HIGHER HOMOTOPY GROUPS JOSHUA BENJAMIN III. CLASS OF 2020

Transcription

1 HIGHER HOMOTOPY GROUPS JOSHUA BENJAMIN III CLASS OF 2020

2 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 offering advice as to what topic I should explore. ii

3 To My Parents iii

4 HIGHER HOMOTOPY GROUPS JOSHUA BENJAMIN III JOE HARRIS iv

5 Contents 1 Introduction 1 2 Higher Homotopy Groups Definition of the homotopy groups A first important result Homotopy groups of Spheres π i (S n ) when i < n π i (S n ) when i = n Stable and unstable π i (S n ) when i > n v

6 Chapter 1 Introduction The study of higher homotopy groups has long been of interest to algebraic topologists. These groups, which will be defined in the next chapter, are quite simple to define, but unfortunately are very difficult to calculate. Especially when compared to the homology groups, the higher homotopy groups are rather elusive in nature. The field of study was opened when Heinz Hopf introduced Hopf fibration (also known as the Hopf map) which was the first non-trivial mapping of S 3 to S 2. The fibers of this map were S 1 and it allowed a calculation of the third homotopy group of the 2-sphere S 2. 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

7 Chapter 2 Higher Homotopy Groups The idea of homotopy and a homotopy group was introduced by Camille Jordan who did so without using the syntax and notation of group theory. The concepts of homology and the fundamental group were intertwined by Henri Poincare, but the higher homotopy groups were not defined until Eduard Cech did so. A major step in both the definition and computation of the homotopy groups came from Witold Hurewicz with the Hurewicz theorem. A later development came with Hans Freudenthal s suspension theorem. This 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 spectral sequences to prove that all the higher homotopy groups of spheres are finite, with the exception of π n (S n ) and π 4n 1 (S 2n ). 2.1 Definition of the homotopy groups For any given space X and base point b, π n (X) is the set of homotopy classes of maps f : S n X that take a base point a b. π n (X) can also be defined as the group of homotopy classes of maps g : I n X that take I n b, where I n denotes the n-cube [0, 1] n and denotes the boundary. 2

8 2.2 A first important result The result the author wishes to show is that all the π n (X, x 0 ) for n > 1 are abelian. This will be done in two ways. The first of which is the Eckmann-Hilton trick and the other is an inductive proof. Theorem π n (X, x 0 ) 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, while taking α = e = δ yields α δ = α δ β γ = γ β Thus and coincide, and define a commutative operation on S As for the inductive method, a few things must first be defined. Let Ω x0 X consist of all loops at x 0 and let C be a constant loop at x 0. This allows us the theorem: Theorem π 1 (Ω x0, C) is abelian Proof. Let f, g be loops in Ω x0 at C. Now define ( f g)(t) = f (t)g(t). f g f g g f rel(0, 1) Now define the higher homotopy groups as π n (X, x 0 ) = π n 1 (Ω x0, C) for n 2 From this we have the corollary Corollary The higher homotopy groups are all abelian 3

9 Chapter 3 Homotopy groups of Spheres The next major patterns come from theorems that are beyond the (current) knowledge of the author. Qualitative explanations will be given where possible. 3.1 π i (S n ) when i < n Nicely, π i (S n ) = 0 when i < n. This 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. This confirms that, its image is contained in S n 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. 3.2 π i (S n ) when i = n π i (S n ) = Z when i = n. This 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). The theorem shows that for a simply-connected space X, the first nonzero homotopy group π k (X), with k > 0, is isomorphic to the first nonzero homology group H k (X). For the n-sphere, this means that for n > 0, π n (S n ) = H n (S n ) = Z. This can also be proven inductively from the Freudenthal suspension theorem which implies that the suspension homomorphism from is an isomorphism for n > k + 1 π n+k (S n ) π n+k+1 (S n+1 ) 4

10 3.3 Stable and unstable The groups π n+k (S n ) with n > k + 1 are called the stable homotopy groups of spheres, and are denoted πk S. These are finite abelian groups for k = 0. The general formula is still unknown. For n k + 1, the groups are called the unstable homotopy groups of spheres. 3.4 π i (S n ) when i > n Unfortunately, these groups are not trivial in general eve though the homology groups of this form are. There is no known general formula for computing these groups. As stated before, most of these groups are finite and all are still abelian. An example of a known pattern is that π 3+j (S 2 ) = π 3+j (S 3 ), j N This is also a consequence of the Hopf fibration. 5

11 Bibliography Cartan, Henri; Serre, Jean-Pierre (1952a), Espaces fibres et groupes d homotopie. I. Constructions generales, Comptes Rendus de l Academie des Sciences. Serie I. Mathematique, Paris: Elsevier, 234: Greenberg, Marvin J., and John R. Harper. Loop Spaces and Higher Homotopy Groups. Algebraic Topology: A First Course. Redwood City, CA: Addison-Wesley Pub., N. pag. Print. Hopf, Heinz (1931), Uber die Abbildungen der dreidimensionalen Sphare auf die Kugelflache, Mathematische Annalen, Berlin: Springer, 104 (1): 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 , in I. M. James, History of Topology, Elsevier Science, pp , 6