Download THE HIGHER HOMOTOPY GROUPS 1. Definitions Let I = [0,1] be

THE HIGHER HOMOTOPY GROUPS
1. Definitions
Let I = [0, 1] be the closed unit interval and let I n = [0, 1] × ... × [, 1] be the product
of n copies of I. Notice that the boundary ∂I n is the set
∂I n = {(t1 , ..., tn ) ∈ I n | some ti is either 0 or 1 }
Given a topological space X, as in the case of the fundamental group, we shall pick
an arbitrary point p ∈ X (which once picked should be fixed) and refer to it as the
base point. Let Sn (X, p) (or simply Sn for short) be the set
Sn = {α : I n → X | α is continuous and α(t1 , ..., tn ) = p,
∀(t1 , ..., tn ) ∈ ∂I n }
Thus, Sn are all the continous functions from I n into X which map all of ∂I n to the
basepoint p. Observe that when n = 1 then ∂I = {0, 1} and so the set S1 agrees with
the loop space at the point p ∈ X.
We define an operation which takes two elements from Sn and produces a new
element in Sn : given α, β ∈ Sn we define α · β as

£ ¤
t1 ∈ 0, 12
 α(2t1 , t2 , t3 , ..., tn )
(α · β)(t1 , ..., tn ) =
£ ¤

β(2t1 − 1, t2 , t3 , ..., tn )
t1 ∈ 12 , 1
It is easy to check that α · β is continuous: when t1 = 1/2 then α(1, t2 , ..., tn ) = p
and β(0, t2 , ..., tn ) = p since (1, t2 , ..., tn ), (0, t2 , .., tn ) ∈ ∂I n . On the other hand, if
(t1 , ..., tn ) ∈ ∂I n then the points (2t1 , t2 , ..., tn ) and (2t1 − 2, t2 , ..., tn ) are also in ∂I n
and so (α · β)(t1 , ..., tn ) = p for any (t1 , ..., tn ) ∈ ∂I n . This shows that α · β ∈ Sn .
To make Sn with this operation into a group, we need to pass to homotopy equivalence.
Definition 1.1. We say that two maps α, β ∈ Sn are homotopic if they are homotopic
F
relative ∂I n . We will write α ' β rel ∂I n if there exists a map F : I n × I → X such
that
F (t1 , ..., tn , 0) = α(t1 , ..., tn )
F (t1 , ..., tn , 1) = β(t1 , ..., tn )
F (t1 , ..., tn , s) = p ∀s ∈ I and (t1 , ..., tn ) ∈ ∂I n
We already know that relative homotopy equivalence is an equivalence relation. Let
[α] denote the equivalence class of α ∈ Sn . Let πn (X, p) be the set
πn (X, p) = {[α] | α ∈ Sn (X, p)}
We shall equip πn (X, p) with the operation
(1)
[α] · [β] = [α · β]
1
2
THE HIGHER HOMOTOPY GROUPS
Lemma 1.2. The operation defined by (2.2) on πn (X, P ) is well defined. That is,
given α1 , α2 ∈ [α] and β1 , β2 ∈ [β], there is a homotopy F between
F
α1 · β1 ' α2 · β2 rel ∂I n
.
Proof. Let G and H be the homotopies between the two α’s and β’s:
G
α1 ' α2 rel ∂I n
H
β1 ' β2 rel ∂I n
Then F is the homotopy given by the formula

£ ¤
t1 ∈ 0, 21
 G(2t1 , t2 , ..., tn , s)
F (t1 , ..., tn , s) =
£ ¤

H(2t1 − 1, t2 , ..., tn , s) t1 ∈ 12 , 1
¤
Theorem 1.3. The set πn (X, p) equipped with the operation (2.2) is a group for any
n ≥ 1. The unit element is the homotopy class [ep ] of the constant map ep : I n → X
and the inverse of [α] is the homotopy class [α−1 ] where α−1 : I n → X is the map
(2)
α−1 (t1 , t2 , ...tn ) = α(1 − t1 , t2 , ..., tn )
For n ≥ 2 these groups are referred to as the higher homotopy groups.
Proof. As in the case of the fundamental group, one needs to check the three group
axioms, each of which comes down to finding homotopies with certain properties. We
shall only focus on last group axiom - the existence of an inverse element. The verification of the other two axioms is left as an exercise.
F
Thus, we need to check that α · α−1 ' ep rel ∂I n with α−1 as defined in (2). Such a
homotopy is easy to find, namely

