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.