<Chapter><Heading>Covering spaces</Heading>

<P/>Let <M>Y</M> denote a finite regular CW-complex.
Let <M>\widetilde Y</M>  denote its universal covering space.
The covering space  inherits a regular CW-structure which
can be computed  and stored using the datatype of a
<M>\pi_1Y</M>-equivariant CW-complex. The
cellular chain complex <M>C_\ast\widetilde Y</M>  of <M>\widetilde Y</M> can
be computed and stored as an equivariant chain complex.
Given an admissible discrete vector field  on
<M> Y,</M> we can endow <M>Y</M> 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
 <M>C_\ast \widetilde Y</M> involving one free generator for each critical cell in the vector field.

<Section><Heading>Cellular chains on the universal cover</Heading>

<P/>The following commands construct a <M>6</M>-dimensional
regular CW-complex
<M>Y\simeq S^1 \times
S^1\times S^1</M>
homotopy equivalent to a product of three circles.

<Example>
<#Include SYSTEM "tutex/3.1.txt">
</Example>

<P/>The CW-somplex <M>Y</M> has <M>110592</M> cells.
The next commands construct a free
<M>\pi_1Y</M>-equivariant chain complex
<M>C_\ast\widetilde Y</M>  homotopy equivalent to the chain complex of the
universal cover of <M>Y</M>. The chain complex <M>C_\ast\widetilde Y</M>
has just <M>8</M> free generators.
<Example>
<#Include SYSTEM "tutex/3.2.txt">
</Example>

<P/>The next commands construct a subgroup <M>H &lt; \pi_1Y</M>
of index <M>50</M> and the chain complex
<M>C_\ast\widetilde Y\otimes_{\mathbb ZH}\mathbb Z</M>  which is
homotopy equivalent to the cellular chain complex
<M>C_\ast\widetilde Y_H</M> of the <M>50</M>-fold cover
<M>\widetilde Y_H</M> of
<M>Y</M> corresponding to <M>H</M>.

<Example>
<#Include SYSTEM "tutex/3.3.txt">
</Example>

<P/>General theory implies that the <M>50</M>-fold covering space
<M>\widetilde Y_H</M>  should again be homotopy equivalent to a
product of three circles. In keeping with this, the following commands
verify that <M>\widetilde Y_H</M> has the same integral homology
as <M>S^1\times S^1\times S^1</M>.

<Example>
<#Include SYSTEM "tutex/3.4.txt">
</Example>

</Section>
<Section><Heading>Spun knots and the Satoh tube map</Heading>

<P/>We'll contruct two spaces  <M>Y,W</M> 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/>By <E>spinning</E> a  link <M>K \subset \mathbb R^3</M> about a plane
<M> P\subset \mathbb R^3</M> with <M>P\cap K=\emptyset</M>, we obtain a  collection
<M>Sp(K)\subset \mathbb R^4</M> of knotted tori. The following commands produce the two tori obtained by spinning the Hopf link <M>K</M>
and
show that the space <M>Y=\mathbb R^4\setminus Sp(K) = Sp(\mathbb R^3\setminus K)</M> is connected with fundamental group <M>\pi_1Y = \mathbb Z\times \mathbb Z</M> and homology groups <M>H_0(Y)=\mathbb Z</M>, <M>H_1(Y)=\mathbb Z^2</M>, <M>H_2(Y)=\mathbb Z^4</M>, <M>H_3(Y,\mathbb Z)=\mathbb Z^2</M>. The space <M>Y</M> is only constructed up to homotopy, and for this reason is <M>3</M>-dimensional.

<Example>
<#Include SYSTEM "tutex/3.5.txt">
</Example>

<P/>An alternative embedding of two tori  <M>L\subset \mathbb R^4 </M> can be
obtained by applying the 'tube map' of Shin Satoh to a welded Hopf link 
<Cite Key="MR1758871"/>.
 The following commands
construct the complement <M>W=\mathbb R^4\setminus L</M> of this alternative
embedding
and show that <M>W </M> has the same fundamental group and integral homology as <M>Y</M> above.

