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called the
Hurewicz homomorphisms
, which generalize the homomorphism for the
fundamental group. (We are again pretending that H
n
(
X
) is well defined.) These
homomorphisms are neither onto nor one-to-one in general though. The homotopy
groups capture the idea of “holes” better than the homology groups. After all, the
n-sphere
S
n
is the prototype of an n-dimensional “hole.”
Nontrivial “higher” homotopy groups of spheres are one important example of
what sets homotopy groups apart from homology groups. The homology groups H
i
(
X
)
are all
0
if i is larger than the dimension of
X
, but this is not necessarily the case for
homotopy groups. For example,
()
ª
2
SZ
p
3
.
One well-known theorem that relates the homotopy and homology groups in a
special case is
7.4.3.4. Theorem.
(The Hurewicz Isomorphism Theorem) If n ≥ 2 and if
X
is a con-
nected polyhedron whose first n - 1 homotopy groups vanish, then the Hurewicz
homorphism
(
)
Æ
()
mp
:
Xx
,
H
X
n
0
n
is an isomorphism.
Proof.
See [Span66].
Theorem 7.4.3.4 is one result that can be used to compute higher homotopy
groups.
7.4.3.5. Theorem.
Let n ≥ 1.
(1) p
i
(
S
n
) = 0 for 0 £ i < n.
(2) p
n
(
S
n
) ª
Z
.
Proof.
To prove (1) consider a map f :
S
i
Æ
S
n
. The map f is homotopic to a map
that misses a point, say
e
n+1
. (To prove this fact, use the simplicial approximation
theorem with respect to some triangulations of the spheres.) But
S
n
-
e
n+1
is home-
orphic to an open disk that is contractible. It follows that f is homotopic to a constant
map and proves (1). The case n = 1 in part (2) is just Corollary 7.4.2.23(1). If n > 1,
then (2) follows from (1), Theorem 7.4.3.4, and Theorem 7.2.3.4.
It should be noted that Theorem 7.4.3.4 implies nothing about the homomor-
phisms
(
)
Æ
()
mp
:
Xx
,
H
X
i
0
i
for i > n (n as in the theorem). In general, homotopy groups are much harder to
compute than homology groups but they are stronger invariants than homology
groups. As an example of the latter, there is the following theorem:
