<Chapter><Heading>Three Manifolds</Heading>
<Section><Heading>Dehn Surgery</Heading>
The following example constructs, as a regular CW-complex, a closed orientable 
3-manifold <M>W</M> obtained from the 3-sphere by drilling out
 a tubular neighbourhood of a
 trefoil knot and  then gluing a solid torus to the boundary of the cavity via a homeomorphism corresponding to a Dehn surgery coefficient <M>p/q=17/16</M>.

<Example>
<#Include SYSTEM "tutex/12.1.txt">
</Example>

The next commands show that this  <M>3</M>-manifold <M>W</M> has integral homology
<P/> <M> H_0(W,\mathbb Z)=\mathbb Z</M>, <M> H_1(W,\mathbb Z)=\mathbb Z_{16}</M>, 
<M> H_2(W,\mathbb Z)=0</M>, 
<M> H_3(W,\mathbb Z)=\mathbb Z</M> 
<P/> and that the fundamental group <M>\pi_1(W)</M> is non-abelian.

<Example>
<#Include SYSTEM "tutex/12.1a.txt">
</Example>

<P/>
The following famous result of Lickorish and (independently) Wallace shows that Dehn surgery on knots leads to an interesting range of spaces.

<P/><B>Theorem:</B> <E> Every closed, orientable, connected <M>3</M>-manifold can be obtained by surgery on a link  in <M>S^3</M>. (Moreover, one can always 
perform the surgery with surgery coefficients <M>\pm 1</M> and with each individual component of the link unknotted.) 
</E>

</Section>

<Section><Heading>Connected Sums</Heading>
The following  example constructs the connected sum <M>W=A\#B</M> of two <M>3</M>-manifolds, where <M>A</M> is obtained from a <M>5/1</M> Dehn surgery on the complement of the first prime knot on 11 crossings and <M>B</M> is obtained by a <M>5/1</M> Dehn surgery on the complement of the 
second prime knot
 on 11 crossings. The homology groups 
<P/><M>H_1(W,\mathbb Z) = \mathbb Z_2\oplus \mathbb Z_{594}</M>,
<M>H_2(W,\mathbb Z) = 0</M>,
<M>H_3(W,\mathbb Z) = \mathbb Z</M>
<P/> are computed.

<Example>
<#Include SYSTEM "tutex/12.6.txt">
</Example>

</Section>


<Section><Heading>Dijkgraaf-Witten Invariant</Heading>
Given a closed connected orientable <M>3</M>-manifold <M>W</M>, a finite group <M>G</M> and
a 3-cocycle <M>\alpha\in H^3(BG,U(1))</M> Dijkgraaf and Witten define the complex number
<P/>$$ Z^{G,\alpha}(W) = \frac{1}{|G|}\sum_{\gamma\in {\rm Hom}(\pi_1W, G)}   \langle \gamma^\ast[\alpha], [M]\rangle \ \in\ \mathbb C\ $$
where <M>\gamma</M> ranges over all group homomorphisms <M>\gamma\colon \pi_1W \rightarrow G</M>.
This complex number is  an invariant of the homotopy type of <M>W</M> and is useful for
  distinguishing between certain homotopically distinct <M>3</M>-manifolds.

<P/>A homology version of the Dijkgraaf-Witten invariant can be defined as the set of homology homomorphisms

$$D_G(W) =\{ \gamma_\ast\colon H_3(W,\mathbb Z) \longrightarrow H_3(BG,\mathbb Z)   \}_{\gamma\in {\rm Hom}(\pi_1W, G)}.$$

Since <M>H_3(W,\mathbb Z)\cong \mathbb Z</M> we represent <M>D_G(W)</M> by the set <M>D_G(W)=\{ \gamma_\ast(1) \}_{\gamma\in {\rm Hom}(\pi_1W, G)}</M> 
where <M>1</M> denotes one of the two possible generators of <M>H_3(W,\mathbb Z)</M>.

<P/> For any coprime integers <M>p,q\ge 1</M> the <E>lens space</E> <M>L(p,q)</M> is 
obtained from the 3-sphere by drilling out
 a tubular neighbourhood of the trivial 
  knot and  then gluing a solid torus to the boundary of the cavity via a homeomorphism corresponding to a Dehn surgery coefficient <M>p/q</M>. Lens spaces have  cyclic fundamental group <M>\pi_1(L(p,q))=C_p</M> and homology <M>H_1(L(p,q),\mathbb Z)\cong \mathbb Z_p</M>,
<M>H_2(L(p,q),\mathbb Z)\cong 0</M>,
<M>H_3(L(p,q),\mathbb Z)\cong \mathbb Z</M>.
It was proved by J.H.C. Whitehead that two lens spaces <M>L(p,q)</M> and <M>L(p',q')</M> are homotopy
equivalent if and only if <M>p=p'</M> and <M>qq'\equiv \pm n^2 \bmod p</M> for some integer <M>n</M>.


