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tutorial/chap6.html

@@ -174,6 +174,34 @@ <h4>6.5 <span class="Heading">Group presentations and homotopical syzygies</span
 
 <p><span class="SimpleMath">[ [x,y], ^yz ] [ [y,z], ^zx ] [ [z,x], ^xy ] = 1 .</span></p>
 
+<p>The homotopical syzygy is special since in this example the edge directions and labels can be understood as specifying three homeomorphisms between pairs of faces. Viewing the syzygy as the boundary of the <span class="SimpleMath">3</span>-ball, by using the homeomorphisms to identify the faces in each face pair we obtain a quotient CW-complex <span class="SimpleMath">M</span> involving one vertex, three edges, three <span class="SimpleMath">2</span>-cells and one <span class="SimpleMath">3</span>-cell. The cell structure on the quotient exists because, under the restrictions of homomorphisms to the edges, any cycle of edges retricts to the identity map on any given edge. The following result tells us that <span class="SimpleMath">M</span> is in fact a closed oriented compact <span class="SimpleMath">3</span>-manifold.</p>
+
+<p><strong class="button">Theorem.</strong> [Seifert u. Threlfall, Topologie, p.208] <em>Let <span class="SimpleMath">S^2</span> denote the boundary of the <span class="SimpleMath">3</span>-ball <span class="SimpleMath">B^3</span> and suppose that the sphere <span class="SimpleMath">S^2</span> is given a regular CW-structure in which the faces are partitioned into a collection of face pairs. Suppose that for each face pair there is an orientation reversing homeomorphism between the two faces that sends edges to edges and vertices to vertices. Suppose that by using these homeomorphisms to identity face pairs we obtain a (not necessarily regular) CW-structure on the quotient <span class="SimpleMath">M</span>. Then <span class="SimpleMath">M</span> is a closed compact orientable manifold if and only if its Euler characteristic is <span class="SimpleMath">χ(M)=0</span>.</em></p>
+
+<p>The next commands construct a presentation and associated unique homotopical syzygy for the free nilpotent group of class <span class="SimpleMath">c=2</span> on <span class="SimpleMath">n=2</span> generators.</p>
+
+
+<div class="example"><pre>
+<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">n:=2;;c:=2;;</span>
+<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">G:=Image(NqEpimorphismNilpotentQuotient(FreeGroup(n),c));;</span>
+<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">R:=ResolutionNilpotentGroup(G,4);;</span>
+<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">P:=PresentationOfResolution(R);;</span>
+<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">P.freeGroup;</span>
+&lt;free group on the generators [ x, y, z ]&gt;
+<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">P.relators;</span>
+[ z*x*y*x^-1*y^-1, z*x*z^-1*x^-1, z*y*z^-1*y^-1 ]
+<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IdentityAmongRelatorsDisplay(R,1);</span>
+
+</pre></div>
+
+<p><img src="images/syznil.gif" align="center" height="160" alt="Homotopical syzygy for the free nilpotent group of class two on two generators"/></p>
+
+<p>The syzygy represents the following relationship between commutators (in a free group).</p>
+
+<p><span class="SimpleMath">[ [x^-1,y][x,y] , [y,x][y^-1,x]y^-1 ] [ [y,x][y^-1,x] , x^-1 ]</span></p>
+
+<p>Again, using the theorem of Seifert and Threlfall we see that the free nilpotent group of class three on two generators arises as the fundamental group of a closed compact orientable <span class="SimpleMath">3</span>-manifold.</p>
+
 <p><a id="X7F719758856A443D" name="X7F719758856A443D"></a></p>
 
 <h4>6.6 <span class="Heading">Bogomolov multiplier</span></h4>

tutorial/chap6.txt

@@ -138,6 +138,51 @@
 
   [ [x,y], ^yz ] [ [y,z], ^zx ] [ [z,x], ^xy ] = 1 .
 
+  The  homotopical syzygy is special since in this example the edge directions
+  and  labels  can  be  understood  as specifying three homeomorphisms between
+  pairs  of  faces. Viewing the syzygy as the boundary of the 3-ball, by using
+  the  homeomorphisms  to  identify  the  faces  in each face pair we obtain a
+  quotient  CW-complex  M involving one vertex, three edges, three 2-cells and
+  one  3-cell.  The  cell  structure on the quotient exists because, under the
+  restrictions  of  homomorphisms to the edges, any cycle of edges retricts to
+  the  identity map on any given edge. The following result tells us that M is
+  in fact a closed oriented compact 3-manifold.
+
+  Theorem.  [Seifert  u.  Threlfall,  Topologie,  p.208]  Let  S^2  denote the
+  boundary  of  the  3-ball  B^3  and  suppose  that the sphere S^2 is given a
+  regular CW-structure in which the faces are partitioned into a collection of
+  face  pairs.  Suppose  that  for  each  face  pair  there  is an orientation
+  reversing  homeomorphism between the two faces that sends edges to edges and
+  vertices to vertices. Suppose that by using these homeomorphisms to identity
+  face  pairs  we  obtain  a  (not  necessarily  regular)  CW-structure on the
+  quotient  M.  Then  M is a closed compact orientable manifold if and only if
+  its Euler characteristic is χ(M)=0.
+
+  The next commands construct a presentation and associated unique homotopical
+  syzygy for the free nilpotent group of class c=2 on n=2 generators.
+
+    Example  
+    gap> n:=2;;c:=2;;
+    gap> G:=Image(NqEpimorphismNilpotentQuotient(FreeGroup(n),c));;
+    gap> R:=ResolutionNilpotentGroup(G,4);;
+    gap> P:=PresentationOfResolution(R);;
+    gap> P.freeGroup;
+    <free group on the generators [ x, y, z ]>
+    gap> P.relators;
+    [ z*x*y*x^-1*y^-1, z*x*z^-1*x^-1, z*y*z^-1*y^-1 ]
+    gap> IdentityAmongRelatorsDisplay(R,1);
+    
+  
+
+  The  syzygy  represents the following relationship between commutators (in a
+  free group).
+
+  [ [x^-1,y][x,y] , [y,x][y^-1,x]y^-1 ] [ [y,x][y^-1,x] , x^-1 ]
+
+  Again,  using  the  theorem  of  Seifert  and Threlfall we see that the free
+  nilpotent  group  of class three on two generators arises as the fundamental
+  group of a closed compact orientable 3-manifold.
+
 
