INTRODUCTION TO ALGEBRAIC TOPOLOGY
MARTIN ˇCADEK
14. Short overview of some further methods in homotopy theory
We start this sections with two examples of computations of homotopy groups.
These computations demonstrate the fact that the possibilities of the methods we have
learnt so far are very restricted. Hence we outline some further (still very classical)
methods which enable us to prove and compute more.
14.1. Homotopy groups of Stiefel manifolds. Let n ≥ 3 and n > k ≥ 1. The
Stiefel manifold Vn,k is (n − k − 1)-connected and
πn−k(Vn,k) =



Z for k = 1,
Z for k = 1 and n − k even,
Z2 for k = 1 and n − k odd.
Proof. The statement about connectivity follows from the long exact sequence for the
fibration
Vn−1,k−1 → Vn,k → Vn,1 = Sn−1
by induction.
As for the second statement, it is sufficient to prove that
πn−2(Vn,2) =
Z for n even,
Z2 for n odd
and to use the induction in the long exact sequence for the fibration above.
We have the fibration
Sn−2
= Vn−1,1 → Vn,2
p
−→ Vn,1 = Sn−1
which corresponds to the tangent vector bundle of the sphere Sn−1
. If n is even, there
is a nonzero vector field on Sn−1
. This field is a map s : Sn−1
→ Vn,2 such that
ps = idSn−1 . Such a map is called a section and its existence ensures that the map
p∗ : πn−1(Vn,2) → πn−1(Sn−1
) is an epimorphism. Hence we get the following part of
the long exact sequence
πn−1(Vn,2)
epi
−−→ πn−1(Sn−1
)
0
−−→ πn−2(Sn−2
)
∼=
−−→ πn−2(Vn,2) → 0.
Consequently, πn−2(Vn,2) = Z.
The case n odd is more complicated. We need the fact that the Euler class of tangent
bundle of Sn−1
is twice a generator ι ∈ Hn−1
(Sn−1
). We obtain the following part of
1
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the Gysin exact sequence for cohomology groups with integer coefficients
0 → Hn−2
(Vn,2)
0
−−→ H0
(Sn−1
)
∪2ι
−−−→ Hn−1
(Sn−1
) → Hn−1
(Vn,2) → 0.
From this sequence and the universal coefficient theorem we get that
0 = Hn−2
(Vn,2; Z) ∼= Hom(Hn−2(Vn,2), Z)
Z2
∼= Hn−1
(Vn,2) ∼= Hom(Hn−1(Vn,2), Z) ⊕ Ext(Hn−2(Vn,2), Z)
which implies that Hn−2(Vn,2; Z) ∼= Z2. The Hurewicz theorem now yields πn−1(Vn,2) ∼=
Z2.
14.2. Hopf fibration. Consider the Hopf fibration
S1
→ S3 η
−−→ S2
defined in 10.5. From the long exact sequence for this fibration we get
πi(S2
) ∼= πi(S3
) for i ≥ 2.
Particularly,
π3(S2
) ∼= Z
with [η] as a generator (since [id] is a generator of π3(S3
)). By the Freudenthal theorem
Z ∼= π3(S2
)
epi
−→ π4(S3
)
∼=
−→ πs
1. The methods we have learnt so far give us only that
π4(S3
) ∼= πs
1 is a factor of Z with Ση as a generator.
Exercise. Try to compute as much as possible from the long exact sequences for the
other two Hopf fibrations in 10.5.
14.3. Composition methods were developed in works of I. James and the Japanese
school of H. Toda in the 1950-ies and are described in the monograph [Toda]. They
enable us to find maps which determine the generators of homotopy groups πn+k(Sn
)
for k not very big (approximately k ≤ 20). For these purposes various types of
compositions and products are used.
Having two maps f : Si
→ Sn
and g : Sn
→ Sm
their composition gf : Si
→ Sm
determines an element [gf] ∈ πi(Sm
) which depends only on [f] and [g]. If the target
of f is different from the source of g, we can use suitable multiple suspensions to be
able to make compositions. For instance, if f : S6
→ S4
and g : S7
→ S3
we can make
composition g ◦ (Σ3
f) : S9
→ S3
. (Here Σ stands for reduced suspension.) In this way
we get a bilinear map πs
a × πs
b → πs
a+b.