p








 α(2t1 − s, t2 , ..., tn )
F (t1 , , , .tn , s) =


α−1 (2t1 + s − 1, t2 , ..., tn )







p
£
¤
t1 ∈ 0, 21 s
t1 ∈
t1 ∈
£1
s, 12
2
¤
£1
¤
1
,
1
−
s
2
2
£
¤
t1 ∈ 1 − 21 s, 1
You should convince yourself that F is indeed a homotopy with the needed properties.
¤
Remark: The homotopies that come up in the proof of theorem 1.3 are in fact the
same homotopies we encountered in the proof for the fundamental group. You should
notice that in the homotopy F above, not much happens in the coordinates t2 , ..., tn ,
all the action is in t1 . Indeed, F is simply the homotopy corresponding to the picture
in figure 1 with the horizontal direction representing the coordinate t1 and the vertical
direction representing the coordinate s.
THE HIGHER HOMOTOPY GROUPS
3
Figure 1. The homotopy F between α · α−1 and the constant map ep .
2. Properties
Unlike the fundamental group π1 (X, p), the higher homotopy groups πn (X, p) are
always Abelian groups.
Theorem 2.1. For n ≥ 2, the group πn (X, p) is Abelian.
Proof. Given any two [α], [β] ∈ πn (X, p) we need to exhibit that there is a relative
homotopy F between
F
α · β ' β · α rel ∂I n
This homotopy can be taken to be of the form F (t1 , t2 , t3 , .., tn , s) = (G(t1 , t2 , s), t3 , ..., tn )
where G : I 2 × I → I 2 is the homotopy modeled on the sequence of pictures in figure
2.
¤
Figure 2. Each of the squares above represents I 2 with the horizontal
direction corresponding to t1 and the vertical direction to t2 . These pictures represent 5 snapshots (think of the parameter s as being “time”)
of the homotopy G. As s goes from 0 to 1, the homotopy G interchanges
the left and right halves of I 2 by first shrinking both (second picture),
rotating them about each other (pictures 3 and 4) and finally expanding
each to their original size (picture 5). The black and white regions correspond to the maps α and β respectively while all of the grey areas (in
pictures 2,3 and 4) get mapped to the basepoint p.
Theorem 2.2. The groups πn (X, p) × πn (Y, q) and πn (X × Y, (p, q)) are isomorphic.
Proof. Consider the function Φ : πn (X, p) × πn (Y, q) → πn (X × Y, (p, q)) defined by
Φ([α], [β]) = [(α, β)]
It is easy to check that Φ is a homomorphism and a bijection, it is left as an exercise. ¤
As was the case with the fundamental group, the higher homotopy groups are independent of the particular choice of basepoint. Namely, given two points p, q ∈ X and
given a path σ : I → X with σ(0) = p and σ(1) = q, there is again a homomorphism
σ# : πn (X, q) → πn (X, p) with σ# ([α]) being the homotopy class of the map σ(α) which
4
THE HIGHER HOMOTOPY GROUPS
Figure 3. The square above represents I n . The map σ(α) : I n → X is
constructed by using the map α on the smaller (shaded) square and using
the path σ along the radial lines connecting the boundary of the outer
square to the boundary of the inner (shaded) square. Convince yourself
that this construction in the case of n = 1 agrees with the construction
we used in the context of π1 .
is defined (and explained) in figure 3. It is not hard to see that σ# is a homomorphism
(see figure 4) whose inverse is the homomorphism (σ −1 )# (where σ−1 is the inverse
path of σ).
Exercise: Give a homotopy between σ −1 (σ(α)) and α.
Thus, as in the case of the fundamental group, σ# is an isomorphism between πn (X, p)
and πn (X, q). Given this observation, if X is path-connected we will simply write
πn (X) to denote πn (X, p) for some p ∈ X.
Figure 4. The two pictures above represent σ# (α)·σ# (β) and σ# (α·β).
The map α is represented by the grey region and β by the black region.
