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<div class="ChapSects"><a href="chap3_mj.html#X87472058788D76C0">3 <span class="Heading">Covering spaces</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X85FB4CA987BC92CC">3.1 <span class="Heading">Cellular chains on the universal cover</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7E5CC04E7E3CCDAD">3.2 <span class="Heading">Spun knots and the Satoh tube map</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7C304A1C7EF0BA60">3.3 <span class="Heading">Cohomology with local coefficients</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7A4F34B780FA2CD5">3.4 <span class="Heading">Distinguishing between two non-homeomorphic homotopy equivalent spaces</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X869FD75B84AAC7AD">3.5 <span class="Heading"> Second homotopy groups of spaces with finite fundamental group</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X87F8F6C3812A7E73">3.6 <span class="Heading">Third homotopy groups of simply connected spaces</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7B506CF27DE54DBE">3.6-1 <span class="Heading">First example: Whitehead's certain exact sequence</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X828F0FAB86AA60E9">3.6-2 <span class="Heading">Second example: the Hopf invariant</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7EAF7E677FB9D53F">3.7 <span class="Heading">Computing the second homotopy group of a space with infinite fundamental group</span></a>
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<h3>3 <span class="Heading">Covering spaces</span></h3>
<p>Let <span class="SimpleMath">\(Y\)</span> denote a finite regular CW-complex. Let <span class="SimpleMath">\(\widetilde Y\)</span> denote its universal covering space. The covering space inherits a regular CW-structure which can be computed and stored using the datatype of a <span class="SimpleMath">\(\pi_1Y\)</span>-equivariant CW-complex. The cellular chain complex <span class="SimpleMath">\(C_\ast\widetilde Y\)</span> of <span class="SimpleMath">\(\widetilde Y\)</span> can be computed and stored as an equivariant chain complex. Given an admissible discrete vector field on <span class="SimpleMath">\( Y,\)</span> we can endow <span class="SimpleMath">\(Y\)</span> with a smaller non-regular CW-structre whose cells correspond to the critical cells in the vector field. This smaller CW-structure leads to a more efficient chain complex <span class="SimpleMath">\(C_\ast \widetilde Y\)</span> involving one free generator for each critical cell in the vector field.</p>
<p><a id="X85FB4CA987BC92CC" name="X85FB4CA987BC92CC"></a></p>
<h4>3.1 <span class="Heading">Cellular chains on the universal cover</span></h4>
<p>The following commands construct a <span class="SimpleMath">\(6\)</span>-dimensional regular CW-complex <span class="SimpleMath">\(Y\simeq S^1 \times S^1\times S^1\)</span> homotopy equivalent to a product of three circles.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=[[1,1,1],[1,0,1],[1,1,1]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S:=PureCubicalComplex(A);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:=DirectProduct(S,S,S);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Y:=RegularCWComplex(T);;</span>
Regular CW-complex of dimension 6
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(Y);</span>
110592
</pre></div>
<p>The CW-somplex <span class="SimpleMath">\(Y\)</span> has <span class="SimpleMath">\(110592\)</span> cells. The next commands construct a free <span class="SimpleMath">\(\pi_1Y\)</span>-equivariant chain complex <span class="SimpleMath">\(C_\ast\widetilde Y\)</span> homotopy equivalent to the chain complex of the universal cover of <span class="SimpleMath">\(Y\)</span>. The chain complex <span class="SimpleMath">\(C_\ast\widetilde Y\)</span> has just <span class="SimpleMath">\(8\)</span> free generators.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Y:=ContractedComplex(Y);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CU:=ChainComplexOfUniversalCover(Y);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List([0..Dimension(Y)],n->CU!.dimension(n));</span>
[ 1, 3, 3, 1 ]
</pre></div>
<p>The next commands construct a subgroup <span class="SimpleMath">\(H < \pi_1Y\)</span> of index <span class="SimpleMath">\(50\)</span> and the chain complex <span class="SimpleMath">\(C_\ast\widetilde Y\otimes_{\mathbb ZH}\mathbb Z\)</span> which is homotopy equivalent to the cellular chain complex <span class="SimpleMath">\(C_\ast\widetilde Y_H\)</span> of the <span class="SimpleMath">\(50\)</span>-fold cover <span class="SimpleMath">\(\widetilde Y_H\)</span> of <span class="SimpleMath">\(Y\)</span> corresponding to <span class="SimpleMath">\(H\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:=LowIndexSubgroupsFpGroup(CU!.group,50);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H:=L[Length(L)-1];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Index(CU!.group,H);</span>
50
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=TensorWithIntegersOverSubgroup(CU,H);</span>
Chain complex of length 3 in characteristic 0 .
