Projections of Immersed Surfaces and Regular Homotopy

Transcription

1 U.U.D.M. Project Report 2009:24 Projections of Immersed Surfaces and Regular Homotopy S. Mohammad Hossein Bani-Hashemian Examensarbete i matematik, 30 hp Handledare och examinator: Tobias Ekholm December 2009 Department of Mathematics Uppsala University

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3 Matematiska Institutionen Master s Thesis Projections of Immersed Surfaces and Regular Homotopy By: S. Mohammad Hossein Bani-Hashemian Advisor: Tobias Ekholm November 2009

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5 Abstract This thesis is based on U. Pinkall s study of the classification of immersions of compact surfaces into R 3 up to regular homotopy. The main idea of the classification is to associate to any immersion f a quadratic form q f on the first homology group of the underlying surface Σ with Z 2 coefficients, whose associated bilinear form is the nondegenerate intersection form in H 1 (Σ, Z 2 ), having the property that it depends only on the regular homotopy class of f. In the case of orientable surfaces q f turns out to be a Z 2 -quadratic form. In this thesis we construct the Z 2 -quadratic form using the notion of Spin - structure, and via D. Johnson s correspondence between Spin - structures on a surface and Z 2 -quadratic forms on the first homology group of the surface. Then by studying the relation between surface immersions into 3-space and their projections to a 2-plane, we give a formula for computing the value of the quadratic form on any homology class c H 1 (Σ, Z 2 ), which we will use to construct an example of two nonregularly homotopic immersions of the 2-dimensional torus T 2 into R 3 with identical plane projections. ii

6 Acknowledgments First, I would like to extend my sincere thanks to my supervisor, Professor Tobias Ekholm, for proposing the thesis and for sharing his knowledge throughout the thesis project. I am truly thankful to him for his kind support, patience and flexibility. I would also like to express my gratitude to all my lecturers at the department of mathematics at Uppsala University, to whom I am greatly indebted. My special thanks go to my father and brothers for their support and constant encouragement while I pursued my master s study, and finally, I dedicate this thesis in loving memory of my mother. iii

7 Contents Acknowledgments iii 1 Introduction 1 2 Preliminaries and Background Stable Maps Fiber Bundles Associated Bundles Characteristic Classes Spin - structures Proof of the Lifting Theorem 17 4 Spin -structures and Classification of Immersions 19 A Quadratic Forms and the Arf -invariant 23 B Spin Groups 24 References 25 iv

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9 1 Introduction Let M and N be differentiable manifolds of dimensions m and n respectively, consider a differentiable (smooth) map f : M N, a point p M is a singular point of f if rankdf p < min (m, n), where df p : T p M T f(p) N is the differential of f. The set of all singular points of f is denoted by S f. f is called immersion if df p at all points p M is injective (i.e. rankdf p = m), then f(m) is called an immersed manifold. Given two immersions f, g : M N, recall that we define a regular homotopy between f and g to be a continuous map H : M [0, 1] N ;(p, t) H t (p) such that for every p M, H 0 (p) = f(p) and H 1 (p) = g(p) and for any t [0, 1], H t is an immersion and its derivative dh t varies continuously with t, in other words, H defines a continuous family of immersions between M and N connecting f and g which are parametrized by t [0, 1]. If f, g are two immersions of a surface into space, then their corresponding immersed surfaces f(m) and g(m) are said to be regularly homotopic if there is a diffeomorphism ϕ of M such that g is regularly homotopic to f ϕ, note that f and f ϕ are just two different parametrizations of the same surface. Now we give another definition for regular homotopy between immersions. Let Imm(M, N) denote the space of all immersions of M into N endowed with the Whitney C 1 topology, a path in Imm(M, N) is called a regular homotopy. Two immersions that belong to the same path connected component of Imm(M, N) are said to be regular homotopic. Theorem (Smale-Hirsch). Let M and N be two smooth manifolds with M compact and dimm < dimn, then the map d : Imm(M, N) Mon(TM, TN) ; f df is a homotopy equivalence, where Mon(T M, T N) is the space of all fiber-wise vector space monomorphisms from TM into TN. According to [15] and based on the above theorem, given a closed surface Σ we can correspond to any f Imm(Σ, R 3 ) a mod 4 quadratic form, q f, on the first homology group of the underlying surface Σ with Z 2 coefficients, whose associated bilinear form is the nondegenerate intersection form in H 1 (Σ, Z 2 ), having the property that it depends only on the regular homotopy class of f as follows : Let c H 1 (Σ, Z 2 ) be a homology class represented by an embedded loop (simple closed curve) α : S 1 Σ. Assume that N α is a tubular neighborhood of α(s 1 ) in Σ, then define q f (α) to be the linking number between f α and f Nα. The equivalence class of the quadratic forms is completely determined by their Arf- invariant g Arf (q f ) := q f (a i )q f (b i ), i=1 where a 1,...a g, b 1,...b g is a symplectic basis for H 1 (Σ, Z 2 ). As the main result, Pinkall in [15] proved that: 1

