HIGHER HOMOTOPY GROUPS
JOSHUA BENJAMIN III
CLASS OF 2020 [email protected] Acknowledgments
I would like to thank:
Joe Harris for lending me a copy of the Greenberg and Harper Algebraic Topology book.
I would also like to thank him for allowing me to create an exposition and for oering advice as to what topic I should explore.
ii To My Parents
iii HIGHER HOMOTOPY
GROUPS
JOSHUA BENJAMIN III
JOE HARRIS
iv Contents
1 Introduction 1
2 Higher Homotopy Groups 2
2.1 Denition of the homotopy groups ...... 2
2.2 A rst important result ...... 3
3 Homotopy groups of Spheres 4
n 3.1 πi(S ) when i < n ...... 4
n 3.2 πi(S ) when i = n ...... 4
3.3 Stable and unstable ...... 5
n 3.4 πi(S ) when i > n ...... 5
v Chapter 1
Introduction
e study of higher homotopy groups has long been of interest to algebraic topologists.
ese groups, which will be dened in the next chapter, are quite simple to dene, but unfortunately are very dicult to calculate. Especially when compared to the homology groups, the higher homotopy groups are rather ”elusive” in nature. e eld of study was opened when Heinz Hopf introduced Hopf bration (also known as the Hopf map) which was the rst non-trivial mapping of S3 to S2. e bers of this map were S1 and it allowed a calculation of the third homotopy group of the 2-sphere S2. For this text, the author will provide some history and some important theorems and results. Due to author’s limited knowledge, proofs of these theorems and results will not be included unless they are within the scope of the author’s ability.
1 Chapter 2
Higher Homotopy Groups
e idea of homotopy and a homotopy group was introduced by Camille Jordan who did so without using the syntax and notation of group theory. e concepts of homol- ogy and the fundamental group were intertwined by Henri Poincare, but the higher homotopy groups were not dened until Eduard Cech did so. A major step in both the denition and computation of the homotopy groups came from Witold Hurewicz with the Hurewicz theorem. A later development came with Hans Freudenthal’s suspen- sion theorem. is development contributed to the study of stable algebraic topology to examine the properties that did not rely on dimension. George Whitehead and Jean- Pierre Serre made the next major advancements. Whitehead proved that there exists a metastable range for the homotopy groups of spheres. Serre employed the use of spec- tral sequences to prove that all the higher homotopy groups of spheres are nite, with n 2n the exception of πn(S ) and π4n−1(S ).
2.1 Denition of the homotopy groups
For any given space X and base point b, πn(X) is the set of homotopy classes of maps
f : Sn → X that take a base point a → b. πn(X) can also be dened as the group of homotopy classes of maps
g : In → X that take ∂In → b, where In denotes the n-cube [0, 1]n and ∂ denotes the boundary.
2 2.2 A rst important result
e result the author wishes to show is that all the πn(X, x0) for n > 1 are abelian. is will be done in two ways. e rst of which is the ”Eckmann-Hilton trick” and the other is an inductive proof.
eorem 2.2.1. πn(X, x0) for n > 1 are abelian ”Eckmann-Hilton trick”
Proof. Let S be a set with two associative operations ∗, ◦ : S × S → S having a common unit e ∈ S. Suppose ∗ and ◦ distribute over each other, in the sense that
(α ∗ β) ◦ (γ ∗ δ) = (α ◦ γ) ∗ (β ◦ δ)
Taking β = e = γ in the distributive law yields
α ◦ δ = α ∗ δ
, while taking α = e = δ yields β ◦ γ = γ ∗ β us ∗ and ◦ coincide, and dene a commutative operation on S
As for the inductive method, a few things must rst be dened. Let Ωx0 ⊂ X consist of all loops at x0 and let C be a constant loop at x0. is allows us the theorem:
eorem 2.2.2. π1(Ωx0 , C) is abelian
Proof. Let f , g be loops in Ωx0 at C. Now dene ( f ∗ g)(t) = f (t)g(t).
f g ' f ∗ g ' g f rel(0, 1)
Now dene the higher homotopy groups as
πn(X, x0) = πn−1(Ωx0 , C) for n ≥ 2 From this we have the corollary
Corollary 2.2.3. e higher homotopy groups are all abelian
3 Chapter 3
Homotopy groups of Spheres
e next major paerns come from theorems that are beyond the (current) knowledge of the author. alitative explanations will be given where possible.
n 3.1 πi(S ) when i < n
n Nicely, πi(S ) = 0 when i < n. is can be proven formally but it can also be seen in the following way. Given any continuous mapping from an i-sphere to an n-sphere with i < n, it can always be deformed so that it is not a surjective map. is conrms that, its image is contained in Sn with a point removed. In all cases, this is a contractible space, and any mapping to a contractible space can be deformed into a one-point mapping. Hence it is trivial.
n 3.2 πi(S ) when i = n
n πi(S ) = Z when i = n. is can be realized in multiple ways. Witold Hurewicz related the homotopy groups to the homology groups by abelianization(i.e. taking a quotient group G/[G, G]). So, the result directly follows from the Hurewicz theorem (which will not be stated or proved here for lack of knowledge of homology). e the- orem shows that for a simply-connected space X, the rst nonzero homotopy group πk(X), with k > 0, is isomorphic to the rst nonzero homology group Hk(X). For the n-sphere, this means that for n > 0,
n n πn(S ) = Hn(S ) = Z . is can also be proven inductively from the Freudenthal suspension theorem which implies that the suspension homomorphism from
n n+1 πn+k(S ) → πn+k+1(S ) is an isomorphism for n > k + 1
4 3.3 Stable and unstable
n e groups πn+k(S ) with n > k + 1 are called the stable homotopy groups of spheres, S and are denoted πk . ese are nite abelian groups for k 6= 0. e general formula is still unknown. For n ≤ k + 1, the groups are called the unstable homotopy groups of spheres.
n 3.4 πi(S ) when i > n Unfortunately, these groups are not trivial in general eve though the homology groups of this form are. ere is no known general formula for computing these groups. As stated before, most of these groups are nite and all are still abelian. An example of a known paern is that 2 3 π3+j(S ) = π3+j(S ), ∀j ∈ N is is also a consequence of the Hopf bration.
5 Bibliography
Cartan, Henri; Serre, Jean-Pierre (1952a), ”Espaces bres et groupes d’homotopie.
I. Constructions generales”, Comptes Rendus de l’Academie des Sciences. Serie I.
Mathematique, Paris: Elsevier, 234: 288-290
Greenberg, Marvin J., and John R. Harper. ”Loop Spaces and Higher Homotopy
Groups.” Algebraic Topology: A First Course. Redwood City, CA: Addison-Wesley
Pub., 1981. N. pag. Print.
Hopf, Heinz (1931), ”Uber die Abbildungen der dreidimensionalen Sphare auf die
Kugelache”, Mathematische Annalen, Berlin: Springer, 104 (1): 637-665
May, J. Peter (1999b), A Concise Course in Algebraic Topology, Chicago lectures
in mathematics (revised ed.), University of Chicago Press
May, J. Peter (1999a), ”Stable Algebraic Topology 1945-1966”, in I. M. James, History
of Topology, Elsevier Science, pp. 665-723,
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