There is an obvious homotopy between them and thus σ# ([α])·σ# ([β]) =
σ# ([α] · [β]).
Any map f : X → Y induces a homomorphism f# : πn (X, p) → πn (Y, f (p)) defined
as f# ([α]) = [f ◦ α].
Theorem 2.3. Let f : X → Y and g : Y → Z be two maps. Then
1. (idX )# = idπn (X)
2. (g ◦ f )# = g# ◦ f#
THE HIGHER HOMOTOPY GROUPS
5
The first part of the theorem is obvious while the second part is proved in much the
same way as the analogous theorem was proved in the case of the fundamental group
and is omitted. An important corollary of the above theorem is
Corollary 2.4. If f : X → Y is a homeomorphism then f# : πn (X, p) → πn (Y, f (p))
is an isomorphism. In particular, if X and Y are two path-connected, homeomorphic
spaces then πn (X) ∼
= πn (Y ) for all n ≥ 1.
Thus the higher homotopy groups are topological invariants. As in the case of the
fundamental group, they too only depend on homotopy type of a space rather than on
its homeomorphism type:
Theorem 2.5. If X and Y are two path-connected, homotopy equivalent spaces then
πn (X) ∼
= πn (Y ) for all n ≥ 1.
Theorem 2.6. If X and Y are path connected and homotopy equivalent then πn (X) ∼
=
πn (Y ) for all n ≥ 1.
3. Examples and applications
The higher homotopy groups are harder to calculate than the fundamental group.
There is no analogue of the Seifert-VanKampem theorem for πn when n is greater
or equal to 2. One calculational tool for the higher homotopy groups is described in
section 4. Nonetheless, here are some examples with some partial proofs.
1. πn (Rm ) ∼
= 0 for any n, m ∈ N. This follows from the fact that Rm is homotopy
equivalent to a point (and clearly a point has trivial homotopy groups).
2. πn (S n+k ) ∼
= 0 for all k ≥ 1. To prove this, we shall rely on a theorem whose proof
falls into the realm of differential topology:
Theorem 3.1. Suppose k ≥ 1 and let α : I n → S n+k be a map with α(∂I n ) = p
(where p is some chosen basepoint in S n+k ). Then there exists a non-surjective map
β : I n → S n+k which is homotopic to α rel ∂I n .
The upshot of the theorem is that any homotopy class [α] ∈ πn (S n+k ) has a nonsurjective representative β : I n → S n+k . Let q ∈ S n+k be any point not in the image
of β. Then S n+k − {q} is homeomorphic to Dn+k and can be contracted to p. Thus,
any non-surjective map is null-homotopic and so πn (S n+k ) ∼
= 0.
3. πn (S n ) ∼
= Z. The outline of a proof of this fact is the subject of section 4. An
immediate and very important consequence is
Corollary 3.2. If Rn ∼
= Rm then n = m.
Proof. Suppose that n < m and that f : Rn → Rm is a homeomorphism. Then
f |Rn −{0} : Rn − {0} −→ Rm − {f (0)}
is also a homeomorphism and thus induces an isomorphism between πn (Rn − {0}) and
πn (Rm − {f (0)}). This however leads to a contradiction since Rn − {0} ' S n and
Rm − {f (0)} ' S m while πn (S n ) ∼
= 0. The assumption n > m also
= Z and πn (S m ) ∼
leads to a contradiction and so the only leftover possibility is n = m.
¤
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THE HIGHER HOMOTOPY GROUPS
4. πn (S 1 ) ∼
= 0 for n ≥ 2. This follows from the theory of covering spaces.
5. Unlike in the case of S 1 , the homotopy groups πn+k (S n ) for n ≥ 2 do not generally
vanish. Not all of them are known and calculating them in general is a difficult task.
Here are some samples results:
π3 (S 2 ) ∼
= Z π7 (S 4 ) ∼
= Z × Z12
π10 (S 4 ) ∼
= Z24 × Z3
π12 (S 5 ) ∼
= Z30
4. Higher homotopy groups and suspensions
Definition 4.1. Let X be some topological space. We shall define a new topological space ΣX, called the suspension of X as the identification space of X × [−1, 1]
