Bott periodicity and exotic spheres: some comments
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 1 The Bott periodicity Theorem
One of the fundamental results in modern topology is Bott periodicity [Bott]. It concerns the homotopy groups of the stable orthogonal or unitary groups resp :
Theorem 1.1 [Bott1958]. For there are isomorphisms
 2 An interpretation in terms of vector bundles over spheres
Most applications of this result concern the interpretation of these groups as stable vector bundles over spheres. Namely if is a continuous map, then one obtains a vector bundle over by taking two copies of and by identifying in the first copy with in the second copy. This map gives an isomorphism from to the set of isomorphism classes of -dimensional vector bundles over . I don't know who observed this first, it can for example be found in Steenrod's book [Steenrod]. Similarly corresponds to the set of isomorphism classes of complex -dimensional vector bundles over . Passing from to (or to ) by the standard inclusion corresponds to stabilization of vector bundles by taking the Whitney sum with the -dimensional trivial bundle. By a general position argument the stabilization map is a bijection if ( in the complex case). If (or in the complex case) one calls such a bundle a stable vector bundle. Actually the -dimensional vector bundles over (not over a general space) form an abelian group, where the sum is given by a connected sum of vector bundles: One choses a trivialization of the vector bundles over an open disk and identifies the resulting boundaries. Bott's theorem implies that for not equal to mod all real vector bundles of dimension over are trivial, that for mod there are precisely such bundles and for mod there are countably many such bundles.
Remark 2.1. I find it remarkable that Bott doesn't mention the relation to vector bundles. Actually Bott does not say a single word, why his result is interesting. He obviously assumes that a reader finds the problem to determine the homotopy groups of such fundamental objects like the stable orthogonal or unitary group interesting in itself and he is of course right. Whether he has foreseen that it is such a fundamental result would be interesting to know, perhaps his friends Atiyah and Hirzebruch can comment on this.
 3 The dates of background papers
The paper with complete proofs appeared in September 1959, it was submitted November 1958. An announcement containing the above statements appeared in 1957 [Bott1957]. The methods of the proof were developed in several earlier papers [Bott1954], [Bott 1956].
 4 The role of Bott periodicity for Kervaire-Milnor's paper
I hope there will be many articles in the atlas explaining more or less immediate applications of the periodicity theorem. Here I comment on the role it played in Kervaire and Milnor's important paper "Groups of homotopy spheres I" [Kervaire&Milnor1963]. This paper appeared May 1963, and was submitted April 1962. By that time the periodicity theorem must have been a standard tool in topology.
A smooth manifold homotopy equivalent to a sphere is called a homotopy sphere. Such manifolds are interesting for many reasons, one of them is the Poincaré conjecture saying that all these manifolds are homeomorphic to the standard sphere. There are no obvious constructions of such manifolds. But Milnor [Milnor1956] constructed a series of -dimensional manifolds which are all homeomorphic to the -sphere (in particular homotopy equivalent) but some of which are not diffeomorphic to it. Such manifolds are called exotic spheres.
The Poincaré conjecture was proven by Smale, Stallings and Zeeman (references) for already before the paper by Kervaire and Milnor, and later by Freedman [Freedman1982] in dimension and by Perelmann [Perelman] in dimension . Thus the diffeomorphism classes of homotopy spheres correspond to the diffemorphism classes of smooth structures on the sphere.
If is a homotopy sphere one can ask for ways to distinguish it from the standard sphere. For example one could look at the stable tangent bundle, which for the standard sphere is trivial. Surprisingly this also holds for homotopy spheres:
Theorem [Kervaire&Milnor1963] 4.1. For all homotopy spheres the stable tangent bundle is trivial.
The proof of this theorem is based on three non-trivial theorems, one of which is Bott periodicity. This makes it in half of the cases a triviality, since for mod there is no non-trivial stable vector bundle over . The remaining cases are not so easy, one needs a way to decide whether two stable vector bundles over are isomorphic. Let's begin with the case .
In the case one has an invariant for stable vector bundles , namely the Pontrjagin classes . It turns out that this map is a homomorphism from the stable vector bundles over to which by choosing an orientation we identify with . Thus we have a homomorphism from a group isomorphic to to , and if it is non-trivial it is an injection implying that two stable vector bundles over are isomorphic if and only their Pontrjagin classes agree. To show that the homomorphism is non-trivial (for ) one needs a single example where this is the case. Such examples where for example constructed by Borel and Hirzebruch (reference).
Kervaire and Milnor proceed slightly differently. They refer to obstruction theory, a theory which can be used to decide whether a vector bundle is trivial. They say (with reference to earlier papers by Kervaire) that the obstruction class and the Pontrjagin class are proportional by a non-zero factor and so, if the Pontrjagin class is trivial, the bundle is trivial. They finish the argument that the stable tangent bundle of a -dimensional homotopy sphere is trivial by one sentence: "But by the Hirzebruch signature theorem the Pontrjagin class is a multiple of the signature , which is zero since ."
Remark 4.2. Let me make a short comment on this sentence. It shows that the signature theorem, which was published by Hirzebruch in his 1956 book, is at the time when Kervaire and Milnor wrote their paper so standard, that neither a reference to Hirzebruch's book is needed (which appears in the list of references but no reference is given at this place) nor the formula of the signature theorem is repeated.
