Embeddings & Immersions of Manifolds: Whitney-Stiefel classes & Smale’s Theorem Hayley Wragg University of Sussex [email protected]

Introduction

Differential Topology is the study of smooth manifolds and their differentiable structures. In this project I will present some of the most oustanding and surprising results obtained in the field over the last fifty years mostly on the problem of embedding and immersions of manifolds. In order to do so I will introduce some technical device including vector bundles and Stiffel-Whitney (Chern-Pontryagin) classes that serve as key ingredients in formulating necessary and sufficient conditions for such immersions to exist. As a remarakble consequence I will present the extraordinarily surprising theorem of Steven Smale (1954) on eversion of spheres in three space which astounded the mathematical community for decades!

Now consider the same Classification of Surfaces, Genus & Euler Whitney-Stiefel Characteristic Classes process with multiple Characteristic In general a Characteristic class is a cohomology class associated guides. to a vector bundle attached to a topological space. What concerns Let us start by looking at surfaces, that is, 2-dimensional smooth us most in this research, and the problem of immerssions of man- Since the surface closed manifolds. Then by a classical result in topology each such ifolds, are primarily the Stiefel-Whitney and the Chern-Pontryagin can pass through itself, surface is diffeomorphic to a sphere with g handles attached to it. classes. To put this into context we present the following funda- each guide can be turned The number g here is a topological invariant and is called the genus mental existence result on the Stiefel-Whitney class: There is a co- inside out at the same of the surface. A related and equally useful notion is that of Euler time. homology class wi(ξ) on each vector bundle ξ of a manifold where characteristic χ defined as χ = 2 − 2g. So the sphere (g = 0) has i i th wi(ξ) ∈ H (B(ξ); Z/2), i = 0, 1, 2, ..., H (Bξ); Z/2) is the i sin- χ = 2 and the torus (g = 1) has χ = 0 and all other surfaces have We then have that gular cohomology groups of B with coefficients in z/2 wi(ξ) is the negative χ. the entire Sphere has Stiefel-Whitney class of ξ wi(ξ) satisfies the following ∗ been turned inside out 1. If a bundle map covers f:B(ξ) → B(η) then : wi(ξ) = f (wi(η)). without making any 2. If ξ and η are vector bundles over the same base space, then holes or tight creases. g=0 g=1 g=2 g=3 g=4 k P wk(ξ + η) = wi(ξ) ∪ wk−i(η). i=0 One can think of these Characteristic classes as obstruction cocyles Embeddings and Immersions of Manifolds asscociated with the extendibility of maps from the manifolds and its corresponding vector bundle to the Stiefel manifold Vn,k.In what An immersion is a mapping of one smooth manifold into another follows we show how this device can be used to solve the eversion Poincare´-Hopf Theorem 2 3 whose differential satisfies a certain non-degeneracy condition. An problem for the 2-sphere S ⊂ R . embedding is an immersion which is additionally injective. This Let X be a smooth vector field on a compact manifold Mn. If X can be easily seen in the case of the circle in the plane. Any smooth has only isolated zeros then, Index(X)=χ(Mn). Here closed curve in the plane is an immersion of the circle where as the n The n-Sphere Inversion Problem X only embeddings of the circle are smooth Jordan curves! One of the χ(Mn) = (−1)iβ (M), (1) fundamental problems in differential topology is to characterise, for i i=0 a given pair of manifolds, all possible classes of immersion of one Take a circle, try to invert it inside out without leaving the plane. manifold into the other. An indespensible tool for doing this are the This challenging task turns out to be impossible! For many years it n i n where βi is the i-th Betti number on M : βi = dimRH (M ). As so-called ”Characteristic classes” described below. was believed that the same is true for the 2-sphere. However much χ(Mn) is a topological invariant of Mn then so is the index of X! to the surprise of the mathematical world, Steven Smale, using tools from differential topology proved that it is possible to invert a 2- Some Classes of n-Manifolds Sphere inside out in the 3-space. Technically speaking this means that there exists a homotopy within the class of immersions of the 3 For the sake of clarity here we list some important classes of mani- 2-Sphere in R , starting from the identity and terminating at the an- folds that frequently occur in the theory: tipodal map. Existence of an immersion n 1. Sphere S . Using similar techniques Smale managed to give a complete proof According to Whitneys embedding theorem every smooth mani- of the Poincare´ conjecture in dimensions n ≥ 5. More precisely: n 2n 2. Projective spaces: n fold M embedds smoothly in R and immerses smoothly into If M is a differentiable homotopy sphere of dimension n ≥ 5, 2n−1 (a) Real Pn(R), (b) Complex Pn(C), (c) Quaternionic Pn(H). n n n R . The device of characterstic classes and the vanishing of the then M is homeomorphic to S . In fact, M is diffeomorphic to a corresponding cohomology co-cyles dictates whether one can re- 3. Grassmann and Stiefel manifolds Gn,k,Vn,k. manifold obtained by gluing together the boundaries of two closed duce the dimensions further in the target Euclidean space (e.g., if 4. Orthogonal and Special Orthogonal Groups O(n), SO(n). n-balls under a suitable diffeomorphism. m m+k wi(M ) =6 0, i < k Then M can not be immersed in R ). 5. Unitary and Special Unitary Groups U(n), SU(n).