<span class="GAPprompt">gap></span> <span class="GAPinput">List([0..3],D!.dimension);</span>
[ 50, 150, 150, 50 ]
</pre></div>
<p>General theory implies that the <span class="SimpleMath">\(50\)</span>-fold covering space <span class="SimpleMath">\(\widetilde Y_H\)</span> should again be homotopy equivalent to a product of three circles. In keeping with this, the following commands verify that <span class="SimpleMath">\(\widetilde Y_H\)</span> has the same integral homology as <span class="SimpleMath">\(S^1\times S^1\times S^1\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(D,0);</span>
[ 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(D,1);</span>
[ 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(D,2);</span>
[ 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(D,3);</span>
[ 0 ]
</pre></div>
<p><a id="X7E5CC04E7E3CCDAD" name="X7E5CC04E7E3CCDAD"></a></p>
<h4>3.2 <span class="Heading">Spun knots and the Satoh tube map</span></h4>
<p>We'll contruct two spaces <span class="SimpleMath">\(Y,W\)</span> with isomorphic fundamental groups and isomorphic intergal homology, and use the integral homology of finite covering spaces to establsh that the two spaces have distinct homotopy types.</p>
<p>By <em>spinning</em> a link <span class="SimpleMath">\(K \subset \mathbb R^3\)</span> about a plane <span class="SimpleMath">\( P\subset \mathbb R^3\)</span> with <span class="SimpleMath">\(P\cap K=\emptyset\)</span>, we obtain a collection <span class="SimpleMath">\(Sp(K)\subset \mathbb R^4\)</span> of knotted tori. The following commands produce the two tori obtained by spinning the Hopf link <span class="SimpleMath">\(K\)</span> and show that the space <span class="SimpleMath">\(Y=\mathbb R^4\setminus Sp(K) = Sp(\mathbb R^3\setminus K)\)</span> is connected with fundamental group <span class="SimpleMath">\(\pi_1Y = \mathbb Z\times \mathbb Z\)</span> and homology groups <span class="SimpleMath">\(H_0(Y)=\mathbb Z\)</span>, <span class="SimpleMath">\(H_1(Y)=\mathbb Z^2\)</span>, <span class="SimpleMath">\(H_2(Y)=\mathbb Z^4\)</span>, <span class="SimpleMath">\(H_3(Y,\mathbb Z)=\mathbb Z^2\)</span>. The space <span class="SimpleMath">\(Y\)</span> is only constructed up to homotopy, and for this reason is <span class="SimpleMath">\(3\)</span>-dimensional.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Hopf:=PureCubicalLink("Hopf");</span>
Pure cubical link.
<span class="GAPprompt">gap></span> <span class="GAPinput">Y:=SpunAboutInitialHyperplane(PureComplexComplement(Hopf));</span>
Regular CW-complex of dimension 3
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(Y,0);</span>
[ 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(Y,1);</span>
[ 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(Y,2);</span>
[ 0, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(Y,3);</span>
[ 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(Y,4);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GY:=FundamentalGroup(Y);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup(GY);</span>
[ f2, f3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RelatorsOfFpGroup(GY);</span>
[ f3^-1*f2^-1*f3*f2 ]
</pre></div>
<p>An alternative embedding of two tori <span class="SimpleMath">\(L\subset \mathbb R^4 \)</span> can be obtained by applying the 'tube map' of Shin Satoh to a welded Hopf link <a href="chapBib_mj.html#biBMR1758871">[Sat00]</a>. The following commands construct the complement <span class="SimpleMath">\(W=\mathbb R^4\setminus L\)</span> of this alternative embedding and show that <span class="SimpleMath">\(W \)</span> has the same fundamental group and integral homology as <span class="SimpleMath">\(Y\)</span> above.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:=HopfSatohSurface();</span>
Pure cubical complex of dimension 4.