10 Theorem (Pinkall). Let Σ be a compact surface, f, g : Σ R 3 two immersions. Then (i) f and g are regularly homotopic if and only if q f = q g ; (ii) The corresponding immersed surfaces f(σ) and g(σ) are regularly homotopic if and only if Arf (q f ) = Arf (q g ). This theorem gives a classification of immersions of compact surfaces into R 3 up to regular homotopy, allowing diffeomorphisms of the source surface. In this thesis we study the relation between surface immersions into 3-space and their projections to a 2-plane. First, we study whether a projection of a smooth surface onto the plane R 2 can be lifted to an immersion into Euclidean three-space or not. In [6] Haefliger established a useful result, which gives necessary and sufficient conditions for the existence of an immersion lift to R 3 over a projection of a smooth surface onto the plane R 2 via the projection map R 3 R 2, for a class of maps called stable maps : Let C (Σ, R 2 ) be the space of all smooth maps of Σ into R 2, endowed with the Whitney C topology. Elements f and g of C (Σ, R 2 ) are said to be equivalent, if there exist diffeomorphisms ϕ : Σ Σ and Φ : R 2 R 2 such that g = ϕ f Φ 1, and f C (Σ, R 2 ) is called stable if there exists a neighborhood of f in C (Σ, R 2 ) such that each member of the neighborhood is equivalent to f. It is well-known that the set of stable maps is open and dense in C (Σ, R 2 ). That is, every smooth map of Σ into R 2 can be approximated arbitrary closely (for any number of derivatives) by a stable map. A theorem due to Whitney states that the singularities of the projection of a smooth surface onto a plane are of three types: (a) a regular 1 point on the smooth curves corresponding to a fold of the surface; (b) a cusp corresponding to a pleat; and (c) a double point corresponding to the projection of two transversal folds. Haefliger proved: Theorem 1. Let f : Σ R 2 be a stable map from a smooth and closed surface Σ into the plane and pr : R 3 R 2 be the projection map. Then there exists an immersion ˆf : Σ R 3 which satisfies f = pr ˆf, if and only if each connected component of its singular set S f, say C i, has an orientable (or non-orientable) neighborhood provided that the number of cusps on C i is even (resp. odd). Now we get back to Pinkall s idea. Let Σ be a compact orientable surface, in this case the Z 4 -quadratic form assigned to each immersion reduces to a Z 2 -quadratic form. Here we 1 Here by regular point we mean a point which is not the intersection of two curves, this shouldn t be mistaken with a regular point at which the map is nonsingular. 2