<P/> The following session constructs the two lens spaces <M>L(5,1)</M> and <M>L(5,2)</M>. The homology version of the Dijkgraaf-Witten invariant is used with <M>G=C_5</M> to demonstrate that the two lens spaces are not homotopy equivalent.

<Example>
<#Include SYSTEM "tutex/12.2.txt">
</Example>


A theorem of Fermat and Euler states that if a prime <M>p</M> is congruent to 3 modulo 4, then for any <M>q</M> exactly one  of <M>\pm q</M> is a 
quadratic residue mod p. For all other primes <M>p</M> either both or neither
of <M>\pm q</M> is a quadratic residue mod <M>p</M>.
Thus for fixed <M>p \equiv 3 \bmod 4</M>  the lens spaces
<M>L(p,q)</M> form a single homotopy class.  There are precisely two
homotopy classes  of lens spaces for other <M>p</M>. 

<P/> The following commands confirm that <M>L(13,1) \not\simeq L(13,2)</M>.
<Example>
<#Include SYSTEM "tutex/12.3.txt">
</Example>

	</Section>

<Section><Heading>Cohomology rings</Heading>
The following commands construct the multiplication table (with respect to some basis) for the cohomology rings <M>H^\ast(L(13,1),\mathbb Z_{13})</M> and
<M>H^\ast(L(13,2),\mathbb Z_{13})</M>. These rings are isomorphic  and so fail to distinguish between the homotopy types of the lens spaces <M>L(13,1)</M>
and <M>L(13,2)</M>.

<Example>
<#Include SYSTEM "tutex/12.4.txt">
</Example>
</Section>

<Section><Heading>Linking Form</Heading>

Given a closed connected <B>oriented</B> <M>3</M>-manifold <M>W</M> let 
<M>\tau H_1(W,\mathbb Z)</M> denote the torsion subgroup of the first integral homology. The <E>linking form</E> is a bilinear mapping
<P/><M>Lk_W\colon \tau H_1(W,\mathbb Z) \times \tau H_1(W,\mathbb Z) \longrightarrow \mathbb Q/\mathbb Z</M>.
<P/>To construct this form note that we have a Poincare duality isomorphism
<P/><M>\rho\colon H^2(W,\mathbb Z) \stackrel{\cong}{\longrightarrow} H_1(W,\mathbb Z), z \mapsto z\cap [W]</M>
<P/>involving the cap product with the fundamental class <M>[W]\in H^3(W,\mathbb Z)</M>. That is,  <M>[M]</M> is the generator of <M>H^3(W,\mathbb Z)\cong \mathbb Z</M> determining the orientation.

The short exact sequence <M>\mathbb Z \rightarrowtail \mathbb Q \twoheadrightarrow \mathbb Q/\mathbb Z</M> gives rise to a cohomology exact sequence
<P/><M> \rightarrow H^1(W,\mathbb Q) \rightarrow H^1(W,\mathbb Q/\mathbb Z) \stackrel{\beta}{\longrightarrow}
H^2(W,\mathbb Z) \rightarrow H^2(W,\mathbb Q) \rightarrow </M> 
<P/> from which we obtain the isomorphism <M>\beta \colon \tau H^1(W,\mathbb Q/\mathbb Z)
\stackrel{\cong}{\longrightarrow} \tau H^2(W,\mathbb Z)</M>. The linking form <M>Lk_W</M> can be defined as the composite 
<P/> <M>Lk_W\colon \tau H_1(W,\mathbb Z) \times \tau H_1(W,\mathbb Z) \stackrel{1\times \rho^{-1}}{\longrightarrow}
\tau H_1(W,\mathbb Z) \times \tau H^2(W,\mathbb Z)
\stackrel{1\times \beta^{-1}}{\longrightarrow}
\tau H_1(W,\mathbb Z) \times \tau H^1(W,\mathbb Q/\mathbb Z)
\stackrel{ev}{\longrightarrow } \mathbb Q/\mathbb Z </M>
<P/>where <M>ev(x,\alpha)</M> evaluates a <M>1</M>-cocycle <M>\alpha</M> on a <M>1</M>-cycle <M>x</M>.