<Example>
<#Include SYSTEM "tutex/3.6.txt">
</Example>

<P/>Despite having the same fundamental group and integral homology groups,
the above two spaces <M>Y</M> and <M>W</M> were shown by  Kauffman and  Martins  <Cite Key="MR2441256"/> 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 <M>5</M>-fold covers of the spaces.

<Example>
<#Include SYSTEM "tutex/3.7.txt">
</Example>

</Section>
<Section><Heading>Cohomology with local coefficients</Heading>

<P/>The <M>\pi_1Y</M>-equivariant cellular chain complex
<M>C_\ast\widetilde Y</M> of the universal cover <M>\widetilde Y</M> of a regular
CW-complex <M>Y</M> can be used to compute the homology
<M>H_n(Y,A)</M> and cohomology <M>H^n(Y,A)</M>
of <M>Y</M> with local coefficients in a
<M>\mathbb Z\pi_1Y</M>-module <M>A</M>.
To illustrate this we consister the space <M>Y</M> arising as the
complement of the trefoil knot, with fundamental group
<M>\pi_1Y = \langle x,y : xyx=yxy \rangle</M>.
We take <M>A= \mathbb Z</M> to be the integers with non-trivial
<M>\pi_1Y</M>-action given by <M>x.1=-1, y.1=-1</M>.
We then compute
<P/><M>\begin{array}{lcl}
H_0(Y,A) &amp;= &amp;\mathbb Z_2\, ,\\
H_1(Y,A) &amp;= &amp;\mathbb Z_3\, ,\\
H_2(Y,A) &amp;= &amp;\mathbb Z\, .\end{array}</M>

<Example>
<#Include SYSTEM "tutex/3.8.txt">
</Example>

</Section>
<Section><Heading>Distinguishing between two non-homeomorphic homotopy equivalent spaces</Heading>

<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 <M>1</M> homology homomorphisms
<P/><M>H_1(p^{-1}(B),\mathbb Z)
\rightarrow H_1(\widetilde X_H,\mathbb Z)</M>
<P/> distinguish between the
homeomorphism types of the complements <M>X\subset \mathbb R^3</M> of the granny knot and 
the reef knot, where <M>B\subset X</M> is the knot boundary, and where
<M>p\colon \widetilde X_H \rightarrow X</M> is the  covering map 
corresponding to the finite index subgroup <M>H &lt; \pi_1X</M>.
 More precisely, <M>p^{-1}(B)</M> is in general a union of path components
<P/><M>p^{-1}(B) = B_1 \cup B_2 \cup \cdots \cup B_t</M> .
<P/> The function
<C>FirstHomologyCoveringCokernels(f,c)</C> inputs an integer <M>c</M> and the inclusion
<M>f\colon B\hookrightarrow X</M> of a knot boundary <M>B</M> into the knot complement <M>X</M>. The function returns the ordered list of the  lists of abelian invariants of cokernels
<P/><M>{\rm coker}(\ H_1(p^{-1}(B_i),\mathbb Z)
\rightarrow H_1(\widetilde X_H,\mathbb Z)\ )</M>
<P/>arising from subgroups <M>H &lt; \pi_1X</M> of index <M>c</M>. To distinguish between the granny and reef knots we use index <M>c=6</M>.

<Example>
<#Include SYSTEM "tutex/3.9.txt">
</Example>

</Section>
<Section><Heading> Second homotopy groups of spaces with finite fundamental group</Heading>

<P/>If <M>p:\widetilde Y \rightarrow Y</M> is the  universal covering map,
 then the fundamental group of
<M>\widetilde Y</M> is trivial and the Hurewicz homomorphism
<M>\pi_2\widetilde Y\rightarrow H_2(\widetilde Y,\mathbb Z)</M> from the second
homotopy group of <M>\widetilde Y</M> to the second integral homology of
<M>\widetilde Y</M> is an isomorphism. Furthermore, the map <M>p</M>
induces an isomorphism  <M>\pi_2\widetilde Y \rightarrow
\pi_2Y</M>. Thus <M>H_2(\widetilde Y,\mathbb Z)</M>
 is isomorphic to
the second homotopy group <M>\pi_2Y</M>.

<P/>
If the fundamental group of <M>Y</M> happens to be finite, then
in principle we
can calculate <M>H_2(\widetilde Y,\mathbb Z) \cong \pi_2Y</M>.
We illustrate this computation for <M>Y</M> equal to the
real projective plane.
The above computation shows that <M>Y</M> has
 second homotopy group <M>\pi_2Y \cong \mathbb Z</M>.

<Example>
<#Include SYSTEM "tutex/3.10.txt">
</Example>

</Section>
<Section>
<Heading>Third homotopy groups of simply connected spaces</Heading>


<Subsection><Heading>First example: Whitehead's certain exact sequence</Heading>

<P/>For any path connected space <M>Y</M> with universal cover
<M>\widetilde Y</M> there is an exact sequence
<P/>
<M>\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
</M>
<P/>
due to J.H.C.Whitehead.  Here
<M>K(\pi_2(\widetilde Y),2)</M> is an Eilenberg-MacLane space with
second homotopy group equal to <M>\pi_2\widetilde Y</M>.

<P/>Continuing with the above example where <M>Y</M> is the real
projective plane, we see that
<M>H_4(\widetilde Y,\mathbb Z) = H_3(\widetilde Y,\mathbb Z) = 0</M>
 since <M>\widetilde Y</M> is a <M>2</M>-dimensional CW-space. The exact sequence implies
 <M>\pi_3\widetilde Y \cong H_4(K(\pi_2\widetilde Y,2), \mathbb Z )</M>. Furthermore, <M>\pi_3\widetilde Y = \pi_3 Y</M>.
 The following commands establish that
<M>\pi_3Y \cong \mathbb Z\, 
</M>.

<Example>
<#Include SYSTEM "tutex/3.11.txt">
</Example>
</Subsection>

<Subsection><Heading>Second example: the Hopf invariant</Heading>

<P/> The following commands construct a <M>4</M>-dimensional simplicial complex
<M>Y</M> with <M>9</M> vertices and <M>36</M> <M>4</M>-dimensional simplices,
and establish that
<P/>
<M>\pi_1Y=0 , \pi_2Y=\mathbb Z , H_3(Y,\mathbb Z)=0, H_4(Y,\mathbb Z)=\mathbb Z
</M>.

<Example>
<#Include SYSTEM "tutex/3.12.txt">
</Example>

<P/>
Previous commands have established <M> H_4(K(\mathbb Z,2), \mathbb Z)=\mathbb Z</M>.
 So Whitehead's sequence reduces to an exact sequence


<P/><M>\mathbb Z \rightarrow \mathbb Z \rightarrow \pi_3Y \rightarrow 0</M>


<P/>in which the first map is
<M>
H_4(Y,\mathbb Z)=\mathbb Z \rightarrow H_4(K(\pi_2Y,2), \mathbb Z )=\mathbb Z
</M>. Hence <M>\pi_3Y</M> is cyclic. 

<P/> HAP is currently unable to compute the order of <M>\pi_3Y</M> directly from Whitehead's sequence. Instead, we can use the <E>Hopf invariant</E>.
 For any map <M>\phi\colon S^3 \rightarrow S^2</M> we consider the space
 <M>C(\phi) = S^2 \cup_\phi e^4</M>  obtained by attaching a <M>4</M>-cell
<M>e^4</M> to <M>S^2</M> via the attaching map <M>\phi</M>. The cohomology groups <M>H^2(C(\phi),\mathbb Z)=\mathbb Z</M>, <M>H^4(C(\phi),\mathbb Z)=\mathbb Z</M> are generated by elements <M>\alpha, \beta</M> say, and the cup product has the form 
<P/><M>- \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</M> 
<P/>for some integer <M>h_\phi</M>.
The integer <M>h_\phi</M> is the <B>Hopf invariant</B>. The function <M>h\colon \pi_3(S^3)\rightarrow \mathbb Z</M> is a homomorphism and there is an isomorphism
<P/><M>\pi_3(S^2\cup e^4) \cong \mathbb Z/\langle h_\phi\rangle</M>.

