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Note that L(1,q) is the sphere S 3 , but this case has to be treated in a slightly
special way if one wants a nice cell decomposition. One has to add another vertex
at - e 1 .
Lens spaces were first defined by Tietze ([Tiet08]). We list some of their proper-
ties below:
(1) They are closed compact three-dimensional manifolds.
(2) They can be triangulated.
(3) L(2,1) is homeomorphic to P 3 .
(4) The homology groups of L(p,q) are
Z
Z
i
i
i
i
=
=
=
=
0
1
Ï
¸
Ô
Ô
p
(
(
)
) ª
HLpq
,
.
Ì
˝
i
0
2
3
Ô
Ô
Ô
Ô
Ó
Z
˛
(5) c(L(p,q)) = 0.
(6) L(p,q) is homeomorphic to L(p,q¢) if and only if
±1
(
)
qq
¢∫±
mod
p
.
(7) L(p,q) and L(p,q¢) have the same homotopy type if and only if qq¢ or -qq¢ is a
quadratic residue modulo p.
Properties (1)-(3) are easy to prove. Figure 7.14(b) should make clear what one
needs to do for (2). For properties (4) and (5) use the cell decomposition of L(p,q)
induced by the cell decomposition of D 3 described above to compute its homology
groups. Property (6) was proved by Reidemeister ([Reid35]). Property (7) was proved
by Whitehead ([Whit41]). More details about lens spaces can also be found in [SeiT80]
and [HilW60].
Although it is obvious that homotopy equivalence is a weaker relation than home-
omorphism (consider a disk and a point or, more generally, any deformation retract),
it is not so obvious with respect to some types of spaces like manifolds without bound-
ary. This makes the next example and lens spaces all the more interesting.
7.2.4.7. Example. A consequence of properties (6) and (7) is that the 3-manifolds
L(7,1) and L(7,2) have the same homotopy type but are not homeomorphic.
7.2.5
Incidence Matrices
This section discusses the so-called incidence matrices. These matrices played an
important role in the history of combinatorial topology. An excellent detailed account
of these matrices can be found in [Cair68]. Computers can easily use them to compute
homology groups.
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