11 construct the mod 2 quadratic form using the notion of Spin-structure. A Spin-structure on Σ is by definition a non-trivial 2-sheeted covering map f : P SO(Σ), where SO(Σ) is the S 1 - bundle of all positively-oriented orthonormal frames on K whose restriction to each fiber of P is isomorphic to the standard 2-sheeted covering S 1 S 1 ; or equivalently is a cohomology class in H 1 (SO(Σ), Z 2 ), whose restriction to each fiber is a generator of H 1 (S 1, Z 2 ). Any immersion f : Σ R 3 induces a Spin-structure on Σ, and via a method suggested by Denis Johnson in [7], a mod 2 quadratic form on H 1 (Σ, Z 2 ) can be associated to the Spin- structure, this is precisely the quadratic form, that we were looking for. In the last step, we give a formula to compute the value of the quadratic form on any homology class c in H 1 (Σ, Z 2 ) (and hence its Arf - invariant): Let α : S 1 Σ be a smooth simple closed curve on Σ and the representative of a homology class c in H 1 (Σ, Z 2 ), and by α we denote its orthogonal projection on the plane R 2. By studying the orthogonal projection of the immersed surface we associate a loop λ(α) in SO(3) to α. Then q f (c) = [λ(α)] + 1, (1.1) where [λ(α)] denotes the homotopy class of λ(α) in π 1 (SO(3)) = Z 2. The rest of this thesis is organized in the following manner: In Section 2 we gather several definitions and results, often without proofs, as prerequisites to our study, in three subsections, the first subsection provides a brief background from singularity theory that we need to understand the proof of Theorem 1. In the second subsection we remind the reader of a few basic concepts aiming at introducing the notion of Spin-structure and some related results, in the last subsection, where we give three equivalent definitions of Spin- structure on a vector bundle, in particular, a definition based on the properties of 2-sheeted covering spaces, which we use in Example 2.2 to find the Spin-structures on the 2 - dimensional torus. We also explain the one to one correspondence between quadratic forms and Spinstructures proved by D. Johnson and use his idea to obtain the values of the quadratic forms associated to the four Spin- structures on the torus. Section 3 deals with the Haefliger s proof to the lifting theorem, Theorem 1, originally proved in [6], here we make this proof more explicit by adding a few simple details; and in the final section after explaining how to associate a mod 2 quadratic form to an immersion using the Spin-structures on the source surface, we derive formula (1.1) for computing the values of the corresponding quadratic form to the induced Spin-structure on mod 2 homology classes. Finally, we use this formula to 3

12 construct an example of two non-regularly homotopic immersions of the torus into R 3, with identical plane projections. 4

13 2 Preliminaries and Background This section presents some relevant background concepts and definitions needed to understand the main work done in this thesis. In the first subsection we review some basic definitions 2 and well-known results from singularity theory. In the second subsection, we recall the standard definition of a fiber bundle as well as some related notions that we will need to introduce Spin-structures in the last subsection. 2.1 Stable Maps Two smooth maps f, g : M N between differentiable manifolds M and N are said to have contact of order r at p M if f(p) = g(p) and in some coordinate charts in M and N centered at p and f(p) respectively, all the partial derivatives at p of f and g up to order r are equal. This is an equivalence relation on C (M, N), whose equivalence classes are called r -jets. An r - jet at a fixed point p M represented by some map f is called the r -jet of f with source p and target f(p) and is denoted by jpf. r We denote the set of r - jets with a fixed source by Jp r(m, N) and call Jr (M, N) := p M Jr p (M, N) the space of r -jets of mappings from M to N. For any subset U of J r (M, N), set M(U) := {f C r (M, N) : j r f(m) U}, where j r f(m) = { jp rf : p M}. Then the family of all sets M(U), where U ranges over all open subsets of J r (M, N) generates a topology on C (M, N) called the Whitney C r topology. If we denote this topology by W r, then W = r=0 W r form a basis for a new topology on C (M, N) called the Whitney C topology. Take the space C (M, N) endowed with the Whitney C topology. Let Diff(M) and Diff(N) denote the group of diffeomorphisms of M and N respectively and define the leftright action of the group Diff(M) Diff(N) on C (M, N) by (ϕ, Φ)f := ϕ f Φ 1, for any f C (M, N), ϕ Diff(M) and Φ Diff(N), then Definition. A smooth map f from a closed (compact, without boundary) manifold M to a manifold N is said to be stable, if its orbit under the left-right action is open. In dimension two we have the following famous theorems and results: (for proofs see [16], [4] or [8]) 2 We borrow some of the definitions mainly from [1] and [10]. 5

14 Theorem (Whitney) A map f : M N between 2-dimensional manifolds is stable at a point p in M if and only if the map can be described in some local coordinates (x 1, x 2 ) centered at p and (y 1, y 2 ) centered at f(p) in one of the following forms: (i) (x 1, x 2 ) (y 1, y 2 ) = (x 1, x 2 ) (ii) (x 1, x 2 ) (y 1, y 2 ) = (x 2 1, x 2) (iii) (x 1, x 2 ) (y 1, y 2 ) = (x x 1 x 2, x 2 ) (p is a regular point); (p is a fold point); (p is a cusp point). Also another theorem due to Whitney states as follows: Theorem Let M be a 2-dimensional manifold, a smooth map f : M R 2 is stable if and only if the following conditions are satisfied: (i) f is stable at every point in M; (ii) images of folds and cusps do not intersect; (iii) f {folds} is an immersion with normal crossings, i.e. has only double self intersections at which the two tangent lines are distinct and form an angle. So in this case, S f consists of smooth curves called general folds on which the cusp points are isolated. The image of general folds are smooth curves except for cusps at the image of cusp points (see figures 1 on the facing page and 2 on page 8). It is known that if M is compact (closed and without boundary), then there are a finite number of general folds and cusps (the interested reader can see [11] for more results regarding the number of cusps). In the space of smooth maps from compact 2- dimensional manifolds into the plane the property of being stable is generic from the topological point of view. 6

