3Four major tools of (co)homology
III Algebraic Topology
3.3 Relative homology
The next result again gives us another exact sequence. This time, the exact
sequence involves another quantity known as relative homology.
Definition
(Relative homology)
.
Let
A ⊆ X
. We write
i
:
A → X
for the
inclusion map. Then the map
i
n
:
C
n
(
A
)
→ C
n
(
X
) is injective as well, and we
write
C
n
(X, A) =
C
n
(X)
C
n
(A)
.
The differential
d
n
:
C
n
(
X
)
→ C
n−1
(
X
) restricts to a map
C
n
(
A
)
→ C
n−1
(
A
),
and thus gives a well-defined differential d
n
: C
n
(X, A) → C
n−1
(X, A), sending
[c] 7→ [d
n
(c)]. The relative homology is given by
H
n
(X, A) = H
n
(C
·
(X, A)).
We think of this as chains in
X
where we ignore everything that happens in
A.
Theorem
(Exact sequence for relative homology)
.
There are homomorphisms
∂ : H
n
(X, A) → H
n−1
(A) given by mapping
[c]
7→ [d
n
c].
This makes sense because if
c ∈ C
n
(
X
), then [
c
]
∈ C
n
(
X
)
/C
n
(
A
). We know
[
d
n
c
] = 0
∈ C
n−1
(
X
)
/C
n−1
(
A
). So
d
n
c ∈ C
n−1
(
A
). So this notation makes
sense.
Moreover, there is a long exact sequence
· · · H
n
(A) H
n
(X) H
n
(X, A)
H
n−1
(A) H
n−1
(X) H
n−1
(X, A) · · ·
· · · H
0
(X) H
0
(X, A) 0
∂
i
∗
q
∗
∂
i
∗
q
∗
q
∗
,
where
i
∗
is induced by
i
:
C
·
(
A
)
→ C
·
(
X
) and
q
∗
is induced by the quotient
q : C
·
(X) → C
·
(X, A).
This, again, is natural. To specify the naturality condition, we need the
following definition:
Definition
(Map of pairs)
.
Let (
X, A
) and (
Y, B
) be topological spaces with
A ⊆ X and B ⊆ Y . A map of pairs is a map f : X → Y such that f(A) ⊆ B.
Such a map induces a map
f
∗
:
H
n
(
X, A
)
→ H
n
(
Y, B
), and the exact sequence
for relative homology is natural for such maps.