<p>A <em>crossed module</em> consists of a homomorphism of groups <span class="SimpleMath">\(\partial\colon M\rightarrow G\)</span> together with an action <span class="SimpleMath">\((g,m)\mapsto\, {^gm}\)</span> of <span class="SimpleMath">\(G\)</span> on <span class="SimpleMath">\(M\)</span> satisfying</p>
<p>A crossed module <span class="SimpleMath">\(\partial\colon M\rightarrow G\)</span> is equivalent to a cat<span class="SimpleMath">\(^1\)</span>-group <span class="SimpleMath">\((H,s,t)\)</span> (see <a href="chap6_mj.html#X78040D8580D35D53"><span class="RefLink">6.11</span></a>) where <span class="SimpleMath">\(H=M \rtimes G\)</span>, <span class="SimpleMath">\(s(m,g) = (1,g)\)</span>, <span class="SimpleMath">\(t(m,g)=(1,(\partial m)g)\)</span>. A cat<span class="SimpleMath">\(^1\)</span>-group is, in turn, equivalent to a simplicial group with Moore complex has length <span class="SimpleMath">\(1\)</span>. The simplicial group is constructed by considering the cat<span class="SimpleMath">\(^1\)</span>-group as a category and taking its nerve. Alternatively, the simplicial group can be constructed by viewing the crossed module as a crossed complex and using a nonabelian version of the Dold-Kan theorem.</p>
<p>The following example concerns the crossed module</p>
<p>associated to the dihedral group <span class="SimpleMath">\(G\)</span> of order <span class="SimpleMath">\(16\)</span>. This crossed module represents, up to homotopy type, a connected space <span class="SimpleMath">\(X\)</span> with <span class="SimpleMath">\(\pi_iX=0\)</span> for <span class="SimpleMath">\(i\ge 3\)</span>, <span class="SimpleMath">\(\pi_2X=Z(G)\)</span>, <span class="SimpleMath">\(\pi_1X = Aut(G)/Inn(G)\)</span>. The space <span class="SimpleMath">\(X\)</span> can be represented, up to homotopy, by a simplicial group. That simplicial group is used in the example to compute</p>
<h4>12.2 <span class="Heading">Eilenberg-MacLane spaces as simplicial groups (not recommended)</span></h4>
<p>The following example concerns the Eilenberg-MacLane space <span class="SimpleMath">\(X=K(\mathbb Z_3,3)\)</span> which is a path-connected space with <span class="SimpleMath">\(\pi_3X=\mathbb Z_3\)</span>, <span class="SimpleMath">\(\pi_iX=0\)</span> for <span class="SimpleMath">\(3\ne i\ge 1\)</span>. This space is represented by a simplicial group, and perturbation techniques are used to compute</p>
<h4>12.3 <span class="Heading">Eilenberg-MacLane spaces as simplicial free abelian groups (recommended)</span></h4>
<p>For integer <span class="SimpleMath">\(n>1\)</span> and abelian group <span class="SimpleMath">\(A\)</span> the Eilenberg-MacLane space <span class="SimpleMath">\(K(A,n)\)</span> is better represented as a simplicial free abelian group. (The reason is that the functorial bar resolution of a group can be replaced in computations by the smaller functorial Chevalley-Eilenberg complex of the group when the group is free abelian, obviating the need for perturbation techniques. When <span class="SimpleMath">\(A\)</span> has torision we can replace it with an inclusion of free abelian groups <span class="SimpleMath">\(A_1 \hookrightarrow A_0\)</span> with <span class="SimpleMath">\(A\cong A_0/A_1\)</span> and again invoke the Chevalley-Eilenberg complex. The current implementation unfortunately handles only free abelian <span class="SimpleMath">\(A\)</span> but the easy extension to non-free <span class="SimpleMath">\(A\)</span> is planned for a future release.)</p>
