MILNOR'S CONSTRUCTION OF EXOTIC 7-SPHERES RACHEL MCENROE Abstract. In this paper, I will provide a detailed explanation of Milnor's construction of exotic 7-spheres. 3 4 The candidate manifolds will be constructed as total spaces of S bundles over S , denoted Mh;l. The subset of these candidates satisfying the condition h + l = ±1 will be shown to be topological spheres by Morse Theory. A subset of these that do not satisfy (h−l)2 ≡ 1 (mod 7) will be shown to not be differential spheres, by the Hirzebruch Signature Theorem and some other results from the theory of characteristic classes. Finally, I will discuss some interesting applications of exotic smooth structure throughout mathematics. Contents 1. Motivation 1 2. Hopf Fibrations 2 2.1. The Complex Hopf fibration 2 2.2. The Quaternionic Hopf fibration 2 3. S3 bundles over S4 3 4. The exotic sphere is a topological 7-sphere 5 4.1. Morse Theory 5 4.2. Reeb's Theorem 6 4.3. Proof that Mh;l is a topological sphere if h + l = −1 6 5. Characteristic Classes 8 5.1. Euler Class 8 5.2. Chern Class 8 5.3. Pontryagin Class 9 6. Mh;l is not Diffeomorphic to the 7-sphere 9 6.1. Calculating p1(ξh;l) 9 6.2. Calculating the first Pontryagin class of Kh;l 12 6.3. Applying the Hirzebruch Signature Theorem 13 7. Conclusion 14 8. Further discussion of Exotic Structure 14 Acknowledgments 15 References 15 1. Motivation The existence of exotic smooth structure is a sign of the limits of human intuition. Geometric intuition is based on generalization of everyday objects. Surfaces, or 2-manifolds, are a class of such objects. One of the most `natural' spaces is a sphere. We can give several different `natural' definitions of spheres. A selection follow. Definition 1.1. A standard sphere is a hypersurface cut out by an equation of the form n X 2 (1.2) xi = 1 i=1 where the xi are all real numbers. Date: DEADLINES: Draft AUGUST 17 and Final version AUGUST 28, 2015. 1 2 RACHEL MCENROE Definition 1.3. A homotopy sphere is a topological manifold M of dimension n with H0(M; Z) = n ∗ H (M; Z) = Z, H (M; Z) = 0 otherwise and π1(M) = 0. Definition 1.4. A topological sphere is a topological manifold that is homeomorphic to a standard sphere. The intuition behind the definition of a topological sphere is that a topological sphere can be continuously deformed into a standard sphere. Definition 1.5. A differential sphere is a smooth manifold that is diffeomorphic to a standard sphere. In other words, a differential sphere can be `smoothly deformed' into a standard sphere or, equivalently, the calculus done on a differential sphere is equivalent to the calculus done on a standard sphere. For the case of surfaces, homotopy spheres, topological spheres, and differential spheres are all equivalent. Therefore, it seems obvious that all of these topological equivalences will hold in higher dimensions. In fact, by the Poincare Conjecture, homotopy spheres and topological spheres are equivalent in all dimensions. On the other hand, topological spheres and differential spheres are not. The simplest known example of a smooth manifold that is a topological sphere, but is not a differential sphere, occurs in 7 dimensions. In this paper, we will construct this example. 2. Hopf Fibrations In this section, we will discuss Hopf fibrations, which are fiber bundles in which the base space, the fibers, and the total space are all spheres. 2.1. The Complex Hopf fibration. We will start with the classical Hopf fibration, which can be realized in 3 dimensions. This is the fiber bundle with fibers homeomorphic to S1 and base space S2. The total space of this fiber bundle is a 3-sphere. So, we represent this fiber bundle with the following diagram: S1 S3 S2 The 1-sphere S1 is just the set of complex numbers with norm 1, and the 3-sphere is the set of ordered pairs of complex numbers with norm 1. The 2-sphere is the one point compactification of the complex plane, 1 which is the complex projective line CP . We can explicitly write down the projection map in the complex Hopf fibration as π :(z1; z2) 7! [z1; z2]. 