<span class="GAPprompt">gap></span> <span class="GAPinput">W:=ContractedComplex(RegularCWComplex(PureComplexComplement(L)));</span>
Regular CW-complex of dimension 3
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(W,0);</span>
[ 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(W,1);</span>
[ 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(W,2);</span>
[ 0, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(W,3);</span>
[ 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(W,4);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GW:=FundamentalGroup(W);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup(GW);</span>
[ f1, f2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RelatorsOfFpGroup(GW);</span>
[ f1^-1*f2^-1*f1*f2 ]
</pre></div>
<p>Despite having the same fundamental group and integral homology groups, the above two spaces <span class="SimpleMath">\(Y\)</span> and <span class="SimpleMath">\(W\)</span> were shown by Kauffman and Martins <a href="chapBib_mj.html#biBMR2441256">[KFM08]</a> to be not homotopy equivalent. Their technique involves the fundamental crossed module derived from the first three dimensions of the universal cover of a space, and counts the representations of this fundamental crossed module into a given finite crossed module. This homotopy inequivalence is recovered by the following commands which involves the <span class="SimpleMath">\(5\)</span>-fold covers of the spaces.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CY:=ChainComplexOfUniversalCover(Y);</span>
Equivariant chain complex of dimension 3
<span class="GAPprompt">gap></span> <span class="GAPinput">LY:=LowIndexSubgroups(CY!.group,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">invY:=List(LY,g->Homology(TensorWithIntegersOverSubgroup(CY,g),2));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CW:=ChainComplexOfUniversalCover(W);</span>
Equivariant chain complex of dimension 3
<span class="GAPprompt">gap></span> <span class="GAPinput">LW:=LowIndexSubgroups(CW!.group,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">invW:=List(LW,g->Homology(TensorWithIntegersOverSubgroup(CW,g),2));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SSortedList(invY)=SSortedList(invW);</span>
false
</pre></div>
<p><a id="X7C304A1C7EF0BA60" name="X7C304A1C7EF0BA60"></a></p>
<h4>3.3 <span class="Heading">Cohomology with local coefficients</span></h4>
<p>The <span class="SimpleMath">\(\pi_1Y\)</span>-equivariant cellular chain complex <span class="SimpleMath">\(C_\ast\widetilde Y\)</span> of the universal cover <span class="SimpleMath">\(\widetilde Y\)</span> of a regular CW-complex <span class="SimpleMath">\(Y\)</span> can be used to compute the homology <span class="SimpleMath">\(H_n(Y,A)\)</span> and cohomology <span class="SimpleMath">\(H^n(Y,A)\)</span> of <span class="SimpleMath">\(Y\)</span> with local coefficients in a <span class="SimpleMath">\(\mathbb Z\pi_1Y\)</span>-module <span class="SimpleMath">\(A\)</span>. To illustrate this we consister the space <span class="SimpleMath">\(Y\)</span> arising as the complement of the trefoil knot, with fundamental group <span class="SimpleMath">\(\pi_1Y = \langle x,y : xyx=yxy \rangle\)</span>. We take <span class="SimpleMath">\(A= \mathbb Z\)</span> to be the integers with non-trivial <span class="SimpleMath">\(\pi_1Y\)</span>-action given by <span class="SimpleMath">\(x.1=-1, y.1=-1\)</span>. We then compute</p>
<p><span class="SimpleMath">\(\begin{array}{lcl} H_0(Y,A) &= &\mathbb Z_2\, ,\\ H_1(Y,A) &= &\mathbb Z_3\, ,\\ H_2(Y,A) &= &\mathbb Z\, .\end{array}\)</span></p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">K:=PureCubicalKnot(3,1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Y:=PureComplexComplement(K);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Y:=ContractedComplex(Y);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Y:=RegularCWComplex(Y);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Y:=SimplifiedComplex(Y);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=ChainComplexOfUniversalCover(Y);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=C!.group;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup(G);</span>
[ f1, f2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RelatorsOfFpGroup(G);</span>
[ f2^-1*f1^-1*f2^-1*f1*f2*f1, f1^-1*f2^-1*f1^-1*f2*f1*f2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">hom:=GroupHomomorphismByImages(G,Group([[-1]]),[G.1,G.2],[[[-1]],[[-1]]]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=function(x); return Determinant(Image(hom,x)); end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=TensorWithTwistedIntegers(C,A); #Here the function A represents </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">#the integers with twisted action of G.</span>
Chain complex of length 3 in characteristic 0 .