<p>The following commands compute the integral homology <span class="SimpleMath">\(H_n(K(\mathbb Z,3),\mathbb Z)\)</span> for <span class="SimpleMath">\( 0\le n \le 16\)</span>. (Note that one typically needs fewer than <span class="SimpleMath">\(n\)</span> terms of the Eilenberg-MacLance space to compute its <span class="SimpleMath">\(n\)</span>-th homology -- an error is printed if too few terms of the space are available for a given computation.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=AbelianPcpGroup([0]);; #infinite cyclic group </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">K:=EilenbergMacLaneSimplicialFreeAbelianGroup(A,3,14);</span>
Simplicial free abelian group of length 14
<span class="GAPprompt">gap></span> <span class="GAPinput">for n in [0..16] do</span>
<span class="GAPprompt">></span> <span class="GAPinput">Print("Degree ",n," integral homology of K is ",Homology(K,n),"\n");</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;</span>
Degree 0 integral homology of K is [ 0 ]
Degree 1 integral homology of K is [ ]
Degree 2 integral homology of K is [ ]
Degree 3 integral homology of K is [ 0 ]
Degree 4 integral homology of K is [ ]
Degree 5 integral homology of K is [ 2 ]
Degree 6 integral homology of K is [ ]
Degree 7 integral homology of K is [ 3 ]
Degree 8 integral homology of K is [ 2 ]
Degree 9 integral homology of K is [ 2 ]
Degree 10 integral homology of K is [ 3 ]
Degree 11 integral homology of K is [ 5, 2 ]
Degree 12 integral homology of K is [ 2 ]
Degree 13 integral homology of K is [ ]
Degree 14 integral homology of K is [ 10, 2 ]
Degree 15 integral homology of K is [ 7, 6 ]
Degree 16 integral homology of K is [ ]
</pre></div>
<p>For an <span class="SimpleMath">\(n\)</span>-connected pointed space <span class="SimpleMath">\(X\)</span> the Freudenthal Suspension Theorem states that the map <span class="SimpleMath">\(X \rightarrow \Omega(\Sigma X)\)</span> induces a map <span class="SimpleMath">\(\pi_k(X) \rightarrow \pi_k(\Omega(\Sigma X))\)</span> which is an isomorphism for <span class="SimpleMath">\(k\le 2n\)</span> and epimorphism for <span class="SimpleMath">\(k=2n+1\)</span>. Thus the Eilenberg-MacLane space <span class="SimpleMath">\(K(A,n+1)\)</span> can be constructed from the suspension <span class="SimpleMath">\(\Sigma K(A,n)\)</span> by attaching cells in dimensions <span class="SimpleMath">\(\ge 2n+1\)</span>. In particular, there is an isomorphism <span class="SimpleMath">\( H_{k-1}(K(A,n),\mathbb Z) \rightarrow H_k(K(A,n+1),\mathbb Z)\)</span> for <span class="SimpleMath">\(k\le 2n\)</span> and epimorphism for <span class="SimpleMath">\(k=2n+1\)</span>.</p>
<p>For instance, <span class="SimpleMath">\( H_{k-1}(K(\mathbb Z,3),\mathbb Z) \cong H_k(K(\mathbb Z,4),\mathbb Z) \)</span> for <span class="SimpleMath">\(k\le 6\)</span> and <span class="SimpleMath">\( H_6(K(\mathbb Z,3),\mathbb Z) \twoheadrightarrow H_7(K(\mathbb Z,4),\mathbb Z) \)</span>. This assertion is seen in the following session.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=AbelianPcpGroup([0]);; #infinite cyclic group </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">K:=EilenbergMacLaneSimplicialFreeAbelianGroup(A,4,11);</span>