1 2 For example, note that the preimage of [1; 0] 2 CP under the projection map π is fz1 2 C kz1k = 1g, which is a copy of S1, and therefore is a fiber embedded in the total space. Note that every two fibers are linked with linking number one in the total space. This is a qualitative difference from the relationship between the fibers in the fiber bundle whose fibers and base space are copies of S1 and whose total space is a torus. In that case, the linking number of any two fibers is zero. 2.2. The Quaternionic Hopf fibration. There are in fact only four Hopf fibrations, which correspond to the four division algebras, which is a result of Adams. The four division algebras are R, C, H, and O. The quaternions are denoted H, and they have non-commutative multiplication. Formally, Definition 2.1. H is the set of numbers of the form a + bi + cj + dk, where ij = k = −ji and addition is component-wise. The octonions are a division algebra with non-associative multiplication. So, as the `size' of the division algebra increases, the division algebra has fewer basic properties like commutativity. The quaternionic Hopf fibration is formed exactly analogously to the complex, or classical, Hopf fibration. So it has the following diagram: S3 S7 S4 MILNOR'S CONSTRUCTION OF EXOTIC 7-SPHERES 3 where S3 is the set of unit quaternions and S7 is the set of pairs of quaternions with unit norm. The maps are also analogous to the complex case: π :(z1; z2) 7! [z1; z2]. The preimage of the point [1; 0] under the 2 projection map is fz1 2 H kz1k = 1g, which is a 3-sphere embedded in the total space. The only difference is in the definition of projective space. Projective space is defined by putting an equivalence relation on any two points that are on the same line through the origin. That is: (2.2) (z1; z2) ∼ (λz1; λz2) Since multiplication in the quaternions is not commutative, the definition of quaternionic projective spaces includes a choice of default multiplication. We choose left-multiplication. This is a degree of freedom that did not occur in the complex Hopf fibration and is the reason why the simplest known example of exotic smooth structure occurs in seven dimensions, instead of three dimensions. 3. S3 bundles over S4 To construct the exotic 7-sphere, we want to look at different ways of constructing sphere bundles of S3 over S4. We will need some tools from algebraic topology to do this, which will be developed in this section. We will construct this family explicitly and show that each element of the family corresponds to an element of the group Z ⊕ Z. We will show that this is π3(SO(4)). First, we note that S4 can be covered by an atlas with only two charts, just like any other topological sphere. An R4 bundle over each chart is trivial since the charts are contractible, so the only way to get a non-trivial bundle is to have some `twisting' in the gluing maps between the fibers. We want to construct a family of such gluing maps that give a variety of total spaces. 4 We will choose the two charts to be all of S except the south pole, which will be denoted U1 and all of 4 4 S except the north pole, which will be denoted U2. Now I define a map that takes each chart to R , which I can identify with the quaternions. I will write this correspondence in the following diagrams to emphasize to the reader the importance of this correspondence for this construction. The maps are called φ1 and φ2 respectively. 4 (3.1) φ1 : U1 ! R = H (3.2) [z; 1] 7! z 4 (3.3) φ2 : U2 ! R = H (3.4) [1; w] 7! w Note that the transition map is: −1 (3.5) φ2 ◦ φ1 : H − f0g ! H − f0g 1 (3.6) z 7! z which is exactly what we expect. We will start with the simplest example, which is that of the standard quaternionic Hopf fibration of an S3 bundle over S4. The total space of this fiber bundle is given by 2 1 (3.7) f (x; y); [z; w] xw = zy; z; w 6= 0g ⊂ H × HP The projection from the total space to S4 is denoted π. We want to construct the local trivializations of the fiber bundle. These need to be done on each chart of the base space and then matched. First, we will consider the preimage of the the first chart of the base space in the total space. −1 (3.8) π (U1) = f((x; y); [z; 1]) j x = λz; y = λ for some λ 2 Hg For the first chart, the local trivialization is: −1 (3.9) ρ1 : π (U1) ! φ1(U1) × H ((x; y); [z; 1]) 7! (z; y)(3.10) Similarly the preimage of U2 in the total space of the Hopf fibration is: −1 (3.11) π (U2) = f((x; y); [1; w]) j x = λ, y = λw for some λ 2 Hg 4 RACHEL MCENROE The local trivialization is: −1 (3.12) ρ2 : π (U2) ! φ2(U2) × H ((x; y); [1; w]) 7! (w; x)(3.13) Now we want to compute the transition map of the trivializations −1 (3.14) ρ2 ◦ ρ1 : φ1(U1 \ U2) × H ! φ2(U1 \ U2) × H ! 1 (3.15) (z; y) 7! ; yz z 1 So what we are doing is gluing (z; y) to ( z ; yz).