Method Vector Bundles Examples and results Consider a Sphere which can be bent, A vector bundle over a manifold is an assignment of a vector spaces n n+1 and stretched, and pass through itself. 1. If M is parallelisable then it can be immersed in R . (real or complex) to each point of the manifold. Whilst locally the But we can not make tight creases. 2. Every closed 3-manifold can be immersed in 4. structure of a vector bundle is dictated by the structure of the vec- R n 2n−2 tor space the picture is completely different globally. The study of 3. If n ≡ 1(4) then M can be immersed in R . We can not simply pass the sphere vector bundles over a manifolds says a lot about the topology and ( ) not 2n−2 n = 2s through itself since this creates a tight 4. Pn R can be immersed in R with . immersions of the manifold. We proceed by first presenting the pre- 3 4 crease. 5. P2(R) can not be embedded in R but can in R . cise definition leaving the discussion and some prominent exmaples Now imagine the Sphere is made up (Note that a manifold is said to be parallelisable iff its tangent bun- of vector bundles to the next section. A real vector bundle ξ over a 1 of a series of circles which we give a dle is trivial. As an example the only parallelisable spheres are S , base space B consists of the following: 3 7 wavy boundary. S and S and no more!) 1. A topological space E=E(ξ) reffered to as the total space. We can then stretch this circle to 2. A projection map π:E→B. create most of our sphere then use a 3. the structure of a vector space ∀ b∈B over the real numbers in dome at the top and bottom to show −1 the poles. the set π (b). References Note that it is the last condition above that describes the local structure of a bundle as that of its corresponding vector space. 1. J. Milnor, J. Stasheff, Characteristic Classes, Princeton Univer- sity Press, 1974. We represent one of these waves with 2. J. Lee, Introduction to Smooth Manifolds, Springer, 1950. The Tangent Bundle a guide, with the poles at the top and bottom, we now want to turn this 3. M. Adachi, Embeddings and Immersions,AMS, 1993. The Tangent bundle τM of a manifold M is a vector bundle in which guide inside out. 4. M. Hirsch, Differential Topology, Springer, 1980. the total space DM is formed of the pairs (x, v) with x ∈ M and v 5. G.Bredon, Topology and Geometry, Spinger, 1997. in the tangent space to M at x. The projection map π : DM → M 6. J. Rotman, Introduction to Algebraic Topology, Springer, 1988. such that π(x, v) = x and the vector space structure π−1(x) defined To start we pass the poles through by t1(x, v1) + t2(x, v2) = (x, t1v1 + t2v2) each other, but not far enough to form 7. J. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer, a crease from the loop. 2006. 8. I. Madsen, J. Tornehave, From calculus to Cohomology, Cam- The Normal Bundle Then rotate the poles once in op- bridge University Press, 1998. n posite directions. The Normal bundle ν of a manifold M ⊂ R is the vector bun- n dle where the total space E ⊂ MxR is formed of the pairs (x, v) This untwists the loop and our Acknowledgements where v is orthogonal to the tangent space of M at x. The projec- guide has been turned inside out. tion map π : E → M. The vector space structore in π−1(x) defined Ali Taheri, Miroslav Chlebnik, Stuart Day, Charles Morris, Abim- by t1(x, v1) + t2(x, v2) = (x, t1v1 + t2v2) bola Abolarinwa.