A non-trivial theorem is also needed in the remaining cases where mod is. The cases and are trivial, but the higher dimensions not. Here the obstruction class sits in (by Bott's theorem) and so again one has to find a way to distinguish the non-trivial element form . There is a homomorphism introduced by Hopf-Whitehead (reference), the -homomorphism, from to the stable homotopy groups of spheres. Rohlin (reference) has shown that under this homomorphism the obstruction class vanishes (the authors don't give a reference to a paper by Rohlin but refer instead to an earlier paper by them [Kervaire&Milnor1958]). The argument, that also in the remaining case the stable tangent bundle is trivial is finished by applying a recent deep theorem by Adams [Adams] saying that this -homomorphism is injective.
 5 Some comments
After repeating the role of Bott periodicity in the proof of this theorem I would like to comment a bit on the role of this theorem in the paper of Kervaire and Milnor and in the further development of analyzing smooth structures on a topological manifold, here the sphere.
On the one hand the message of the theorem is negative, one cannot use the stable tangent bundle to distinguish different smooth structures on the sphere. This leads to the interesting question, whether the same is true for arbitrary manifolds. Later in the sixties some very deep theorems in this direction were proved.
On the other hand the theorem was a good start to develop a method to study the different smooth structures on spheres or equivalently the homotopy spheres in dimension , a method which in the following years was generalized to arbitrary manifolds. This is not the place to describe this method, which is called surgery theory. But the following can be said. The most successful way to decide whether two closed smooth simply connected manifolds and of dimension are diffeomorphic is to try to find an -cobordism between them, that is a compact manifold with boundary the disjoint union of and , such that the inclusions from and to are homotopy equivalent (up to homotopy is cylinder). Then Smale's -cobordism theorem [Smale1961] implies that if the dimension of is larger than , then is diffeomorphic to the cylinder , and so and are diffeomorphic.
Thus one tries to find an -cobordism between two homotopy spheres. Since the stable tangent bundle of the homotopy spheres is trivial the same holds for . Thus one can ask a weaker question, namely whether there is any bordism between the two homotopy spheres which has trivial stable tangent bundle. This is equivalent to a question in the stable homotopy groups of spheres, thus a very difficult question, but it is an unavoidable problem. If such a bordism exists Kervaire and Milnor try to improve it until it is an -cobordism. They solve this problem completely by surgery theory.
Remark 5.1. Suppose that a topological manifold is given. Then one can ask whether admits a smooth structure. A necessary condition is that has a tangent bundle, and since this is easier to analyze and essentially the same, that it has a stable tangent bundle. It turns out that in a certain sense, which should be made precise elsewhere, this is the only obstruction, again in dimension , but false in dimension by the fundamental work of Donaldson (reference), and again true in dimension by different methods (Kirby-Sibenmann (reference)).
If we assume that has a smooth structure, we choose one and compare all possible other smooth structures with this. Then again the stable tangent bundle plays the deciding role (in dimension ). Roughly speaking the different smooth structures on correspond to the different ways to impose a stable tangent bundle on (Kirby-Sibenmann (reference)). Thus the understanding of stable vector bundles is what at the end is needed. This is the content of a very important theory, called -theory, invented by Atiyah and Hirzebruch [Atiyah-Hirzebruch]. This is a generalized cohomology theory (meaning that the Eilenberg-Steenrod axioms for ordinary cohomology are fulfilled except the dimension axiom). This is the first generalized cohomology theory and - besides stable homotopy - the most important one. To construct it is up to a certain point rather elementary. But then one comes at a point where the arguments are highly non-trivial and the central tool, which one has to apply, is Bott perodicity. It does not only give the fundamental input for completing the proof that it is a generalized homology theory, it also is the central tool for all computations. In particular, in the few cases where one can give detailed information about the different smooth structures on a manifold, always Bott's theorem is in the background - like we indicated in one aspect for the spheres.
 6 References
- [Adams] Template:Adams
- [Atiyah-Hirzebruch] Template:Atiyah-Hirzebruch
- [Bott] Template:Bott
- [Bott 1956] Template:Bott 1956
- [Bott1954] Template:Bott1954
- [Bott1957] R. Bott, The stable homotopy of the classical groups, Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 933–935. MR0102802 (21 #1588) Zbl 0129.15601
- [Bott1958] Template:Bott1958
- [Freedman1982] M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no.3, 357–453. MR679066 (84b:57006) Zbl 0528.57011
- [Kervaire&Milnor1958] Template:Kervaire&Milnor1958
- [Kervaire&Milnor1963] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR0148075 (26 #5584) Zbl 0115.40505
- [Milnor1956] J. Milnor, On manifolds homeomorphic to the -sphere, Ann. of Math. (2) 64 (1956), 399–405. MR0082103 (18,498d) Zbl 0072.18402
- [Perelman] Template:Perelman
- [Smale1961] S. Smale, Generalized Poincaré's conjecture in dimensions greater than four, Ann. of Math. (2) 74 (1961), 391–406. MR0137124 (25 #580) Zbl 0099.39202
- [Steenrod] Template:Steenrod