Simplicial free abelian group of length 11
<span class="GAPprompt">gap></span> <span class="GAPinput">for n in [0..13] do</span>
<span class="GAPprompt">></span> <span class="GAPinput">Print("Degree ",n," integral homology of K is ",Homology(K,n),"\n");</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;</span>
Degree 0 integral homology of K is [ 0 ]
Degree 1 integral homology of K is [ ]
Degree 2 integral homology of K is [ ]
Degree 3 integral homology of K is [ ]
Degree 4 integral homology of K is [ 0 ]
Degree 5 integral homology of K is [ ]
Degree 6 integral homology of K is [ 2 ]
Degree 7 integral homology of K is [ ]
Degree 8 integral homology of K is [ 3, 0 ]
Degree 9 integral homology of K is [ ]
Degree 10 integral homology of K is [ 2, 2 ]
Degree 11 integral homology of K is [ ]
Degree 12 integral homology of K is [ 5, 12, 0 ]
Degree 13 integral homology of K is [ 2 ]
<h4>12.4 <span class="Heading">Elementary theoretical information on
<span class="SimpleMath">\(H^\ast(K(\pi,n),\mathbb Z)\)</span></span></h4>
<p>The cup product is not implemented for the cohomology ring <span class="SimpleMath">\(H^\ast(K(\pi,n),\mathbb Z)\)</span>. Standard theoretical spectral sequence arguments have to be applied to obtain basic information relating to the ring structure. To illustrate this the following commands compute <span class="SimpleMath">\(H^n(K(\mathbb Z,2),\mathbb Z)\)</span> for the first few values of <span class="SimpleMath">\(n\)</span>.</p>
<p>There is a fibration sequence <span class="SimpleMath">\(K(\pi,n) \hookrightarrow \ast \twoheadrightarrow K(\pi,n+1)\)</span> in which <span class="SimpleMath">\(\ast\)</span> denotes a contractible space. For <span class="SimpleMath">\(n=1, \pi=\mathbb Z\)</span> the terms of the <span class="SimpleMath">\(E_2\)</span> page of the Serre integral cohomology spectral sequence for this fibration are</p>
</li>
</ul>
<p>Since <span class="SimpleMath">\(K(\mathbb Z,1)\)</span> can be taken to be the circle <span class="SimpleMath">\(S^1\)</span> we know that it has non-trivial cohomology in degrees <span class="SimpleMath">\(0\)</span> and <span class="SimpleMath">\(1\)</span> only. The first few terms of the <span class="SimpleMath">\(E_2\)</span> page are given in the following table.</p>
<p>Let <span class="SimpleMath">\(x\)</span> denote the generator of <span class="SimpleMath">\(H^1(K(\mathbb Z,1),\mathbb Z)\)</span> and <span class="SimpleMath">\(y\)</span> denote the generator of <span class="SimpleMath">\(H^2(K(\mathbb Z,2),\mathbb Z)\)</span>. Since <span class="SimpleMath">\(\ast\)</span> has zero cohomology in degrees <span class="SimpleMath">\(\ge 1\)</span> we see that the differential must restrict to an isomorphism <span class="SimpleMath">\(d_2\colon E_2^{0,1} \rightarrow E_2^{2,0}\)</span> with <span class="SimpleMath">\(d_2(x)=y\)</span>. Then we see that the differential must restrict to an isomorphism <span class="SimpleMath">\(d_2\colon E_2^{2,1} \rightarrow E_2^{4,0}\)</span> defined on the generator <span class="SimpleMath">\(xy\)</span> of <span class="SimpleMath">\(E_2^{2,1}\)</span> by</p>