More complicated tool is the Toda bracket. Consider three maps
W
f
−→ X
g
−→ Y
h
−→ Z
preserving distinquished points such that gf ∼ 0 and hg ∼ 0. Then gf can be extended
to a map F : CW → Y and hg can be extended to a map G : CX → Z. (C stands
for reduced cone.) Define f, g, h : ΣW = C+W ∪ C−W → Z as G ◦ Cf on C+W and
h ◦ F on C−W.
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W X Y Z
f g h
F
Cf G
Figure 14.1. Definition of Toda bracket < f, g, h >.
This definition depends on homotopies gf ∼ 0 and hg ∼ 0. So it defines a map
from πs
i × πs
j × πs
k to cosets of πs
i+j+k+1. See [Toda] and also Exercise 39 in [Hatcher],
Chapter 4.2.
The Whitehead product [ , ] : πi(X) × πj(X) → πi+j−1(X) is defined as follows:
f : Ii
→ X and g : Ij
→ X define the map f × g : Ii+j
= Ii
× Ij
→ X and we put
[f, g] = (f × g)/∂Ii+j
.
Having a map f : S2n−1
→ Sn
, n ≥ 2, we can construct a CW-complex Cf =
Sn
∪f e2n
with just one cell in the dimensions 0, n and 2n. Denote the generators of
Hn
(Cf ; Z) and H2n
(Cf ; Z) by α and β, respectively. Then the Hopf invariant of f is
the number H(f) such that
α2
= H(f)β.
The Hopf invariant determines a homomorphism H : π2n−1(Sn
) → Z.
For the Hopf map η : S3
→ S2
we have Cη
∼= CP2
, consequently
H(η) = 1.
For id : S2
→ S2
we can make the Whitehead product [id, id] : S3
→ S2
and compute
(see [Hatcher], page 474) that
H([id, id]) = ±2.
Since π3(S2
) ∼= Z, we get [id, id] = ±2η. One can show (see [Hatcher], page 474 and
Corollary 4J.4) that the kernel of the suspension homomorphism π3(S2
) → π4(S3
) is
generated just by [id, id]. By the Freudental theorem this suspension homomorphism
is an epimorphism which implies that
π4(S3
) ∼= Z2.
Consequently, πs
1
∼= Z2.
Remark. J. F. Adams proved in [Adams1] that the only maps with the odd Hopf
invariant are the maps coming from the Hopf fibrations S3
→ S2
, S7
→ S4
and
S15
→ S8
.
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Another important tool for composition methods is the EHP exact sequence for the
homotopy groups of Sn
, Sn+1
and S2n
:
π3n−2(Sn
)
E
−→ π3n−1(Sn+1
)
H
−→ π3n−2(S2n
)
P
−→ π3n−3(Sn
) → . . .
· · · → πi(Sn
)
E
−→ πi+1(Sn+1
)
H
−→ πi(S2n
)
P
−→ πi−1(Sn
) → . . .
Here E stands for suspension, H refers to a generalized Hopf invariant and P is defined
with connection to the Whitehead product. See [Whitehead], Chapter XII or [Hatcher],
page 474.
For n = 2 the EHP exact sequence yields
π4(S2
)
E
−→ π5(S3
)
H
−→ π4(S4
)
P
−→ π3(S2
)
E
−→ π4(S3
) → 0.
Since π4(S3
) ∼= Z2, π3(S2
) ∼= Z and π4(S4
) ∼= Z, we obtain that P is a multiplication
by 2 and H = 0. From the long exact sequence for the Hopf fibration (see 14.2) we
get that π4(S2
) ∼= π4(S3
) ∼= Z2 with the generator η(Ση). So π5(S3
) is either Z2 or 0.
By a different methods one can show that
π5(S3
) ∼= Z2
with the generator (Ση)(Σ2
η).
14.4. Cohomological methods have been playing an important role in homotopy
theory since they were introduced in the 1950-ies.
By the methods used in proofs in Section 12 we can construct so called EilenbergMcLane
spaces K(G, n) for any n ≥ 0 and any group G, Abelian if n ≥ 2. These spaces
are up to homotopy equivalence uniquely determined by their homotopy groups
πi(K(G, n)) =
0 for i = n,
G for i = n.
Moreover, these spaces provide the following homotopy description of reduced singular
cohomology groups
[(X, ∗), (K(G, n), ∗)]
∼=
−−→ Hn
(X; G).