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(D,0);</span>
[ 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(D,1);</span>
[ 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(D,2);</span>
[ 0 ]
</pre></div>
<p><a id="X7A4F34B780FA2CD5" name="X7A4F34B780FA2CD5"></a></p>
<h4>3.4 <span class="Heading">Distinguishing between two non-homeomorphic homotopy equivalent spaces</span></h4>
<p>The granny knot is the sum of the trefoil knot and its mirror image. The reef knot is the sum of two identical copies of the trefoil knot. The following commands show that the degree <span class="SimpleMath">\(1\)</span> homology homomorphisms</p>
<p><span class="SimpleMath">\(H_1(p^{-1}(B),\mathbb Z) \rightarrow H_1(\widetilde X_H,\mathbb Z)\)</span></p>
<p>distinguish between the homeomorphism types of the complements <span class="SimpleMath">\(X\subset \mathbb R^3\)</span> of the granny knot and the reef knot, where <span class="SimpleMath">\(B\subset X\)</span> is the knot boundary, and where <span class="SimpleMath">\(p\colon \widetilde X_H \rightarrow X\)</span> is the covering map corresponding to the finite index subgroup <span class="SimpleMath">\(H < \pi_1X\)</span>. More precisely, <span class="SimpleMath">\(p^{-1}(B)\)</span> is in general a union of path components</p>
<p><span class="SimpleMath">\(p^{-1}(B) = B_1 \cup B_2 \cup \cdots \cup B_t\)</span> .</p>
<p>The function <code class="code">FirstHomologyCoveringCokernels(f,c)</code> inputs an integer <span class="SimpleMath">\(c\)</span> and the inclusion <span class="SimpleMath">\(f\colon B\hookrightarrow X\)</span> of a knot boundary <span class="SimpleMath">\(B\)</span> into the knot complement <span class="SimpleMath">\(X\)</span>. The function returns the ordered list of the lists of abelian invariants of cokernels</p>
<p><span class="SimpleMath">\({\rm coker}(\ H_1(p^{-1}(B_i),\mathbb Z) \rightarrow H_1(\widetilde X_H,\mathbb Z)\ )\)</span></p>
<p>arising from subgroups <span class="SimpleMath">\(H < \pi_1X\)</span> of index <span class="SimpleMath">\(c\)</span>. To distinguish between the granny and reef knots we use index <span class="SimpleMath">\(c=6\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">K:=PureCubicalKnot(3,1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:=ReflectedCubicalKnot(K);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">granny:=KnotSum(K,L);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">reef:=KnotSum(K,K);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fg:=KnotComplementWithBoundary(ArcPresentation(granny));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fr:=KnotComplementWithBoundary(ArcPresentation(reef));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=FirstHomologyCoveringCokernels(fg,6);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:=FirstHomologyCoveringCokernels(fr,6);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a=b;</span>
false
</pre></div>
<p><a id="X869FD75B84AAC7AD" name="X869FD75B84AAC7AD"></a></p>
<h4>3.5 <span class="Heading"> Second homotopy groups of spaces with finite fundamental group</span></h4>
<p>If <span class="SimpleMath">\(p:\widetilde Y \rightarrow Y\)</span> is the universal covering map, then the fundamental group of <span class="SimpleMath">\(\widetilde Y\)</span> is trivial and the Hurewicz homomorphism <span class="SimpleMath">\(\pi_2\widetilde Y\rightarrow H_2(\widetilde Y,\mathbb Z)\)</span> from the second homotopy group of <span class="SimpleMath">\(\widetilde Y\)</span> to the second integral homology of <span class="SimpleMath">\(\widetilde Y\)</span> is an isomorphism. Furthermore, the map <span class="SimpleMath">\(p\)</span> induces an isomorphism <span class="SimpleMath">\(\pi_2\widetilde Y \rightarrow \pi_2Y\)</span>. Thus <span class="SimpleMath">\(H_2(\widetilde Y,\mathbb Z)\)</span> is isomorphic to the second homotopy group <span class="SimpleMath">\(\pi_2Y\)</span>.</p>
