.representativeCocycle(x);<br>
Standard 3-cocycle<br>
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