<p>Hence <span class="SimpleMath">\(E_2^{4,0} \cong H^4(K(\mathbb Z,2),\mathbb Z)\)</span> is generated by <span class="SimpleMath">\(y^2\)</span>. The argument extends to show that <span class="SimpleMath">\(H^6(K(\mathbb Z,2),\mathbb Z)\)</span> is generated by <span class="SimpleMath">\(y^3\)</span>, <span class="SimpleMath">\(H^8(K(\mathbb Z,2),\mathbb Z)\)</span> is generated by <span class="SimpleMath">\(y^4\)</span>, and so on.</p>
<p>In fact, to obtain a complete description of the ring <span class="SimpleMath">\(H^\ast(K(\mathbb Z,2),\mathbb Z)\)</span> in this fashion there is no benefit to using computer methods at all. We only need to know the cohomology ring <span class="SimpleMath">\(H^\ast(K(\mathbb Z,1),\mathbb Z) =H^\ast(S^1,\mathbb Z)\)</span> and the single cohomology group <span class="SimpleMath">\(H^2(K(\mathbb Z,2),\mathbb Z)\)</span>.</p>
<p>A similar approach can be attempted for <span class="SimpleMath">\(H^\ast(K(\mathbb Z,3),\mathbb Z)\)</span> using the fibration sequence <span class="SimpleMath">\(K(\mathbb Z,2) \hookrightarrow \ast \twoheadrightarrow K(\mathbb Z,3)\)</span> and, as explained in Chapter 5 of <a href="chapBib_mj.html#biBhatcher">[Hat01]</a>, yields the computation of the group <span class="SimpleMath">\(H^i(K(\mathbb Z,3),\mathbb Z)\)</span> for <span class="SimpleMath">\(4\le i\le 13\)</span>. The method does not directly yield <span class="SimpleMath">\(H^3(K(\mathbb Z,3),\mathbb Z)\)</span> and breaks down in degree <span class="SimpleMath">\(14\)</span> yielding only that <span class="SimpleMath">\(H^{14}(K(\mathbb Z,3),\mathbb Z) = 0 {\rm ~or~} \mathbb Z_3\)</span>. The following commands provide <span class="SimpleMath">\(H^3(K(\mathbb Z,3),\mathbb Z)= \mathbb Z\)</span> and <span class="SimpleMath">\(H^{14}(K(\mathbb Z,3),\mathbb Z) =0\)</span>.</p>
<p>However, the implementation of these commands is currently a bit naive, and computationally inefficient, since they do not currently employ any homological perturbation techniques.</p>
<p>As a CW-complex the Eilenberg-MacLane space <span class="SimpleMath">\(K=K(\mathbb Z,n)\)</span> can be obtained from an <span class="SimpleMath">\(n\)</span>-sphere <span class="SimpleMath">\(S^n=e^0\cup e^n\)</span> by attaching cells in dimensions <span class="SimpleMath">\(\ge n+2\)</span> so as to kill the higher homotopy groups of <span class="SimpleMath">\(S^n\)</span>. From the inclusion <span class="SimpleMath">\(\iota\colon S^n\hookrightarrow K(\mathbb Z,n)\)</span> we can form the mapping cone <span class="SimpleMath">\(X=C(\iota)\)</span>. The long exact homotopy sequence</p>
<p>implies that <span class="SimpleMath">\(\pi_k(K,S^n)=0\)</span> for <span class="SimpleMath">\(0 \le k\le n+1\)</span> and <span class="SimpleMath">\(\pi_{n+2}(K,S^n)\cong \pi_{n+1}(S^n)\)</span>. The relative Hurewicz Theorem gives an isomorphism <span class="SimpleMath">\(\pi_{n+2}(K,S^n) \cong H_{n+2}(K,S^n,\mathbb Z)\)</span>. The long exact homology sequence</p>
<p>arising from the cofibration <span class="SimpleMath">\(S^n \hookrightarrow K \twoheadrightarrow X\)</span> implies that <span class="SimpleMath">\(\pi_{n+1}(S^n)\cong \pi_{n+2}(K,S^n) \cong H_{n+2}(K,S^n,\mathbb Z) \cong H_{n+2}(K,\mathbb Z)\)</span>. From the <strong class="button">GAP</strong> computations in <a href="chap12_mj.html#X7D91E64D7DD7F10F"><span class="RefLink">12.3</span></a> and the Freudenthal Suspension Theorem we find:</p>
