Graphics Reference
In-Depth Information
7.4.3.6. Theorem.
A continuous map f :
X
Æ
Y
between CW complexes is a homo-
topy equivalence if and only if it induces isomorphisms on all (n ≥ 0) homotopy
groups.
Proof.
See [LunW69].
The best that one can do for homology is
7.4.3.7. Theorem.
Let
X
and
Y
be simply connected CW complexes. A continuous
map f :
X
Æ
Y
is a homotopy equivalence if and only it induces isomorphisms on all
homology groups.
Proof.
See [Span66].
Theorem 7.4.3.7 is false if
X
and
Y
are not simply connected. For a counter-
example, see [Span66], page 420. There is an analog of the theorem when spaces are
not simply connected, but things get much more involved. It is based on the notion
of a
simple homotopy equivalence
. See [Dieu89].
We finish with one last application.
Definition.
A topological space that has the homotopy type of the n-sphere
S
n
is
called a
homotopy n-sphere
. A polyhedron is called a
homology n-sphere
if it has the
same homology groups as
S
n
.
7.4.3.8. Theorem.
Every simply connected homology n-sphere, n ≥ 2, is a homo-
topy n-sphere.
By Theorems 7.4.3.4 we get a map between the
S
n
Proof.
and the space that is an
isomorphism on homology. Now use Theorem 7.4.3.7.
Theorem 7.4.3.8 is also false if we drop the simply connected hypothesis. In
[SeiT80] one can find an example of a three-dimensional space, called a
Poincaré
space
, that is a homology 3-sphere but that has a nontrivial fundamental group and
hence cannot be of the same homotopy type as
S
3
(nor homeomorphic to it). The nice
thing about Theorems 7.4.3.7 and 7.4.3.8 is that, in order to prove something about
homotopy type, we do not have to mess around with complicated homotopy groups
but can simply work with homology groups, which is a much easier task. We do have
to check that spaces are simply connected though.
7.5
Pseudomanifolds
This section specializes to manifold-like spaces. We shall define what it means for
them to be orientable and relate this concept to homology. Some applications of this
can be found in the next section.
Definition.
A polyhedron
X
is called an
n-dimensional pseudomanifold
or
n-
pseudomanifold
or simply
pseudomanifold
if it admits a triangulation (K,j) satisfying