(equipped with the product topology) associated to the partition P:
P = {(x, t), X × {−1}, X × {1} | x ∈ X, t ∈ h−1, 1i}
Thus ΣX is the space X × [−1, 1] with X × {−1} and X × {1} each collapsed to a
point. The space X itself is contained in ΣX as X × {0}.
Examples: 1. ΣS n ∼
= S n+1 . This is easily seen in the case of n = 1 (see figure 5)
while in the case of n ≥ 2 one can give an explicit homeomorphism between ΣS n and
S n+1 .
Figure 5. The suspension of S 1 is homeomorphic to S 2 .
2. ΣI n ∼
= I n+1 and ∂(ΣI n ) ∼
= Σ(∂I n ). Prove this!
In order to use suspensions to analyze the higher homotopy groups, we shall use an
equivalent but slightly different description of πn (X, p). Namely, rather than thinking
of elements of πn (X, p) as homotopy classes of maps α : I n → X with α(∂I n ) = p, we
shall regard πn (X, p) as homotopy classes of maps α : S n → X with α(P ) = p where
P ∈ S n is a marked (but otherwise arbitrary) point in S n . The connection between
the two descriptions is of course that the identification space I n /∂I n = S n and P ∈ S n
is the image of ∂I n under the quotient map. The entire formalism developed thus
far works equally well in this new setting. With this understood, we now proceed as
follows:
Any map f : X → Y induces a map Σf : ΣX → ΣY given by
Σf (x, t) = (f (x), t)
THE HIGHER HOMOTOPY GROUPS
7
In particular, given a map α : S n → X with α(P ) = p, the map Σα : S n+1 → ΣX
sends (P, 0) to (p, 0) ∈ ΣX. This observation gives rise to a function φn : πn (X, p) →
πn+1 (ΣX, (p, 0)) defined as φn ([α]) = [Σα].
Lemma 4.2. The function φn : πn (X, p) → πn+1 (ΣX, (p, 0)) defined above is a group
homomorphism and is called the suspension homomorphism.
The importance of the suspension homomorphism lies in the following theorem.
Theorem 4.3. The suspension homomorphism φi : πi (S n ) → πi+1 (S n+1 ) is an isomorphism for i < 2n − 1 and a surjection for i = 2n − 1.
Corollary 4.4. The suspension map φn : πn (S n ) → πn+1 (S n+1 ) is an isomorphism for
n ≥ 2.
Thus knowing that π2 (S 2 ) ∼
= Z we get from the above corollary that πn (S n ) ∼
= Z
2
for all n ≥ 3. The group π2 (S ) can be calculated from the Hurewicz theorem relating
π2 (S 2 ) to the so called homology groups of S 2 which are very easy to calculate.
5. Eilenberg-MacLane spaces
Definition 5.1. Let G be a group and n ∈ N a natural number. A path-connected
topological space X is said to be a K(G, n) space if
πn (X) ∼
and
πk (X) ∼
=G
= 1 for all k 6= n
If n ≥ 2, the group G has to be Abelian according to theorem 2.1. A space X is called
an Eilenberg-MacLane space if it is a K(G, n) space for some G and n.
The main result of this section is the following theorem:
Theorem 5.2. Let G be any group and n ∈ N any natural number. Assume further
that G is Abelian if n ≥ 2. Then K(G, n) spaces exist.
Corollary 5.3. Every group G is the fundamental group of some path-connected topological space.
The K(G, n) spaces whose existence is asserted by the above theorem are usually
rather abstract spaces. There are however some concrete examples. For instance, S 1
is a K(Z, 1), RP∞ is a K(Z2 , 1) and CP∞ is a K(Z, 2)1.
Given a K(G, n) and a K(H, n), the product K(G, n) × K(H, n) is a K(G × H, n)
(this fact follows directly from theorem 2.2). Thus the n-dimensional torus T n is an
example of a K(Zn , 1).
1The
infinite dimensional real and complex projective spaces RP∞ and CP∞ are the identification
spaces of R∞ and C∞ associated to the partitions
PR = { {λ · x | λ ∈ R − {0}} | x ∈ R∞ }}
and
PC = { {µ · z | µ ∈ C − {0}} | z ∈ C∞ }}