<p>The Hopf fibration <span class="SimpleMath">\(S^3\rightarrow S^2\)</span> has fibre <span class="SimpleMath">\(S^1 = K(\mathbb Z,1)\)</span>. It can be constructed by viewing <span class="SimpleMath">\(S^3\)</span> as all pairs <span class="SimpleMath">\((z_1,z_2)\in \mathbb C^2\)</span> with <span class="SimpleMath">\(|z_1|^2+|z_2|^2=1\)</span> and viewing <span class="SimpleMath">\(S^2\)</span> as <span class="SimpleMath">\(\mathbb C\cup \infty\)</span>; the map sends <span class="SimpleMath">\((z_1,z_2)\mapsto z_1/z_2\)</span>. The homotopy exact sequence of the Hopf fibration yields <span class="SimpleMath">\(\pi_k(S^3) \cong \pi_k(S^2)\)</span> for <span class="SimpleMath">\(k\ge 3\)</span>, and in particular</p>
<p>It will require further techniques (such as the Postnikov tower argument in Section <a href="chap12_mj.html#X83EAC40A8324571F"><span class="RefLink">12.7</span></a> below) to establish that <span class="SimpleMath">\(\pi_5(S^3) \cong \mathbb Z_2\)</span>. Once we have this isomorphism for <span class="SimpleMath">\(\pi_5(S^3)\)</span>, the generalized Hopf fibration <span class="SimpleMath">\(S^3 \hookrightarrow S^7 \twoheadrightarrow S^4\)</span> comes into play. This fibration is contructed as for the classical fibration, but using pairs <span class="SimpleMath">\((z_1,z_2)\)</span> of quaternions rather than pairs of complex numbers. The Hurewicz Theorem gives <span class="SimpleMath">\(\pi_3(S^7)=0\)</span>; the fibre <span class="SimpleMath">\(S^3\)</span> is thus homotopic to a point in <span class="SimpleMath">\(S^7\)</span> and the inclusion of the fibre induces the zero homomorphism <span class="SimpleMath">\(\pi_k(S^3) \stackrel{0}{\longrightarrow} \pi_k(S^7) ~~(k\ge 1)\)</span>. The exact homotopy sequence of the generalized Hopf fibration then gives <span class="SimpleMath">\(\pi_k(S^4)\cong \pi_k(S^7)\oplus \pi_{k-1}(S^3)\)</span>. On taking <span class="SimpleMath">\(k=6\)</span> we obtain <span class="SimpleMath">\(\pi_6(S^4)\cong \pi_5(S^3) \cong \mathbb Z_2\)</span>. Freudenthal suspension then gives</p>
<h4>12.6 <span class="Heading">The first two non-trivial homotopy groups of the suspension and double suspension of a <span class="SimpleMath">\(K(G,1)\)</span></span></h4>
<p>For any group <span class="SimpleMath">\(G\)</span> we consider the homotopy groups <span class="SimpleMath">\(\pi_n(\Sigma K(G,1))\)</span> of the suspension <span class="SimpleMath">\(\Sigma K(G,1)\)</span> of the Eilenberg-MacLance space <span class="SimpleMath">\(K(G,1)\)</span>. On taking <span class="SimpleMath">\(G=\mathbb Z\)</span>, and observing that <span class="SimpleMath">\(S^2 = \Sigma K(\mathbb Z,1)\)</span>, we specialize to the homotopy groups of the <span class="SimpleMath">\(2\)</span>-sphere <span class="SimpleMath">\(S^2\)</span>.</p>
<p>via the isomorphisms <span class="SimpleMath">\(\pi_2(\Sigma K(G,1)) \cong H_2(\Sigma K(G,1),\mathbb Z) \cong H_1(K(G,1),\mathbb Z) \cong G_{ab}\)</span>. R. Brown and J.-L. Loday <a href="chapBib_mj.html#biBbrownloday">[BL87]</a> obtained the formulae</p>
<p>involving the nonabelian tensor square and nonabelian symmetric square of the group <span class="SimpleMath">\(G\)</span>. The following commands use the nonabelian tensor and symmetric product to compute the third and fourth homotopy groups for <span class="SimpleMath">\(G =Syl_2(M_{12})\)</span> the Sylow <span class="SimpleMath">\(2\)</span>-subgroup of the Mathieu group <span class="SimpleMath">\(M_{12}\)</span>.</p>