15 x y x y z z y z x y x z Figure 1: The Whitney cusp, which is the singularity of the map (x,y) (z,w) = (x 3 + xy,y). The pictures on the bottom represent the projection of the boundary of a small neighborhood of the general fold on the xy-plane (left) and zw-plane (right). 7

16 z x y z y x y z x Figure 2: A surface with two cusp shaped regions. The two pictures on the top-right represent the singularity of the map f : R 2 R 2 ; (x,y) ((0.05y x)(x 2 + y 2 3y),y) and the projection of the boundary of a neighborhood of the general fold on two coordinate planes. The picture on the bottom-right shows the bundle of kernels of df on the fold curve. 8

17 2.2 Fiber Bundles Let E, B and F be topological spaces and assume that B is connected and let G be a subgroup of Homeo(F). By a fiber bundle over the base space B, with total space E, fiber space F and structure group G, we shall understand a continuous surjective map π : E B such that for every point x in B there is an open neighborhood U x of x called a trivializing neighborhood and a homeomorphism Φ Ux : π 1 (U x ) U x F called a local trivialization of the bundle so that the following diagram commutes: Φ Ux p 1 (U x ) U x F π pr 1 U x and there is an open cover {U α } α A of B with a system of local trivializations {U α, Φ α } α A such that for each α, β A, the composition Φ β Φ a 1 : (U α U β ) F (U α U β ) F ; (x, v) (x, ψ αβ (x)v) gives the transition function ψ αβ : U α U β Homeo(F) ; x U α U β ψ αβ (x) taking values in G. From any fiber bundle consisting the above information one can get a sort of exact sequence of spaces F E B giving rise to a long exact sequence of homotopy groups. Now let F be a field, a F-vector bundle of rank k is a fiber bundle with fiber space F k, whose fibers, E x := π 1 (x), x B, are equipped with the structure of k - dimensional F-vector spaces and moreover the restriction of each local trivialization to any fiber is an isomorphism of vector spaces. In this case the structure group of the bundle is a subgroup of GL(k, F), in particular, if the structure group of a rank k vector bundle can be reduced to the group SO(k), it is said to be orientable. By a section of a fiber bundle π : E B we mean a continuous map σ : B E such that π σ = id B. Also recall that a fiber map ( φ, φ) between two fiber bundles π : E B and π : E B is a pair of maps φ : E E and φ : B B such that π φ = φ π. If B = B, we simply denote the fiber map by E φ E. If φ and φ are both continuous maps (homeomorphisms) the fiber bundles are said to be homomorphic (isomorphic). we denote two isomorphic fiber bundles by (E π B) (E π B ). One of the most important examples of fiber bundles is a principal bundle: Definition. Let M and P be differentiable manifolds and G be a Lie group, a principal G- bundle over M is a differentiable fiber bundle π : P M, whose fiber space is identical to the structure group G, together with a right action µ of G on P, µ : P G P ; (p, g) µ g (p) := pg, such that the following holds: (i) µ is a fiber-wise action. That is, for all p P and g G, π(p) = π(pg) ; 9

