0.5. Higher homotopy groups 9
0.4. The Vietoris-Begle Theorem
Another algebraic result needed much later on is the Vietoris-Begle Map-
ping Theorem. It implies that a closed, surjective mapping for which all
point preimages are cohomologically acyclic induces isomorphisms of
Cechˇ
or Alexander cohomology. See (Spanier, 1966, p. 344) for the proof.
Theorem 0.4.1. Let f : X X be a closed surjective map between para-
compact Hausdorff spaces, and let G be an abelian group. Assume that there
is some m 0 such that
ˇ
H
q(f −1(x);
G) = 0 for all x X and q m. Then
f

:
ˇ
H
q(X;
G)
ˇ
H
q(X
; G)
is an isomorphism for q m and a monomorphism for q = m.
0.5. Higher homotopy groups
Let X be a space with basepoint x0 X. For each integer n 0 there is
a group πn(X, x0), called the nth homotopy group of X with basepoint x0.
The definition of πn(X, x0) closely parallels that of the fundamental group,
only with the role of the unit interval taken over by the unit n-cube. For
that reason, throughout this section
In
will be used to denote [0,
1]n.
Let F
n
denote the set of all maps
(In,∂In)
(X, x0). Define two maps
in F
n
to be equivalent if they are homotopic relative to
∂In;
i.e., f, f F
n
are equivalent if there exists a homotopy Ψt :
In
X between f and f
with
Ψt(∂In)
= {x0} for all t I. As a set, πn(X, x0) is just the set of
equivalence classes.
Define the group operation on F
n,
n 1, as follows: for f, g F
n
their
sum f + g F
n
is the function determined by
(f + g)(t1,t2,...,tn) =
f(2t1,t2,...,tn) if 0 t1 1/2
g(2t1 1,t2,...,tn) if 1/2 t1 1.
Since the homotopy class [f + g] depends only on the classes [f] and [g],
addition in F
n
induces a well-defined addition in πn(X, x0) via [f] + [g] =
[f + g]. One can easily verify that this makes πn(X, x0) a group; its identity
element is the class of the (only possible) constant map.
The 0-dimensional cube
I0
is just a point and
∂I0
= ∅. With this
understanding the definition of the set π0(X, x0) makes sense (it consists of
the set of path components of X), but there is no group structure defined
on it. Even though π0(X, x0) does not have a group structure, we still
designate the equivalence class of the map
I0
x0 as the identity element.
This allows us to include π0(X, x0) in an important exact sequence to be
described below.
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