<h4>12.7 <span class="Heading">Postnikov towers and <span class="SimpleMath">\(\pi_5(S^3)\)</span> </span></h4>
<p>A Postnikov system for the sphere <span class="SimpleMath">\(S^3\)</span> consists of a sequence of fibrations <span class="SimpleMath">\(\cdots X_3\stackrel{p_3}{\rightarrow} X_2\stackrel{p_2}{\rightarrow} X_1\stackrel{p_1}{\rightarrow} \ast\)</span> and a sequence of maps <span class="SimpleMath">\(\phi_n\colon S^3 \rightarrow X_n\)</span> such that</p>
</li>
<li><p>The map <span class="SimpleMath">\(\phi_n\colon S^3 \rightarrow X_n\)</span> induces an isomorphism <span class="SimpleMath">\(\pi_k(S^3)\rightarrow \pi_k(X_n)\)</span> for all <spanclass="SimpleMath">\(k\le n\)</span></p>
</li>
<li><p><span class="SimpleMath">\(\pi_k(X_n)=0\)</span> for <span class="SimpleMath">\(k > n\)</span></p>
</li>
<li><p>and consequently each fibration <span class="SimpleMath">\(p_n\)</span> has fibre an Eilenberg-MacLane space <span class="SimpleMath">\(K(\pi_n(S^3),n)\)</span>.</p>
</li>
</ul>
<p>The space <span class="SimpleMath">\(X_n\)</span> is obtained from <span class="SimpleMath">\(S^3\)</span> by adding cells in dimensions <span class="SimpleMath">\(\ge n+2\)</span> and thus</p>
<ul>
<li><p><span class="SimpleMath">\(H_k(X_n,\mathbb Z)=H_k(S^3,\mathbb Z)\)</span> for <span class="SimpleMath">\(k\le n+1\)</span>.</p>
</li>
</ul>
<p>So in particular <span class="SimpleMath">\(X_1=X_2=\ast, X_3=K(\mathbb Z,3)\)</span> and we have a fibration sequence <span class="SimpleMath">\(K(\pi_4(S^3),4) \hookrightarrow X_4 \twoheadrightarrow K(\mathbb Z,3)\)</span>. The terms in the <span class="SimpleMath">\(E_2\)</span> page of the Serre integral cohomology spectral sequence of this fibration are</p>
</li>
</ul>
<p>The first few terms in the <span class="SimpleMath">\(E_2\)</span> page can be computed using the commands of Sections <a href="chap12_mj.html#X795E339978B42775"><span class="RefLink">12.2</span></a> and <a href="chap12_mj.html#X7D91E64D7DD7F10F"><span class="RefLink">12.3</span></a> and recorded as follows.</p>
<p>Since we know that <span class="SimpleMath">\(H^5(X_4,\mathbb Z) =0\)</span>, the differentials in the spectral sequence must restrict to an isomorphism <span class="SimpleMath">\(E_2^{0,5}=\pi_4(S^3) \stackrel{\cong}{\longrightarrow} E_2^{6,0}=\mathbb Z_2\)</span>. This provides an alternative derivation of <span class="SimpleMath">\(\pi_4(S^3) \cong \mathbb Z_2\)</span>. We can also immediately deduce that <span class="SimpleMath">\(H^6(X_4,\mathbb Z)=0\)</span>. Let <span class="SimpleMath">\(x\)</span> be the generator of <span class="SimpleMath">\(E_2^{0,5}\)</span> and <span class="SimpleMath">\(y\)</span> the generator of <span class="SimpleMath">\(E_2^{3,0}\)</span>. Then the generator <span class="SimpleMath">\(xy\)</span> of <spanclass="SimpleMath">\(E_2^{3,5}\)</span> gets mapped to a non-zero element <span class="SimpleMath">\(d_7(xy)=d_7(x)y -xd_7(y)\)</span>. Hence the term <span class="SimpleMath">\(E_2^{0,7}=\mathbb Z_2\)</span> must get mapped to zero in <span class="SimpleMath">\(E_2^{3,5}\)</span>. It follows that <span class="SimpleMath">\(H^7(X_4,\mathbb Z)=\mathbb Z_2\)</span>.</p>
<p>The integral cohomology of Eilenberg-MacLane spaces yields the following information on the <span class="SimpleMath">\(E_2\)</span> page <span class="SimpleMath">\(E_2^{p,q}=H_p(\,X_4,\,H^q(K(\pi_5S^3,5),\mathbb Z)\,)\)</span> for the fibration <span class="SimpleMath">\(K(\pi_5(S^3),5) \hookrightarrow X_5 \twoheadrightarrow X_4\)</span>.</p>
<p>Since we know that <span class="SimpleMath">\(H^6(X_5,\mathbb Z)=0\)</span>, the differentials in the spectral sequence must restrict to an isomorphism <span class="SimpleMath">\(E_2^{0,6}=\pi_5(S^3) \stackrel{\cong}{\longrightarrow} E_2^{7,0}=H^7(X_4,\mathbb Z)\)</span>. We can conclude the desired result:</p>
<p>Note that the fibration <span class="SimpleMath">\(X_4 \twoheadrightarrow K(\mathbb Z,3)\)</span> is determined by a cohomology class <span class="SimpleMath">\(\kappa \in H^5(K(\mathbb Z,3), \mathbb Z_2) = \mathbb Z_2\)</span>. If <span class="SimpleMath">\(\kappa=0\)</span> then we'd have \(X_4 =K(\mathbb Z_2,4)\times K(\mathbb Z,3)\) and, as the following commands show, we'd then have <span class="SimpleMath">\(H_4(X_4,\mathbb Z)=\mathbb Z_2\)</span>.</p>
<p>Since we know that <span class="SimpleMath">\(H_4(X_4,\mathbb Z)=0\)</span> we can conclude that the Postnikov invariant <span class="SimpleMath">\(\kappa\)</span> is the non-zero class in <span class="SimpleMath">\(H^5(K(\mathbb Z,3), \mathbb Z_2) = \mathbb Z_2\)</span>.</p>
<p>Consider the suspension <span class="SimpleMath">\(X=\Sigma K(G,1)\)</span> of a classifying space of a group <span class="SimpleMath">\(G\)</span> once again. This space has a Postnikov system in which <span class="SimpleMath">\(X_1 = \ast\)</span>, <span class="SimpleMath">\(X_2= K(G_{ab},2)\)</span>. We have a fibration sequence <span class="SimpleMath">\(K(\pi_3 X, 3) \hookrightarrow X_3 \twoheadrightarrow K(G_{ab},2)\)</span>. The corresponding integral cohomology Serre spectral sequence has <span class="SimpleMath">\(E_2\)</span> page with terms</p>
</li>
</ul>
<p>As an example, for the Alternating group <span class="SimpleMath">\(G=A_4\)</span> of order <span class="SimpleMath">\(12\)</span> the following commands of Section <a href="chap12_mj.html#X81E2F80384ADF8C2"><span class="RefLink">12.6</span></a> compute <span class="SimpleMath">\(G_{ab} = \mathbb Z_3\)</span> and <span class="SimpleMath">\(\pi_3 X = \mathbb Z_6\)</span>.</p>
<p>The first terms of the <span class="SimpleMath">\(E_2\)</span> page can be calculated using the commands of Sections <a href="chap12_mj.html#X795E339978B42775"><span class="RefLink">12.2</span></a> and <a href="chap12_mj.html#X7D91E64D7DD7F10F"><span class="RefLink">12.3</span></a>.</p>
<p>We know that <span class="SimpleMath">\(H^1(X_3,\mathbb Z)=0\)</span>, <span class="SimpleMath">\(H^2(X_3,\mathbb Z)=H^1(G,\mathbb Z) =0\)</span>, <span class="SimpleMath">\(H^3(X_3,\mathbb Z)=H^2(G,\mathbb Z) =\mathbb Z_3\)</span>, and that <span class="SimpleMath">\(H^4(X_3,\mathbb Z)\)</span> is a subgroup of <span class="SimpleMath">\(H^3(G,\mathbb Z) = \mathbb Z_2\)</span>. It follows that the differential induces a surjection <span class="SimpleMath">\(E_2^{0,4}=\mathbb Z_6 \twoheadrightarrow E_2^{5,0}=\mathbb Z_3\)</span>. Consequently <span class="SimpleMath">\(H^4(X_3,\mathbb Z)=\mathbb Z_2\)</span> and <span class="SimpleMath">\(H^5(X_3,\mathbb Z)=0\)</span> and <span class="SimpleMath">\(H^6(X_3,\mathbb Z)=\mathbb Z_2\)</span>.</p>