<P/> The linking form can be used to define the set
<P/> <M>I^O(W) = \{Lk_W(g,g) \ \colon \ g\in \tau H_1(W,\mathbb Z)\}</M>
<P/>which is an oriented-homotopy invariant of <M>W</M>. Letting <M>W^+</M> and <M>W^-</M> denote the two possible orientations on the manifold, the set
<P/> <M>I(W) =\{I^O(W^+), I^O(W^-)\}</M>
<P/>is a homotopy invariant of <M>W</M> which in this manual we refer to as
 the <E>linking form homotopy invariant</E>.

<P/> The following commands compute the linking form homotopy invariant for the lens spaces <M>L(13,q)</M> with <M>1\le q\le 12</M>. This invariant distinguishes between the two homotopy types that arise. 

<Example>
<#Include SYSTEM "tutex/12.5.txt">
</Example>

</Section>

<Section><Heading>Determining the homeomorphism type of a lens space</Heading>

In 1935 K. Reidemeister <Cite Key="reidemeister"/> classified
 lens spaces up to orientation preserving PL-homeomorphism. This was generalized by E. Moise <Cite Key="moise"/> in 1952 to a classification up to homeomorphism -- his method requred the proof of the Hauptvermutung for <M>3</M>-dimensional manifolds. In 1960, following a suggestion of R. Fox, a proof was given
by
E.J. Brody <Cite Key="brody"/> that avoided the need for the Hauptvermutung.
Reidemeister's method, using what is know termed <E>Reidermeister torsion</E>,
and Brody's method, using  tubular neighbourhoods of <M>1</M>-cycles, both require identifying a suitable "preferred" generator of <M>H_1(L(p,q),\mathbb Z)</M>. In 2003 J. Przytycki and A. Yasukhara <Cite Key="przytycki"/> provided an alternative method for classifying lens spaces, which uses the linking form and
 again requires the identification of a "preferred" generator of <M>H_1(L(p,q),\mathbb Z)</M>.

<P/>   Przytycki and Yasukhara proved the following.

<P/>
<B>Theorem.</B> <E>Let <M>\rho\colon  S^ 3 \rightarrow L(p, q)</M> be the <M>p</M>-fold cyclic cover and <M>K</M> a knot in
<M>L(p, q)</M> that represents a generator of <M>H_1 (L(p, q), \mathbb Z)</M>. If  <M>\rho ^{-1} (K)</M> is the trivial knot, then
<M>Lk_{ L(p,q)} ([K], [K]) = q/p</M> or <M>= \overline q/p \in \mathbb Q/\mathbb Z</M> where <M>q\overline q \equiv 1 \bmod p</M>. </E>

<P/>The ingredients of this theorem can be applied in HAP, but at present only
 to small examples of lens spaces. 
The  obstruction to handling large examples is that the current default 
method for computing the linking form involves
 barycentric subdivision to produce a simplicial complex from a regular CW-complex, and then a homotopy equivalence from this typically large 
simplicial complex to a smaller  non-regular CW-complex. However, for homeomorphism invariants that are not homotopy invariants there is a need to avoid homotopy equivalences. In the current version of HAP this
means that in order to obtain delicate homeomorphism invariants we have to perform homology computations on typically large simplicial complexes. In a future version of HAP we hope to avoid the obstruction by  implementing cup products, cap products and linking forms entirely within the category of regular CW-complexes. 

<P/>The following commands construct a small lens space <M>L=L(p,q)</M> with unknown values of <M>p,q</M>. Subsequent commands will determine the homeomorphism type of <M>L</M>.
<Example>
<#Include SYSTEM "tutex/12.7.txt">
</Example>

We can readily determine the value of <M>p=11</M> by calculating the order of <M>\pi_1(L)</M>.

<Example>
<#Include SYSTEM "tutex/12.8.txt">
</Example>

<P/> The next commands take the default edge path <M>\gamma\colon S^1\rightarrow L</M> representing a generator of the cyclic group <M>\pi_1(L)</M> and 
lift it to an edge path <M>\tilde\gamma\colon S^1\rightarrow \tilde L</M>.

<Example>
<#Include SYSTEM "tutex/12.9.txt">
</Example>

<P/>The next commands check that the path <M>\tilde\gamma</M> is unknotted in <M>\tilde L\cong S^3</M> by checking that <M>\pi_1(\tilde L\setminus {\rm image}(\tilde\gamma))</M> is infinite cyclic.
<Example>
<#Include SYSTEM "tutex/12.10.txt">
</Example>

<P/>Since <M>\tilde\gamma</M> is unkotted the cycle <M>\gamma</M> represents the preferred generator <M>[\gamma]\in H_1(L,\mathbb Z)</M>. The next commands compute <M>Lk_L([\gamma],[\gamma])= 7/11 </M>.

