0.1 The Hurewicz Theorem
For a pointed space X, : π → H is the Hurewicz map, sending a homotopy-class
of maps [f : S
n
→ X] to the cycle [f] ∈ H
n
(X).
Theorem 0.1. Let X be a pointed, n-connected space, with n > 0.
1. The Hurewicz map π
0
(X) → H
0
(X) is a free Abelian group on π
0
(X).
2. If X is n-connected, with n > 1, then the Hurewicz map is an isomorphism
in degree n + 1 and an epimorphism in degree n + 2.
3. The Hurewicz map π
1
(X) →
˜
H
1
(X) is the Abelianisation of π
1
(X).
Proof. 1. This is clear. H
0
(X) is the free Abelian group on the points of X
modulo the relation [x ∼ y if and only ∃e ∈ X
1
such that ∂(e) = x − y],
so choosing a single point from each component of X, the claim follows.
2. We may assume that X = X
∼
n
, so that sk
n
(X) = pt . Then every n-cell
of X is an n-sphere, and H
n
(X) = Z⟨hom(S
n
, X)⟩/∂
∗
Z⟨hom(D
n+1
, X)⟩.
So H
n
is generated by non-contractible homotopy classes of maps, and
n
is surjective.
On the other hand, a homotopy class of maps f : S
n
→ X maps to
zero if in H
n
(X) and only if f is homotopic to a wedge sum of n-spheres
f ∼ ∨
i
f
i
, such that each f
i
is contractible. If n > 1, then π
n
is abelian,
so the natural map ⨿
i
S
n
→ ∨
i
S
n
induces an isomorphism of groups
[∨
i
S
n
, X]
∼
=
⊕
i
π
n
(X)
1
The additivity of the Hurewicz map then proves that [f] = 0 =⇒ [f] = 0
⊕[f
i
] Σ[f
i
] = 0
⊕
i
π
n
(X) ⊕
i
H
n
(X)
π
n
(X) H
n
(X)
[f] [f] = 0
+ +
3. To study : π
1
(X) → H(X, pt .), we may assume that X is connected
and 0-reduced X
0
= pt .. Then the same argument as above proves that
π
1
(X
)
→ H
1
(X, pt .) is surjective. A class of maps f : S → X has [f] = 0
if and only if [f] factors as a product [f
1
]...[f
p
] such that for some permu-
tation σ ∈ Σ
p
, [f
σ(1)
]...[f
σ(p)
] = pt. The kernel of this relation is a normal
subgroup contained in the kernel of any map π
1
(X) → A, where A is an
Abelian group. The result follows.
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