18 (ii) µ is a smooth right action, i.e. the induced map µ : G Diff(P) ; g µ g s a group homomorphism; (iii) G acts freely and transitively on each fiber P x, x M, i.e. for each p, p P x, there is a unique g G such that µ g (p) = p ; (iv) There is an open cover {U α } α A of M with local trivializations { ΦUα : π 1 (U α ) U α G } α A, such that for each α A, g G and p π 1 (U α ) P, Φ Uα (pg) = (x, h g), where π(p) = x, Φ Uα (p) = (x, h) and h g denotes the group action. A bundle map ( φ, φ) between a pair of principal G-bundles is called a principal bundle map if φ(pg) = φ(p)g. Examples 2.1. If π : E M is a real vector bundle of rank k, M an n-dimensional smooth manifold, and for any x in B, F x (E) denotes the set of all k -frames (ordered basis) of the fiber E x, then one can show that the projection map π F : F(E) := x M F x(e) M ;v F x (E) x can be viewed as a principal GL(k, R)-bundle called the principal frame bundle associated to the vector bundle. Similarly, if M is a Riemannian manifold, and O(M) := x M O x(m), where O x (M) is the set of all orthonormal frames of T x M, the map τ O : O(M) M ;v O x (M) x gives a structure of a principal O(n)-bundle called the orthonormal frame bundle of M. In the case of an oriented Riemannian manifold M, taking SO(M), which is defined to be the disjoint union of all oriented orthonormal frames of tangent spaces over M, as the total space, we obtain a principal SO(n)-bundle, τ SO : SO(M) M, called the oriented orthonormal frame bundle. As another example, if M is a regular covering space of M, then the corresponding covering map π : M M is a principal A( M)-bundle, where A( M) denotes its group of covering automorphisms. In particular, any universal covering π : M M is a principal π 1 (M)-bundle, where π 1 (M) is the fundamental group of M Associated Bundles Given a vector bundle π : E M with structure group G and transition functions {ψ αβ }, determined by the given local trivialization {U α } α A of E, we construct a principal G-bundle ˆπ : P G M where P G := ( α A U α G)/ with (x, g) U α G (x, ψ βα (x) g) U β G, for any x U α U β. This principal bundle is called the principal G-bundle associated to the given vector bundle. Conversely, given a principal G-bundle π : P M and a vector space V, we can construct a vector bundle as follows: 10

19 Let ρ : G GL(V ) be a linear representation of G in V, E ρ (V ) := P ρ V = P V/ with (p, v) (pg, ρ g 1(v)) and q : P V E ρ (V ) be the quotient map, then π ρ : E ρ (V ) M, that makes the following diagram commute : P V pr 1 P q π E ρ (V ) is a vector bundle with fiber V called the vector bundle associated to the principal G-bundle. M π ρ For example, the (oriented) orthonormal frame bundle of a (oriented) Riemannian manifold is the associated principal G-bundle to its tangent bundle.(where G = SO(n) or O(n), depending on whether the base manifold is oriented or not.) Note that the above constructions are inverse to each other, for instance, let M be an n-dimensional smooth manifold. If ρ is given by ρ A (v) = Av, for any n n real matrix A and v R n, then the vector bundle associated to the principal frame bundle for the tangent bundle τ : TM M of M is isomorphic to TM. That is: (F(TM) ρ R n τρ M) (TM τ M). In the same way, we can recover the tangent bundle from O(M) or SO(M). More generally, via the representation of GL(k, R) in R k given as the above natural way, for any k - dimensional real vector bundle π : E M we have (F(E) ρ R k πρ M) (E π M) Characteristic Classes A characteristic class c of principal G- bundles is a correspondence that associates to every principal G-bundle π : P M over a manifold M a cohomology class c(p) H (M) with some fixed coefficient group, such that for any principal bundle map ( φ, φ) between two principal G-bundles π 1 : P 1 M 1 and π 2 : P 2 M 2 we have φ (c(p 2 )) = c(p 1 ), where φ denotes the induced natural homomorphism of cohomology groups. Similarly we can define the characteristic classes of vector bundles. The characteristic classes of a manifold are, by definition, the characteristic classes of its tangent bundle. For any real vector bundle π : E B (or principal O(n)-bundle, for an arbitrary n) there is a sequence w i (E) H (B, Z 2 ), i 0, of characteristic classes with coefficient in Z 2 (mod 2 characteristic classes) called the Stiefel-Whitney classes satisfying the following conditions: (i) w 0 = 1 H 0 (B, Z 2 ) and w i = 0 for i > rank (E); (ii) if π : E B is a real vector bundle over the same base space, then the Whitney product formula holds for the Stiefel-Whitney classes of the direct sum of the bundles, i.e. k w k (E E ) = w j (E) w k j (E ), j=0 11