<Example>
<#Include SYSTEM "tutex/12.11.txt">
</Example>

<P/> The classification of Moise/Brody states that <M>L(p,q)\cong L(p,q')</M> if and only if  <M>qq'\equiv \pm 1 \bmod p</M>. Hence the lens space <M>L</M> has the homeomorphism type
<P/> <M>L\cong L(11,7) \cong L(11,8) \cong L(11,4) \cong L(11,3)</M>.

</Section>

<Section><Heading>Surgeries on distinct knots can yield homeomorphic manifolds</Heading>

The lens space <M>L(5,1)</M> is a quotient of the <M>3</M>-sphere <M>S^3</M>
by a certain  action of the cyclic group <M>C_5</M>. It can be realized by a <M>p/q=5/1</M> Dehn filling of the complement of the trivial knot. It can also
be realized by Dehn fillings of other knots.  To see this, the following commands compute the manifold <M>W</M> obtained from a <M>p/q=1/5</M> Dehn filling of the complement of the trefoil and show that <M>W</M> at least has the same integral homology and same fundamental group as <M>L(5,1)</M>.

<Example>
<#Include SYSTEM "tutex/12.12.txt">
</Example>

<P/>The next commands construct the universal cover <M>\widetilde W</M> and show that it has the same homology as <M>S^3</M> and trivivial fundamental group
<M>\pi_1(\widetilde W)=0</M>.

<Example>
<#Include SYSTEM "tutex/12.13.txt">
</Example>

By construction the space <M>\widetilde W</M> is a manifold.
 Had we not known how the regular CW-complex  <M>\widetilde W</M> 
had been constructed then we could prove that it is a closed <M>3</M>-manifold by
creating its barycentric subdivision <M>K=sd\widetilde W</M>, which is 
homeomorphic to <M>\widetilde W</M>, and verifying that the link of each vertex in the simplicial complex <M>sd\widetilde W</M> is a <M>2</M>-sphere. 
The following commands carry out this proof: each link is shown to admit a discrete vector field with precisely two critical cells -- one <M>0</M>-cell and one <M>2</M>-cell.

<Example>
<#Include SYSTEM "tutex/12.14.txt">
</Example>



The Poincare conjecture (now proven) implies that <M>\widetilde W</M> is homeomorphic
to <M>S^3</M>.
 Hence <M>W=S^3/C_5</M> is a quotient of the <M>3</M>-sphere  by an action of <M>C_5</M> and is hence a lens space <M>L(5,q)</M> for some <M>q</M>.

<P/> The next commands determine that <M>W</M> is homeomorphic to
<M>L(5,4)\cong L(5,1)</M>.

<Example>
<#Include SYSTEM "tutex/12.15.txt">
</Example>

<P/> Moser <Cite Key="lmoser"/> gives a precise decription of the lens 
spaces arising from surgery on the trefoil knot and more generally from 
surgery on torus knots. Greene <Cite Key="greene"/> determines the lens spaces that arise by integer Dehn surgery along a knot in the three-sphere
</Section>

<Section><Heading>Finite fundamental groups of <M>3</M>-manifolds</Heading>

Lens spaces are examples of <M>3</M>-manifolds with finite fundamental groups. The complete list of finite groups <M>G</M>
arising as fundamental groups of closed connected
<M>3</M>-manifolds is recalled in <Ref Sect="Secfinitefundman"/> where one method for computing their cohomology rings is presented. Their cohomology could also be computed from explicit <M>3</M>-manifolds <M>W</M> with <M>\pi_1W=G</M>.
 For instance, the following commands realize a closed connected
<M>3</M>-manifold <M>W</M> with <M>\pi_1W = C_{11}\times SL_2(\mathbb Z_5)</M>.

<Example>
<#Include SYSTEM "tutex/12.16.txt">
</Example>

Hence the group <M>G=C_{11}\times SL_2(\mathbb Z_5)</M> of order <M>1320</M>
acts freely on the <M>3</M>-sphere
<M>\widetilde W</M>. It thus  has periodic cohomology with
<Display>
H_n(G,\mathbb Z) = \left\{ \begin{array}{ll}
\mathbb Z_{11} &amp; n\equiv 1  \bmod 4  \\
0  &amp; n\equiv 2 \bmod 4 \\
\mathbb Z_{1320} &amp; n \equiv 3\bmod 4\\
\mathbb 0 &amp; n\equiv 0  \bmod 4   \\
\end{array}\right.
</Display>
for <M>n &gt; 0</M>.

</Section>
</Chapter>