   6.6 Bogomolov multiplier
 

tutorial/chap6_mj.html

@@ -177,6 +177,34 @@ <h4>6.5 <span class="Heading">Group presentations and homotopical syzygies</span
 
 <p><span class="SimpleMath">\( [\ [x,y],\ {^yz}\ ]\ \ [\ [y,z],\ {^zx}\ ]\ \ [\ [z,x],\ {^xy}\ ]\ \ =\ \ 1\ \ .\)</span></p>
 
+<p>The homotopical syzygy is special since in this example the edge directions and labels can be understood as specifying three homeomorphisms between pairs of faces. Viewing the syzygy as the boundary of the <span class="SimpleMath">\(3\)</span>-ball, by using the homeomorphisms to identify the faces in each face pair we obtain a quotient CW-complex <span class="SimpleMath">\(M\)</span> involving one vertex, three edges, three <span class="SimpleMath">\(2\)</span>-cells and one <span class="SimpleMath">\(3\)</span>-cell. The cell structure on the quotient exists because, under the restrictions of homomorphisms to the edges, any cycle of edges retricts to the identity map on any given edge. The following result tells us that <span class="SimpleMath">\(M\)</span> is in fact a closed oriented compact <span class="SimpleMath">\(3\)</span>-manifold.</p>
+
+<p><strong class="button">Theorem.</strong> [Seifert u. Threlfall, Topologie, p.208] <em>Let <span class="SimpleMath">\(S^2\)</span> denote the boundary of the <span class="SimpleMath">\(3\)</span>-ball <span class="SimpleMath">\(B^3\)</span> and suppose that the sphere <span class="SimpleMath">\(S^2\)</span> is given a regular CW-structure in which the faces are partitioned into a collection of face pairs. Suppose that for each face pair there is an orientation reversing homeomorphism between the two faces that sends edges to edges and vertices to vertices. Suppose that by using these homeomorphisms to identity face pairs we obtain a (not necessarily regular) CW-structure on the quotient <span class="SimpleMath">\(M\)</span>. Then <span class="SimpleMath">\(M\)</span> is a closed compact orientable manifold if and only if its Euler characteristic is <span class="SimpleMath">\(\chi(M)=0\)</span>.</em></p>
+
+<p>The next commands construct a presentation and associated unique homotopical syzygy for the free nilpotent group of class <span class="SimpleMath">\(c=2\)</span> on <span class="SimpleMath">\(n=2\)</span> generators.</p>
+
+
+<div class="example"><pre>
+<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">n:=2;;c:=2;;</span>
+<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">G:=Image(NqEpimorphismNilpotentQuotient(FreeGroup(n),c));;</span>
+<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">R:=ResolutionNilpotentGroup(G,4);;</span>
+<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">P:=PresentationOfResolution(R);;</span>
+<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">P.freeGroup;</span>
+&lt;free group on the generators [ x, y, z ]&gt;
+<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">P.relators;</span>
+[ z*x*y*x^-1*y^-1, z*x*z^-1*x^-1, z*y*z^-1*y^-1 ]
+<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IdentityAmongRelatorsDisplay(R,1);</span>
+
+</pre></div>
+
+<p><img src="images/syznil.gif" align="center" height="160" alt="Homotopical syzygy for the free nilpotent group of class two on two generators"/></p>
+
+<p>The syzygy represents the following relationship between commutators (in a free group).</p>
+
+<p><span class="SimpleMath">\( [\ [x^{-1},y][x,y]\ ,\ [y,x][y^{-1},x]y^{-1}\ ]\ [\ [y,x][y^{-1},x]\ , \ x^{-1} \ ] \)</span></p>
+
+<p>Again, using the theorem of Seifert and Threlfall we see that the free nilpotent group of class three on two generators arises as the fundamental group of a closed compact orientable <span class="SimpleMath">\(3\)</span>-manifold.</p>
+
 <p><a id="X7F719758856A443D" name="X7F719758856A443D"></a></p>
 
 <h4>6.6 <span class="Heading">Bogomolov multiplier</span></h4>