<p>If the fundamental group of <span class="SimpleMath">\(Y\)</span> happens to be finite, then in principle we can calculate <span class="SimpleMath">\(H_2(\widetilde Y,\mathbb Z) \cong \pi_2Y\)</span>. We illustrate this computation for <span class="SimpleMath">\(Y\)</span> equal to the real projective plane. The above computation shows that <span class="SimpleMath">\(Y\)</span> has second homotopy group <span class="SimpleMath">\(\pi_2Y \cong \mathbb Z\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">K:=[ [1,2,3], [1,3,4], [1,2,6], [1,5,6], [1,4,5], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [2,3,5], [2,4,5], [2,4,6], [3,4,6], [3,5,6]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">K:=MaximalSimplicesToSimplicialComplex(K);</span>
Simplicial complex of dimension 2.
<span class="GAPprompt">gap></span> <span class="GAPinput">Y:=RegularCWComplex(K); </span>
Regular CW-complex of dimension 2
<span class="GAPprompt">gap></span> <span class="GAPinput"># Y is a regular CW-complex corresponding to the projective plane.</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">U:=UniversalCover(Y);</span>
Equivariant CW-complex of dimension 2
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=U!.group;; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput"># G is the fundamental group of Y, which by the next command </span>
<span class="GAPprompt">gap></span> <span class="GAPinput"># is finite of order 2.</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(G);</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">U:=EquivariantCWComplexToRegularCWComplex(U,Group(One(G))); </span>
Regular CW-complex of dimension 2
<span class="GAPprompt">gap></span> <span class="GAPinput">#U is the universal cover of Y</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(U,0);</span>
[ 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(U,1);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(U,2);</span>
[ 0 ]
</pre></div>
<p><a id="X87F8F6C3812A7E73" name="X87F8F6C3812A7E73"></a></p>
<h4>3.6 <span class="Heading">Third homotopy groups of simply connected spaces</span></h4>
<p><a id="X7B506CF27DE54DBE" name="X7B506CF27DE54DBE"></a></p>
<h5>3.6-1 <span class="Heading">First example: Whitehead's certain exact sequence</span></h5>
<p>For any path connected space <span class="SimpleMath">\(Y\)</span> with universal cover <span class="SimpleMath">\(\widetilde Y\)</span> there is an exact sequence</p>
<p><span class="SimpleMath">\(\rightarrow \pi_4\widetilde Y \rightarrow H_4(\widetilde Y,\mathbb Z) \rightarrow H_4( K(\pi_2\widetilde Y,2), \mathbb Z ) \rightarrow \pi_3\widetilde Y \rightarrow H_3(\widetilde Y,\mathbb Z) \rightarrow 0 \)</span></p>
<p>due to J.H.C.Whitehead. Here <span class="SimpleMath">\(K(\pi_2(\widetilde Y),2)\)</span> is an Eilenberg-MacLane space with second homotopy group equal to <span class="SimpleMath">\(\pi_2\widetilde Y\)</span>.</p>
<p>Continuing with the above example where <span class="SimpleMath">\(Y\)</span> is the real projective plane, we see that <span class="SimpleMath">\(H_4(\widetilde Y,\mathbb Z) = H_3(\widetilde Y,\mathbb Z) = 0\)</span> since <span class="SimpleMath">\(\widetilde Y\)</span> is a <span class="SimpleMath">\(2\)</span>-dimensional CW-space. The exact sequence implies <span class="SimpleMath">\(\pi_3\widetilde Y \cong H_4(K(\pi_2\widetilde Y,2), \mathbb Z )\)</span>. Furthermore, <span class="SimpleMath">\(\pi_3\widetilde Y = \pi_3 Y\)</span>. The following commands establish that <span class="SimpleMath">\(\pi_3Y \cong \mathbb Z\, \)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=AbelianPcpGroup([0]);</span>
Pcp-group with orders [ 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">K:=EilenbergMacLaneSimplicialGroup(A,2,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=ChainComplexOfSimplicialGroup(K);</span>
Chain complex of length 5 in characteristic 0 .