To each [f] ∈ [(X, ∗), (K(G, n), ∗)] we assign
f∗
(ι) ∈ Hn
(X; G)
where ι is the generator of
Hn
(K(G, n); G) ∼= Hom(Hn(K(G, n); Z), G) ∼= Hom(G, G)
corresponding to idG.
A system of homomorphisms θX : Hn
(X; G1) → Hm
(X; G2) which is natural, i. e.
f∗
θY = θXf∗
for all f : X → Y , is called a cohomology operation. A system of cohomology
operations θj : Hn+j
→ Hm+j
is called stable if it commutes with suspensions
Σθj = θj+1Σ.
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The most important stable cohomology operations for singular cohomology are the
Steenrod squares and the Steenrod powers:
Sqi
: Hn
(X; Z2) → Hn+i
(X; Z2)
Pi
p : Hn
(X; Zp) → Hn+2i(p−1)
(X; Zp) for p = 2 a prime.
For their definition and properties see [SE] or [Hatcher], Section 4.L. These operations
can be also interpreted as homotopy classes of maps between Eilenberg-McLane spaces,
for instance
Sqi
: K(Z2, n) → K(Z2, n + i).
Example A. We show how the Steenrod squares can be used to prove that some
maps are not homotopic to a trivial one. Consider the Hopf map η : S3
→ S2
. We
know that Cη = CP2
and H2
(CP2
; Z2) and H4
(CP2
; Z2) have generators α and α2
.
Since one of the properties of the Steenrod squares is
Sqn
x = x2
for x ∈ Hn
(X; Z2),
we get that Sq2
α = α2
= 0. Using this fact we show that [Ση] ∈ π4(S3
) is nontrivial.
For reduced cones and reduced suspensions one can prove that
CΣη = ΣCη ΣCP2
.
Then Σα : ΣCP2
→ K(Z2, 3) and Σα2
: ΣCP2
→ k(Z2; 5) represent generators in
H3
(ΣCP2
; Z2) and H5
(ΣCP2
; Z2), respectively. Now
Sq2
(Σα) = Σ(Sq2
α) = Σα2
= 0.
If Ση were homotopic to a constant map, we would have CΣη = S3
∨ S5
, and consequently,
Sq2
(Σα) = 0 since Sq2
is trivial on S3
.
Example B. We outline how to compute πn+1(Sn
) using cohomological methods.
A generator α ∈ Hn
(Sn
) induces up to homotopy a map Sn
→ K(Z, n). Further,
Hn
(K(Z, n); Z) ∼= Z with a generator ι and Hn+2
(K(Z, n); Z2) ∼= Z2 with the generator
Sq2
ρι where ρ : Hn
(X; Z) → Hn
(X; Z2) is induced by reduction mod 2. Sq2
ρι induces
up to homotopy a map
K(Z, n)
Sq2ρι
−−−→ K(Z2, n + 2).
Consider the fibration
ΩK(Z2, n + 2) → PK(Z2, n + 2) → K(Z2, n + 2)
where PX is the space of all maps p : I → X, p(1) = x0 and ΩX is the space of all
maps ω : I → X, ω(0) = ω(1) = x0. (These maps are called loops in X.) One can
show that ΩK(Z2, n + 2) has a homotopy type of K(Z2, n + 1). The pullback of the
fibration above by the map Sq2
ρι : K(Z, n) → K(Z2, n + 2) is the fibration
K(Z2, n + 1) → E
p
−→ K(Z, n).
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Since Sq2
ρα = 0 in Hn+2
(Sn
; Z), one can show that the map α : Sn
→ K(Z, n) can
be lifted to a map f : Sn
→ E.
E
p

Sn
α
//
f
::vvvvvvvvvv
K(Z, n)
One can compute f∗
in cohomology (using so called long Serre exact sequence) and then
also f∗ in homology. A modified version of the homology Whitehead theorem implies
that f is an (n + 2)-equivalence. Hence f∗ : πn+1(Sn
) → πn+1(E) is an isomorphism.
Using the long exact sequence for the fibration (E, K(Z, n), p) we get
Z2
∼= πn+1(K(Z2, n + 1))
∼=
−−→ πn+1(E) ∼= πn+1(Sn
).
For more details see [MT].
The Steenrod operations form a beginning for the second course in algebraic topology
which should contain spectral sequences, other homology and cohomology theories,
spectra. We refer the reader to [Adams2], [Kochman], [MT], [Switzer], [Whitehead] or
to the last sections of [Hatcher].
CZ.1.07/2.2.00/28.0041
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