3Four major tools of (co)homology
III Algebraic Topology
3.1 Homotopy invariance
The first result is pretty easy to state, but it is a really powerful result that
really drives what we are doing:
Theorem
(Homotopy invariance theorem)
.
Let
f ' g
:
X → Y
be homotopic
maps. Then they induce the same maps on (co)homology, i.e.
f
∗
= g
∗
: H
·
(X) → H
·
(Y )
and
f
∗
= g
∗
: H
·
(Y ) → H
·
(X).
Corollary.
If
f
:
X → Y
is a homotopy equivalence, then
f
∗
:
H
·
(
X
)
→ H
·
(
Y
)
and f
∗
: H
·
(Y ) → H
·
(X) are isomorphisms.
Proof. If g : Y → X is a homotopy inverse, then
g
∗
◦ f
∗
= (g ◦ f)
∗
= (id
X
)
∗
= id
H
·
(X)
.
Similarly, we have
f
∗
◦ g
∗
= (
id
Y
)
∗
=
id
H
·
(Y )
. So
f
∗
is an isomorphism with an
inverse g
∗
.
The case for cohomology is similar.
Since we know that
R
n
is homotopy equivalent to a point, it immediately
follows that:
Example. We have
H
n
(R
n
) = H
n
(R
n
) =
(
Z n = 0
0 n > 0
.