Graphics Reference
In-Depth Information
is a triangulation (K,j) of
S
and subcomplexes L
i
of K such that j(ΩL
i
Ω) =
X
i
. We shall
use this notation in the argument below.
Define a number d(
S
) by
()
=
{
(
)
d
S
max
k
S
-
X
»
X
»
. . .
»
X
is connected
,
1
2
k
}
where
the
X
are distinct simple closed curves in
S
.
i
We need to prove that k
1
(
S
) =d(
S
). It is easy to see from the normal form for a surface
that was given in Chapter 6 and Lemma 7.2.6.3(3) that
()
≥
()
=-
()
d
S
k
S
2
c
S
.
(7.6)
1
Conversely, let d(
S
) = k and assume that k >k
1
(
S
). Let S
i
be the generator of
H
1
(L
i
;
Z
2
) = Z
1
(L
i
;
Z
2
) Õ Z
1
(K,
Z
2
), that is, S
i
is the sum of the 1-simplices in L
i
. Since
k
1
(
S
) is the dimension of the vector space H
1
(K;
Z
2
) and k >k
1
(
S
), the 1-cycles S
i
deter-
mine a linearly dependent set of elements in H
1
(K;
Z
2
). Therefore, there must be a 2-
chain c ΠC
2
(K;
Z
2
), so that
()
=
∂
2
ca
SS
+
a
+◊◊◊+
kk
S,
1
1
2
2
where a
i
Π{0,1} and not all a
i
are zero. The chain c cannnot be the sum of all the 2-
simplices of K, because it would be easy to check that ∂
2
(c) = 0 in that case. Since we
are assuming that ∂
2
(c) π 0, at least one 2-simplex s of K does not belong to c. Using
this fact one shows that
(
)
S
-
XX
»
»◊◊◊»
X
1
2
k
is not connected. This contradicts our initial hypothesis and proves that k >k
1
(
S
) is
impossible, and so
()
£
()
d
S
k
S
.
(7.7)
1
Inequalities (7.6) and (7.7) prove the theorem.
Finally, the mod 2 homology groups can be computed using “mod 2” incidence
matrices.
7.3
Cohomology Groups
In addition to homology groups there are also cohomology groups associated to a
space. These groups are a kind of dual of the homology groups. We shall run into
them in Section 7.5.2 when we discuss the Poincaré duality theorem for manifolds.
They provide a formal setting that makes proving facts about homology easier, even
though they are closely related to the latter and add nothing new as far as the
group
structures are concerned. However, it is possible to define a natural
ring
structure for