<P/>The following commands begin by simplifying the cell structure on the above CW-complex <M>Y\cong K</M> to obtain a homeomorphic CW-complex <M>W</M> with fewer cells. They then create a space <M>S</M> by removing one <M>4</M>-cell from <M>W</M>. The space <M>S</M> is seen to be homotopy equivalent to a CW-complex <M>e^2\cup e^0</M> with a single <M>0</M>-cell and single <M>2</M>-cell. Hence <M>S\simeq S^2</M> is homotopy equivalent to the <M>2</M>-sphere. Consequently
 <M>Y \simeq C(\phi ) = S^2\cup_\phi e^4 </M> for some map <M>\phi\colon S^3 \rightarrow S^2</M>.


<Example>
<#Include SYSTEM "tutex/3.12a.txt">
</Example>
 
<P/> The next commands  show that the map <M>\phi</M> in the construction
<M>Y \simeq C(\phi) </M> has Hopf invariant -1. Hence <M>h\colon \pi_3(S^3)\rightarrow \mathbb Z</M> is an isomorphism. Therefore  <M>\pi_3Y=0</M>.

<Example>
<#Include SYSTEM "tutex/3.12b.txt">
</Example>

<P/>
[The simplicial complex <M>K</M> in this second example is due to W. Kuehnel and
T. F. Banchoff and is homeomorphic to the complex projective plane.
]


</Subsection>
</Section>

<Section>
<Heading>Computing the second homotopy group of a space with infinite fundamental group</Heading>

The following commands compute the second integral homology
<P/>
<M>H_2(\pi_1W,\mathbb Z) = \mathbb Z</M>

<P/>of the fundamental group <M>\pi_1W</M> of the complement <M>W</M> of the Hopf-Satoh surface.

<Example>
<#Include SYSTEM "tutex/3.20.txt">
</Example>

<P/>From Hopf's exact sequence
<P/>
<M> \pi_2W \stackrel{h}{\longrightarrow} H_2(W,\mathbb Z) \twoheadrightarrow H_2(\pi_1W,\mathbb Z) \rightarrow 0</M>
<P/>
and the computation <M>H_2(W,\mathbb Z)=\mathbb Z^4</M> we see that the image of the Hurewicz homomorphism is

<M>{\sf im}(h)= \mathbb Z^3</M> .  
The image of <M>h</M> is referred to as the subgroup of <E>spherical homology classes</E> and often denoted by <M>\Sigma^2W</M>.

<P/>The following command computes the presentation of <M>\pi_1W</M> corresponding to the <M>2</M>-skeleton <M>W^2</M> 
and establishes that <M>W^2 = S^2\vee S^2 \vee S^2 \vee (S^1\times S^1)</M> is a wedge of three spheres and a torus.

<Example>
<#Include SYSTEM "tutex/3.21.txt">
</Example>

<P/>The next command shows that the <M>3</M>-dimensional space <M>W</M> has two <M>3</M>-cells each of which is attached to the base-point of <M>W</M> with trivial boundary (up to homotopy in <M>W^2</M>). Hence <M>W = S^3\vee S^3\vee S^2 \vee S^2 \vee S^2 \vee (S^1\times S^1)</M>.

<Example>
<#Include SYSTEM "tutex/3.22.txt">
</Example>

<P/> Therefore <M>\pi_1W</M> is the free abelian group on two generators, and
<M>\pi_2W</M> is the free <M>\mathbb Z\pi_1W</M>-module on three free generators.
</Section>
</Chapter>