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(C,4);</span>
[ 0 ]
</pre></div>
<p><a id="X828F0FAB86AA60E9" name="X828F0FAB86AA60E9"></a></p>
<h5>3.6-2 <span class="Heading">Second example: the Hopf invariant</span></h5>
<p>The following commands construct a <span class="SimpleMath">\(4\)</span>-dimensional simplicial complex <span class="SimpleMath">\(Y\)</span> with <span class="SimpleMath">\(9\)</span> vertices and <span class="SimpleMath">\(36\)</span> <span class="SimpleMath">\(4\)</span>-dimensional simplices, and establish that</p>
<p><span class="SimpleMath">\(\pi_1Y=0 , \pi_2Y=\mathbb Z , H_3(Y,\mathbb Z)=0, H_4(Y,\mathbb Z)=\mathbb Z \)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">smp:=[ [ 1, 2, 4, 5, 6 ], [ 1, 2, 4, 5, 9 ], [ 1, 2, 5, 6, 8 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 2, 6, 4, 7 ], [ 2, 3, 4, 5, 8 ], [ 2, 3, 5, 6, 4 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 2, 3, 5, 6, 7 ], [ 2, 3, 6, 4, 9 ], [ 3, 1, 4, 5, 7 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 3, 1, 5, 6, 9 ], [ 3, 1, 6, 4, 5 ], [ 3, 1, 6, 4, 8 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 4, 5, 7, 8, 3 ], [ 4, 5, 7, 8, 9 ], [ 4, 5, 8, 9, 2 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 4, 5, 9, 7, 1 ], [ 5, 6, 7, 8, 2 ], [ 5, 6, 8, 9, 1 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 5, 6, 8, 9, 7 ], [ 5, 6, 9, 7, 3 ], [ 6, 4, 7, 8, 1 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 6, 4, 8, 9, 3 ], [ 6, 4, 9, 7, 2 ], [ 6, 4, 9, 7, 8 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 7, 8, 1, 2, 3 ], [ 7, 8, 1, 2, 6 ], [ 7, 8, 2, 3, 5 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 7, 8, 3, 1, 4 ], [ 8, 9, 1, 2, 5 ], [ 8, 9, 2, 3, 1 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 8, 9, 2, 3, 4 ], [ 8, 9, 3, 1, 6 ], [ 9, 7, 1, 2, 4 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 9, 7, 2, 3, 6 ], [ 9, 7, 3, 1, 2 ], [ 9, 7, 3, 1, 5 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">K:=MaximalSimplicesToSimplicialComplex(smp);</span>
Simplicial complex of dimension 4.