<p>The <span class="SimpleMath">\(E_2\)</span> page for the fibration <span class="SimpleMath">\(K(\pi_4 X,4) \hookrightarrow X_4 \twoheadrightarrow X_3\)</span> contains the following terms.</p>
<p>We know that <span class="SimpleMath">\(H^5(X_4,\mathbb Z)\)</span> is a subgroup of <span class="SimpleMath">\(H^4(G,\mathbb Z)=\mathbb Z_6\)</span>, and hence that there is a homomorphisms <span class="SimpleMath">\(\pi_4X \rightarrow \mathbb Z_2\)</span> whose kernel is a subgroup of <span class="SimpleMath">\(\mathbb Z_6\)</span>. If follows that <span class="SimpleMath">\(|\pi_4 X|\le 12\)</span>.</p>
<p>A <em>2-type</em> is a CW-complex <span class="SimpleMath">\(X\)</span> whose homotopy groups are trivial in dimensions <span class="SimpleMath">\(n=0 \)</span> and <span class="SimpleMath">\(n>2\)</span>. As explained in <a href="chap6_mj.html#X78040D8580D35D53"><span class="RefLink">6.11</span></a> the homotopy type of such a space can be captured algebraically by a cat<span class="SimpleMath">\(^1\)</span>-group <span class="SimpleMath">\(G\)</span>. Let <span class="SimpleMath">\(X\)</span>, <span class="SimpleMath">\(Y\)</span> be <span class="SimpleMath">\(2\)</span>-tytpes represented by cat<span class="SimpleMath">\(^1\)</span>-groups <span class="SimpleMath">\(G\)</span>, <span class="SimpleMath">\(H\)</span>. If <span class="SimpleMath">\(X\)</span> and <span class="SimpleMath">\(Y\)</span> are homotopy equivalent then there exists a sequence of morphisms of cat<span class="SimpleMath">\(^1\)</span>-groups</p>
<p>in which each morphism induces isomorphisms of homotopy groups. When such a sequence exists we say that <span class="SimpleMath">\(G\)</span> is <em>quasi-isomorphic</em> to <span class="SimpleMath">\(H\)</span>. We have the following result.</p>
<p><strong class="button">Theorem.</strong> The <span class="SimpleMath">\(2\)</span>-types <span class="SimpleMath">\(X\)</span> and <span class="SimpleMath">\(Y\)</span> are homotopy equivalent if and only if the associated cat<span class="SimpleMath">\(^1\)</span>-groups <span class="SimpleMath">\(G\)</span> and <span class="SimpleMath">\(H\)</span> are quasi-isomorphic.</p>
<p>The following commands produce a list <span class="SimpleMath">\(L\)</span> of all of the <span class="SimpleMath">\(62\)</span> non-isomorphic cat<span class="SimpleMath">\(^1\)</span>-groups whose underlying group has order <span class="SimpleMath">\(16\)</span>.</p>
<p>The next commands use the first and second homotopy groups to prove that the list <span class="SimpleMath">\(L\)</span> contains at least <span class="SimpleMath">\(37\)</span> distinct quasi-isomorphism types.</p>
<p>The following additional commands use second and third integral homology in conjunction with the first two homotopy groups to prove that the list <span class="SimpleMath">\(L\)</span> contains <strong class="button">at least</strong> <span class="SimpleMath">\(49\)</span> distinct quasi-isomorphism types.</p>
<p>The following commands show that the above list <span class="SimpleMath">\(L\)</span> contains <strong class="button">at most</strong> <span class="SimpleMath">\(51\)</span> distinct quasi-isomorphism types.</p>
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