Graphics Reference
In-Depth Information
7.2.6.1. Theorem.
Let K be a simplicial complex. Then
dim
K
Â
q
()
=
()
()
c
K
1
k
K
.
q
q=0
Proof.
The proof of this theorem is the same as that of Theorem 7.2.3.10. The only
difference is that one uses the mod 2 groups now and works with the dimensions of
these vector spaces rather than the ranks of the corresponding groups in the usual
homology theory with integer coefficients. One also needs to use the fact that
(
)
=
()
dim
CK
;
Z
2
nK
.
q
q
7.2.6.2. Corollary.
If
X
is a polyhedron, then
dim
X
Â
1
q
()
=
()
()
c
X
k
X
.
q
q=0
Connectivity numbers have a simple geometric interpretation in the special case
of surfaces.
7.2.6.3. Lemma.
Let
S
be a closed and compact combinatorial surface. Then
(1) k
0
(
S
) = 1.
(2) k
2
(
S
) = 1.
(3) k
1
(
S
) = 2 -c(
S
).
Proof.
Exercise 7.2.6.1(b) proves (1). Next, it is easy to see that H
2
(
S
;
Z
2
) ª
Z
2
because
the sum of all the 2-simplices of
S
is a mod 2 2-cycle that generates H
2
(
S
;
Z
2
). (We
should point out that this fact is actually a special case of Theorem 7.5.1(1) in Section
7.5.) This proves (2). Parts (1), (2), and Corollary 7.2.6.2 now imply (3).
Lemma 7.2.6.3 shows that the first connectivity number k
1
(
S
) (which is the only
one that is interesting for surfaces) does not depend on the orientability of the surface
S
. This was certainly not the case with the first Betti number b
1
(
S
) and more evidence
that mod 2 homology theory does not detect orientability properties of spaces. We
shall also see this later with respect to the top dimensional homology groups of
pseudomanifolds. The next theorem is the main result that we are after right now.
7.2.6.4. Theorem.
The first connectivity number k
1
(
S
) of a combinatorial surface
S
equals the maximum number of distinct, but not necessarily disjoint, simple closed
curves in
S
along which one can cut and still have a connected set left over.
Outline of Proof.
First of all, we need to explain the expression “simple closed
curves” in our current context. A collection of subsets
X
i
of
S
will be called a collec-
tion of
simple closed curves
in
S
if each
X
i
is homeomorphic to the circle
S
1
and there