20 where denotes the cup product on H (B, Z 2 ). (iii) The Stiefel-Whitney class w 1 (E γ 1) of the canonical line bundle γ 1 : E γ 1 RP 1, where E γ 1 := {(l, v) RP 1 R 2 : v l} is nonzero. the i th element of the sequence is called the i th Stiefel-Whitney class of the vector bundle. The Stiefel- Whitney classes are uniquely determined by the above conditions (see [12]) and provide important criteria for some topological structures of vector bundles, for example it is known that a k - dimentional vector bundle (or a manifold) is orientable if and only if its first Stiefel-Whitney class is zero. It can be also shown an orientable vector bundle (or an orientable manifold) has a Spin-structure (to be defined) if and only if its second Stiefel-Whitney class is zero (for the original proof see [3]). 2.3 Spin - structures Let us assume λ : Spin(n) SO(n) is a 2-sheeted covering (or a universal 2-sheeted covering, if n 3) of the rotation group SO(n). Spin(n) := SO(n) is called the spin group ( see Appendix B ). Note that this is a principal Z 2 - bundle as well. Definition 1. Let π : E M be an oriented 3 n-dimensional real vector bundle and ˆπ : P SO(n) M be its associated principal SO(n)-bundle. A Spin-structure (η, f) on π : E M is a principal Spin(n)-bundle η : P Spin(n) M together with a non-trivial 2-sheeted covering map f : P Spin(n) P SO(n), which restricts to the covering map λ on each fiber. That is the following diagram commutes : P Spin(n) Spin(n) µ P Spin(n) η f λ P SO(n) SO(n) ν covering f M ˆπ P SO(n) where µ and ν denote the corresponding right group actions. Two Spin-structures (η, f) and (η, f ) are said to be equivalent if there exists a principal bundle isomorphism P Spin(n) φ P Spin(n) which is also an equivalence of covering spaces, i.e. f φ = f. By a Spin-structure on an oriented Riemannian 4 manifold we mean a Spinstructure on its tangent bundle, whose associated principal SO(n)- bundle turns out to be 3 We didn t really need to assume that the bundle was oriented, we could have used the 2-sheeted cover Pin(n) of O(n). 4 Here again we don t actually need to choose a Riemannian metric, then it is enough to assume that λ is the 2-sheeted cover of GL n (R). 12

21 the oriented orthonormal frame bundle of the manifold. The following result will be useful later, see e.g. [2] for a proof: If an oriented vector bundle can be given a spin-structure (equivalently, if its second Stiefel-Whitney class is zero), then its distinct Spin-structures are in one-to-one correspondence with the elements of H 1 (M, Z 2 ), particularly, there are exactly 2 2g distinct Spin-structures on a closed Riemann surface of genus g. Here we present two other alternative definitions of Spin- structure, but equivalent to the above. The first one is due to Milnor (see [13] or [14]): Definition 2. A Spin-structure on an oriented n-dimensional real vector bundle π : E M is a cohomology class s H 1 (P SO(n), Z 2 ) Hom(H 1 (P SO(n), Z 2 ), Z 2 ) whose restriction to each fiber of the associated principal bundle ˆπ : P SO(n) M is non-zero, or alternatively, s,f = 1, where we denote the generator of H 1 (SO(n), Z 2 ), the fiber class, by f and the dual pairing H 1 (P SO(n), Z 2 ) H 1 (P SO(n), Z 2 ) Z 2 by,. One can also give the following definition based on the properties of 2-sheeted covering spaces, see e.g. Gonçalves, Hayat, and Mello [5]: Definition 3. Let π : E M be an oriented n-dimensional real vector bundle and i SO(n) P ˆπ SO(n) M be its associated principal SO(n)-bundle. A Spin-structure on π : E M is an epimorphism ϕ : π 1 (P SO(n) ) Z 2 such that ϕ i : π 1 (SO(n)) Z 2 is an epimorphism, where i denotes the induced homomorphism between fundamental groups. Note that, the abelianization of the fundamental group of a topological space is isomorphic to its first homology group, and since for a closed, connected oriented manifold M, we have ϕ Hom (π 1 (SO(M)), Z 2 ) Hom (H 1 ((SO(M)), Z 2 ), Z 2 ) H 1 (SO(M), Z 2 ), and considering the Serre exact sequence of the fibration SO(n) SO(M) M, i.e. 0 H 1 (M, Z 2 ) H 1 (SO(M), Z 2 ) i H 1 (SO(n), Z 2 ) H 2 (M, Z 2 ), any Spin-structure on M in the sense of definition 3 actually gives rise to a Spin-structure as defined in definition 2. According to Johnson [7] one can associate to every Spin-structure s on a Riemann surface a mod 2 quadratic form q s : H 1 (M, Z 2 ) Z 2, whose associated bilinear form is the non-degenerate intersection form in H 1 (M, Z 2 ) ( i.e. the dual of the cup product defined on H 1 (M, Z 2 ) via Poincaré duality ), in the following way: Let c H 1 (M, Z 2 ) be a homology class represented by a collection of non-intersecting smooth simple closed curves α 1,...α m : S 1 M, i.e. c = m i=1 [α i], and let α i be the framed closed curve equipped with the positive frame field consisting of the unit tangent vector to 13

