3Four major tools of (co)homology
III Algebraic Topology
3.4 Excision theorem
Now the previous result is absolutely useless, because we just introduced a new
quantity
H
n
(
X, A
) we have no idea how to compute again. The main point of
relative homology is that we want to think of
H
n
(
X, A
) as the homology of
X
when we ignore
A
. Thus, one might expect that the relative homology does
not depend on the things “inside
A
”. However, it is not true in general that,
say,
H
n
(
X, A
) =
H
n
(
X \ A
). Instead, what we are allowed to do is to remove
subspaces of A that are “not too big”. This is given by excision:
Theorem
(Excision theorem)
.
Let (
X, A
) be a pair of spaces, and
Z ⊆ A
be
such that Z ⊆
˚
A (the closure is taken in X). Then the map
H
n
(X \ Z, A \ Z) → H
n
(X, A)
is an isomorphism.
X
A
Z
While we’ve only been talking about homology, everything so far holds anal-
ogously for cohomology too. It is again homotopy invariant, and there is a
Mayer-Vietoris sequence (with maps
∂
MV
:
H
n
(
A ∩ B
)
→ H
n+1
(
X
)). The rela-
tive cohomology is defined by
C
·
(
X, A
) =
Hom
(
C
·
(
X, A
)
, Z
) and so
H
∗
(
X, A
)
is the cohomology of that. Similarly, excision holds.