Exotic

Malik Obeidin May 2, 2014

1 Introduction

One of the biggest shocks in the history of differential geometry and topology occured in 1956, after published the first constructed examples of manifolds which topologically were the same, but differentiably were not. In fact, all of these manifolds are topologically the 7-, but distinguished by an invariant. This bizarre result opened up new areas studying the relationship between topological, smooth, and PL manifolds, and from that point, all results had to be careful about the category in which one was working. Since then, many more examples are known, and many classified, but the original construction due to Milnor is actually quite simple and elegant. We outline his reasoning as given in the original 1956 paper [1]. For the remainder of this paper, all manifolds considered will be differentiable (of class C∞), orientable, and compact, and all cohomology groups will have integer coefficients.

2 Constructing the Invariant

Let M 7 be a differentiable, oriented, closed manifold satisfying

H3(M 7) = H4(M 7) = 0 (1)

On any such manifold, we can define a residue class λ(M 7) modulo 7, which will be the invariant which will distinguish between smooth structures on our constructed 7-spheres. By a result of Thom [2], every closed 7-manifold M 7 is the boundary of some 8-manifold B8; in particular, we can choose an ori- 8 7 7 entation ν ∈ H8(B ,M ) with the property that ∂ν = µ, where µ ∈ H7(M ) is the chosen generator for the orientation. We define a map on the free part H4(B8,M 7)/(torsion) by sending α → hν, α2i; this is a quadratic map Zn → Z, hence a quadratic form. Now we call τ(B8) the index of this form; that is, after diagonalizing over R, we take the number of positive terms minus the number of negative terms. This is the first of two numbers which will be used to define λ.

1 4 8 The second is given as follows: let p1 ∈ H (B ) be the first Pontrjagin class of the tangent bundle of B8. The assumption (1) above means that, by the long exact sequence of a pair in cohomology, the quotient map

q∗ : H4(B8,M 7) → H4(B8) is sandwiched between zero maps, and hence is an isomorphism. So, similarly to τ, we can define what Milnor calls a ”Pontrjagin number”

8 ∗ −1 2 q(B ) = hν, ((q ) p1) i

With these two numbers, we define λ(M 7) to be the residue class modulo 7 of 2q(B8) − τ(B8) for the following reason: Theorem 1. The residue class of λ modulo 7 does not depend on the choice of the manifold B8.

8 8 To prove this theorem, we choose two possibly different manifolds B1 , B2 with boundary M 7. Then, sewing these along the boundary M 7 gives a closed 8 8 8 8-manifold C = B1 ∪ B2 , possessing a differentiable structure compatible with 8 8 both B1 and B2 , and an orientation ν which is a consistent with the orientation 8 8 2 8 ν1 on B1 . We can then define q(C ) = hν, p1(C )i. The index τ(C8) defined as above was previous computed by Thom in [2] to be 1 τ(C8) = hν, (7p (C8) − p2(C8)i 45 2 1 8 where p2(C ) is the second Pontrjagin class. Then, we have that

8 8 8 2 8 2 8 45τ(C ) + q(C ) = hν, 7p2(C ) − p1(C )i + hν, p1(C )i 8 = 7hν, p2(C )i ≡ 0 (mod 7)

So, reducing everything modulo 7 and multiplying by 2 gives

2q(C8) − τ(C8) ≡ 0 (mod 7) (2)

But, the following lemma gives immediately the result: Lemma 1. The given manifold C8 satisfies

8 8 8 τ(C ) = τ(B1 ) − τ(B2 )

8 8 8 q(C ) = q(B1 ) − q(B2 ) as one might guess from the construction. Combined with (2), this proves Theorem 1. The proof of Lemma 1 is not hard but moderately tedious, so we’ll skip it. Note the following facts we get immediately from Theorem 1: Corollary 1. If λ(M 7) 6= 0 then M 7 is not the boundary of any 8-manifold with fourth Betti number zero.

2 Corollary 2. If λ(M 7) 6= 0 then M 7 possesses no orientation reversing self diffeomorphism. The first fact is obvious as such an 8-manifold B8 would force both τ(B8) and q(B8) to be zero. The second is also straightforward; an orientation reversing self diffeomorphism takes both q and τ (hence λ) to their negatives, though λ is diffeomorphism invariant as constructed. In the end, constructing a topological 7-sphere with nonzero λ will give the exotic sphere which we are looking for. We will be able to show the constructed manifolds are homeomorphic to S7 by using the following result of , (a version of) the Reeb : Theorem 2. Let M n be a closed manifold. Suppose there exists a differentiable function f : M n → R such that f has only two critical points, both nondegener- ate (i.e., the Hessian is nonsingular). Then, M n is homeomorphic to the sphere Sn.

3 Constructing the 7-Manifolds

Consider the isomorphism classes of S3 bundles over S4, with structure group SO(4). In fact, it can be shown that this set of isomorphism classes is in one- ∼ to-one correspondence with the group π3(SO(4)) = Z ⊕ Z, with an explicit isomorphism given as follows: for any pair (h, j) ∈ Z ⊕ Z, define a map fhj : S3 → SO(4) by the formula

h j fhj(u) · v = u vu where v is an element of R4. The multiplication between u and v is multiplication. 3 So, let ξhj be the S bundle corresponding the (h, j) and fhj. Then, if we denote the standard orientation ι ∈ H4(S4), we have the following formula:

Lemma 2. The Pontrjagin class p1(ξhj) is equal to ±2(h − j).

It is easy to see that the Pontrjagin class p1(ξhj) is linear as a function of h and j, and it is well known that it is independent of the orientation on the 3 fiber. But, reversing the orientation of S switches ξhj to ξ−h−j, leaving only linear functions of the form c(h − j) for some c to be determined later. Given any odd k ∈ Z, we have a unique (h, j) such that the equations 7 h + j = 1 and h − j = k are satisfied. For any such k, we define Mk to be the total space of ξhj, which is clearly a topological manifold. In fact, we will show 7 Mk has a natural and orientation, and is in fact topologically a sphere.

7 Lemma 3. Mk has a natural smooth structure and orientation, and a smooth 7 function f : Mk → R with exactly two nondegenerate critical points. 7 So, by applying Theorem 2, Mk is homeomorphic to the standard 7-sphere.

3 Proof. Take coordinate charts in the base S4 given by the complements of the north and south poles, which are each identified with R4 by stereographic pro- jection. By elementary geometry, a point corresponding to u ∈ R4 in one chart 0 u is given by u = kuk2 in the other. Over each of these charts, the bundle is necessarily trivial, so we describe 7 4 3 the total space Mk by taking two copies of R × S and identifying the subsets (R4 − 0) × S3 with the diffeomorphism u uhvuj (u, v) → (u0, v0) = ( , ) kuk2 kuk which gives us back the bundle ξhj. Again here we have exploited the quaternion multiplication on R4, and fixed a smooth structure on the total space by starting with smooth manifolds and identifying via diffeomorphisms. Making the change of coordinates (u0, v0) by (u00, v0), where u00 = u0(v0)−1, 7 we can now define the function f : Mk → R by Re(v) Re(u00) f(x) = 1 = 1 (1 + kuk2) 2 (1 + ku00k2) 2 which can be easily verified to have the desired property. The critical points will be at (u, v) = (0, ±1) and will be non-degenerate, completing the proof of the lemma.

7 So, Theorem 2 now concludes that Mk is in fact homeomorphic to a sphere. To prove the existence of manifolds homeomorphic but not diffeomorphic to 7 7 S , we need only to find our previous invariant λ(Mk ), which is now defined, 3 7 4 7 7 since H (Mk ) = H (Mk ) = 0 as Mk is homeomorphic to the sphere. In fact, we have the following:

7 2 Lemma 4. The invariant λ(Mk ) is the residue class of k − 1 modulo 7. 7 4 Proof. With any 3-sphere bundle Mk → S we can extend the total space to a 8 4 8 4-cell bundle ρk : Bk → S , where Bk is a differentiable manifold with boundary 7 4 8 ∗ Mk . The fourth cohomology H (Bk) is generated by the pullback α = ρk(ι). Choosing proper orientations, we can fix hν, ((q∗)−1α)2i = 1, hence we will have 7 the index τ(Bk) = 1. 7 The tangent bundle TBk will decompose as a direct sum of the bundle of vectors tangent to the fiber and the bundle of vectors normal to the fiber. The ∗ first bundle is simply the pullback ρk(ξhj), so by naturality and the proof of ∗ Lemma 2 has the Pontrjagin class p1 = ρk(c(h − j)ι) = ckα, as k = h − j. The 4 second bundle is the pullback of the tangent bundle of S , which has p1 = 0, as any sphere will have trivial Pontrjagin classes due to the fact that the tangent bundle TSn sums with the normal bundle N(Sn) in Rn+1 to a trivial bundle n+1 8  . So, the Whitney product theorem gives that p1(Bk) = ckα. The case k = 1 was already somewhat studied at the time Milnor published 8 the paper - it can be checked that B1 is actually the quaternion projective plane P2(K) with an 8-cell removed, which is known to have first Pontrjagin

4 4 class p1(P2(K)) twice a generator in H (P2(K)), see Hirzebruch [3], which ne- cessitates that the constant c above, which did not depend on k to be ±2, proving Lemma 2. 8 ∗ −1 2 2 So, finally, q(Bk) = hν, ((q ) (±2kα)) i = 4k , and hence 2q − τ = 8k2 − 1 ≡ k2 − 1 (mod 7)

Hence, Lemma 4 has been proven.

7 Notice that now we can distinguish Mk from the standard sphere: 2 7 7 Theorem 3. If k − 1 6≡ 0 (mod 7), the manifold Mk is homeomorphic to S but not diffeomorphic.

7 7 Proof. By Lemma 4, λ(Mk ) 6≡ 0 (mod 7), hence by Corollary 1, Mk is not the boundary of an 8-manifold with vanishing fourth Betti number, and by Corollary 7 2 Mk has no orientation reversing self diffeomorphism. However, any manifold diffeomorphic to S7 clearly has both of these properties; either fact suffices to 7 7 prove that Mk is not diffeomorphic to S . However, as shown from Lemma 3, 7 7 Mk is homeomorphic to S .

4 Moving Forward

Milnor ends his paper with a miscellaneous result, a consequence of his con- struction that showed that one of two unexpected properties must hold: Theorem 4. Either (a) there exists a closed topological 8-manifold which does not possess any differentiable structure; or (b) the Pontrjagin class p1 of an open 8-manifold is not a topological invariant.

He proves this simply by considering the topological manifold given by col- 8 lapsing the boundary of Bk to a point. He states that he has no idea which of the two alternatives holds; however, in fact, in principle both are true. Shortly after the publishing of the landmark paper, examples were constructed of first topo- logical 10-manifolds by Kervaire [4] and then topological 8-manifolds [5] that did not admit any differentiable structure. Moreover, the topological invariance of Pontrjagin classes was also shown to be false, for example in [6]. The existence of these new, bizarre kinds of objects paved the way for nu- merous advancements; classification of exotic spheres and other exotic smooth structures became an important area of research. In the 1960s, exotic spheres were classified in a number of dimensions. One can regard the set of smooth structures as a , with the underlying set being the diffeomorphism classes of smooth manifolds homeomorphic to the sphere, and the operation being . With this setup, the monoid of smooth structures is in fact an , isomorphic to the monoid of h-cobordism classes of oriented homotopy spheres [7]:

5 Definition 1. Two n-manifolds M and N are h-cobordant if there is an (n+1)- manifold W such that the boundary of W is the disjoint union: ∂W = M t N, and M and N are deformation retracts of W . A homotopy n-sphere is, of course, a closed manifold with the homotopy type of Sn. It is easy to show that h-cobordism is an equivalence relation. The set of h-cobordism classes of oriented homotopy n-spheres forms an abelian group under the connected sum operation [7], which is denoted Θn. The main classification result is due to Smale [8], essentially proving that the monoid of n smooth structures on S is isomorphic to Θn for n 6= 3, 4. The fortunate thing about this isomorphism is that the orders of the Θn can be calculated via results in stable homotopy theory giving us the following sequence:

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 |Θn| 1 1 1 1 1 1 28 2 8 6 992 1 3 2 16256 2 16 16 523264 24

In particular, we see our case n = 7 is actually cyclic, of order 28, giving us a full classification. The only unsolved case is the case n = 4, for which it is not even known if the number of smooth structures is finite.

5 Conclusions

Milnor’s classic paper is wholly readable, and extremely elegant, and the subject is fascinating. For a really cool animation of the fiber bundles in question, one should take a look at Niles Johnson’s website [9].

References

[1] J. Milnor, On Manifolds Homeomorphic to the 7-Sphere, Ann. Math., Vol. 64 No. 2, (1956) pp. 399-405 [2] R. Thom, Quelques propri´et´esglobale des vari´et´esdiff´erentiables, Comment. Math. Helv., 28, (1954), pp. 17-86 [3] F. Hirzenbruch, Ueber die quaternionalen projektiven R¨aume, S.-Ber. math.- naturw. Kl. Bayer. Akad. Wiss. M¨unchen (1953), pp. 301-312

[4] M. Kervaire, A Manifold which does not admit any Differentiable Structure, Comment. Math. Helv., 35, (1961) pp. 1-14 [5] I. Tamura, 8-Manifolds admitting no differentiable structure, J. Math. Soc. Japan, Vol. 13, No.4, (1961) pp. 377-382

[6] N. Shimada, Differentiable structures on the 15-sphere and Pontrjagin classes of certain manifolds, Nagoya Math. J. 12 (1957) pp. 59-69

6 [7] M. Kervaire; J. Milnor, Groups of Homotopy Spheres: I, Ann. Math., Vol. 77, No. 3. (1963), pp. 504-537 [8] S. Smale, On the Structure of Manifolds Amer. J. Math., Vol. 84, No. 3, (1962) pp. 387-399

[9] N. Johnson, http://www.nilesjohnson.net/seven-manifolds.html

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