<span class="GAPprompt">gap></span> <span class="GAPinput">Y:=RegularCWComplex(Y);</span>
Regular CW-complex of dimension 4
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(FundamentalGroup(Y));</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(Y,2);</span>
[ 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(Y,3);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(Y,4);</span>
[ 0 ]
</pre></div>
<p>Previous commands have established <span class="SimpleMath">\( H_4(K(\mathbb Z,2), \mathbb Z)=\mathbb Z\)</span>. So Whitehead's sequence reduces to an exact sequence</p>
<p><span class="SimpleMath">\(\mathbb Z \rightarrow \mathbb Z \rightarrow \pi_3Y \rightarrow 0\)</span></p>
<p>in which the first map is <span class="SimpleMath">\( H_4(Y,\mathbb Z)=\mathbb Z \rightarrow H_4(K(\pi_2Y,2), \mathbb Z )=\mathbb Z \)</span>. Hence <span class="SimpleMath">\(\pi_3Y\)</span> is cyclic.</p>
<p>HAP is currently unable to compute the order of <span class="SimpleMath">\(\pi_3Y\)</span> directly from Whitehead's sequence. Instead, we can use the <em>Hopf invariant</em>. For any map <span class="SimpleMath">\(\phi\colon S^3 \rightarrow S^2\)</span> we consider the space <span class="SimpleMath">\(C(\phi) = S^2 \cup_\phi e^4\)</span> obtained by attaching a <span class="SimpleMath">\(4\)</span>-cell <span class="SimpleMath">\(e^4\)</span> to <span class="SimpleMath">\(S^2\)</span> via the attaching map <span class="SimpleMath">\(\phi\)</span>. The cohomology groups <span class="SimpleMath">\(H^2(C(\phi),\mathbb Z)=\mathbb Z\)</span>, <span class="SimpleMath">\(H^4(C(\phi),\mathbb Z)=\mathbb Z\)</span> are generated by elements <span class="SimpleMath">\(\alpha, \beta\)</span> say, and the cup product has the form</p>
<p><span class="SimpleMath">\(- \cup -\colon H^2(C(\phi),\mathbb Z)\times H^2(C(\phi),\mathbb Z) \rightarrow H^4(C(\phi),\mathbb Z), (\alpha,\alpha) \mapsto h_\phi \beta\)</span></p>
<p>for some integer <span class="SimpleMath">\(h_\phi\)</span>. The integer <span class="SimpleMath">\(h_\phi\)</span> is the <strong class="button">Hopf invariant</strong>. The function <span class="SimpleMath">\(h\colon \pi_3(S^3)\rightarrow \mathbb Z\)</span> is a homomorphism and there is an isomorphism</p>
<p><span class="SimpleMath">\(\pi_3(S^2\cup e^4) \cong \mathbb Z/\langle h_\phi\rangle\)</span>.</p>
<p>The following commands begin by simplifying the cell structure on the above CW-complex <span class="SimpleMath">\(Y\cong K\)</span> to obtain a homeomorphic CW-complex <span class="SimpleMath">\(W\)</span> with fewer cells. They then create a space <span class="SimpleMath">\(S\)</span> by removing one <span class="SimpleMath">\(4\)</span>-cell from <span class="SimpleMath">\(W\)</span>. The space <span class="SimpleMath">\(S\)</span> is seen to be homotopy equivalent to a CW-complex <span class="SimpleMath">\(e^2\cup e^0\)</span> with a single <span class="SimpleMath">\(0\)</span>-cell and single <span class="SimpleMath">\(2\)</span>-cell. Hence <span class="SimpleMath">\(S\simeq S^2\)</span> is homotopy equivalent to the <span class="SimpleMath">\(2\)</span>-sphere. Consequently <span class="SimpleMath">\(Y \simeq C(\phi ) = S^2\cup_\phi e^4 \)</span> for some map <span class="SimpleMath">\(\phi\colon S^3 \rightarrow S^2\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:=SimplifiedComplex(Y);</span>
Regular CW-complex of dimension 4
<span class="GAPprompt">gap></span> <span class="GAPinput">S:=RegularCWComplexWithRemovedCell(W,4,6);</span>
Regular CW-complex of dimension 4
<span class="GAPprompt">gap></span> <span class="GAPinput">CriticalCells(S);</span>
[ [ 2, 6 ], [ 0, 5 ] ]
</pre></div>
<p>The next commands show that the map <span class="SimpleMath">\(\phi\)</span> in the construction <span class="SimpleMath">\(Y \simeq C(\phi) \)</span> has Hopf invariant -1. Hence <span class="SimpleMath">\(h\colon \pi_3(S^3)\rightarrow \mathbb Z\)</span> is an isomorphism. Therefore <span class="SimpleMath">\(\pi_3Y=0\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IntersectionForm(K);</span>
[ [ -1 ] ]
</pre></div>
<p>[The simplicial complex <span class="SimpleMath">\(K\)</span> in this second example is due to W. Kuehnel and T. F. Banchoff and is homeomorphic to the complex projective plane. ]</p>
<p><a id="X7EAF7E677FB9D53F" name="X7EAF7E677FB9D53F"></a></p>
<h4>3.7 <span class="Heading">Computing the second homotopy group of a space with infinite fundamental group</span></h4>
<p>The following commands compute the second integral homology</p>
<p><span class="SimpleMath">\(H_2(\pi_1W,\mathbb Z) = \mathbb Z\)</span></p>
<p>of the fundamental group <span class="SimpleMath">\(\pi_1W\)</span> of the complement <span class="SimpleMath">\(W\)</span> of the Hopf-Satoh surface.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:=HopfSatohSurface();</span>
Pure cubical complex of dimension 4.
<span class="GAPprompt">gap></span> <span class="GAPinput">W:=ContractedComplex(RegularCWComplex(PureComplexComplement(L)));</span>
Regular CW-complex of dimension 3
<span class="GAPprompt">gap></span> <span class="GAPinput">GW:=FundamentalGroup(W);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAspherical(GW);</span>
Presentation is aspherical.
true
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ResolutionAsphericalPresentation(GW);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(TensorWithIntegers(R),2);</span>
[ 0 ]
</pre></div>
<p>From Hopf's exact sequence</p>
<p><span class="SimpleMath">\( \pi_2W \stackrel{h}{\longrightarrow} H_2(W,\mathbb Z) \twoheadrightarrow H_2(\pi_1W,\mathbb Z) \rightarrow 0\)</span></p>
<p>and the computation <span class="SimpleMath">\(H_2(W,\mathbb Z)=\mathbb Z^4\)</span> we see that the image of the Hurewicz homomorphism is <span class="SimpleMath">\({\sf im}(h)= \mathbb Z^3\)</span> . The image of <span class="SimpleMath">\(h\)</span> is referred to as the subgroup of <em>spherical homology classes</em> and often denoted by <span class="SimpleMath">\(\Sigma^2W\)</span>.</p>
<p>The following command computes the presentation of <span class="SimpleMath">\(\pi_1W\)</span> corresponding to the <span class="SimpleMath">\(2\)</span>-skeleton <span class="SimpleMath">\(W^2\)</span> and establishes that <span class="SimpleMath">\(W^2 = S^2\vee S^2 \vee S^2 \vee (S^1\times S^1)\)</span> is a wedge of three spheres and a torus.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=FundamentalGroupOfRegularCWComplex(W,"no simplification");</span>
< fp group on the generators [ f1, f2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">RelatorsOfFpGroup(F);</span>
[ < identity ...>, f1^-1*f2^-1*f1*f2, < identity ...>, <identity ...> ]
</pre></div>
<p>The next command shows that the <span class="SimpleMath">\(3\)</span>-dimensional space <span class="SimpleMath">\(W\)</span> has two <span class="SimpleMath">\(3\)</span>-cells each of which is attached to the base-point of <span class="SimpleMath">\(W\)</span> with trivial boundary (up to homotopy in <span class="SimpleMath">\(W^2\)</span>). Hence <span class="SimpleMath">\(W = S^3\vee S^3\vee S^2 \vee S^2 \vee S^2 \vee (S^1\times S^1)\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CriticalCells(W);</span>
[ [ 3, 1 ], [ 3, 3148 ], [ 2, 6746 ], [ 2, 20510 ], [ 2, 33060 ],
[ 2, 50919 ], [ 1, 29368 ], [ 1, 50822 ], [ 0, 21131 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CriticalBoundaryCells(W,3,1);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CriticalBoundaryCells(W,3,3148);</span>
[ -50919, 50919 ]
</pre></div>
<p>Therefore <span class="SimpleMath">\(\pi_1W\)</span> is the free abelian group on two generators, and <span class="SimpleMath">\(\pi_2W\)</span> is the free <span class="SimpleMath">\(\mathbb Z\pi_1W\)</span>-module on three free generators.</p>
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