22 α i and [ α i ] H 1 (SO(M), Z 2 ) be the homology class that it represents. Then there is a canonical lifting defined by the map H 1 (M, Z 2 ) H 1 (SO(M), Z 2 ) ; c c := m [ α i ] + mf, i=1 where f H 1 (SO(M), Z 2 ) is the homology class represented by the tangential framing on the fiber S 1. The map is well-defined and we have ã + b = ã + b + (a b)f. Now to a given Spin-structure s H 1 (SO(M), Z 2 ) we associate a mod 2 quadratic form by: q s (a) := s, ã. And as usual the equivalence class of the quadratic forms is completely determined by the Arf - invariant. Using the first definition, the above method can be stated as follows: Assume (η, f), where η : P Spin(n) M and f : P Spin(n) P SO(n), is a Spin-structure. Let α : S 1 M be a smooth simple closed curve and the representative of a homology class c in H 1 (M, Z 2 ). If α : S 1 P SO(n) lifts to a path with distinct endpoints in P Spin(n), then we define q η (c) = 1, otherwise we define q η (c) = 0. In the following example, we explicitly compute the Spin- structures on the 2- dimensional torus and their corresponding quadratic forms. Example There are 2 2 = 4 distinct Spin-structures on the 2-dimensional torus T 2 : The tangent bundle of the torus τ : TT 2 = T 2 R 2 T 2 is trivial, so is its orthonormal frame bundle τ SO : SO(T 2 ) T 2, i.e. SO(T 2 ) = T 2 SO(2) = T 2 S 1. All the surjective homomorphisms from π 1 (SO(T 2 )) = π 1 (T 2 S 1 ) = (Z Z) Z to Z 2 are 5 In this example, we use the same method as the one used by the authors in [5] to compute the Spinstructures on S 1. 14

23 as follows: ϕ 1 : (Z Z) Z Z 2 ; ((1, 0),0) 0, ((0, 1),0) 0, ((0, 0),1) 1 ; ϕ 2 : (Z Z) Z Z 2 ; ((1, 0),0) 0, ((0, 1),0) 1, ((0, 0),1) 1 ; ϕ 3 : (Z Z) Z Z 2 ; ((1, 0),0) 1, ((0, 1),0) 0, ((0, 0),1) 1 ; ϕ 4 : (Z Z) Z Z 2 ; ((1, 0),0) 1, ((0, 1),0) 1, ((0, 0),1) 1 ; ϕ 5 : (Z Z) Z Z 2 ; ((1, 0),0) 0, ((0, 1),0) 1, ((0, 0),1) 0 ; ϕ 6 : (Z Z) Z Z 2 ; ((1, 0),0) 1, ((0, 1),0) 0, ((0, 0),1) 0 ; ϕ 7 : (Z Z) Z Z 2 ; ((1, 0),0) 1, ((0, 1),0) 1, ((0, 0),1) 0. Furthermore, we have i : Z (Z Z) Z ; x ((0, 0), x), and since just ϕ k i, k = 1, 2, 3, 4 are surjective, by the definition, the four Spin- structures on the torus are : ϕ 1, ϕ 2, ϕ 3, ϕ 4, whose kernels: ker ϕ 1 = Z Z 2Z, ker ϕ 2 = {(a, b, c) : b + c 2Z}, ker ϕ 3 = {(a, b, c) : a + c 2Z}, ker ϕ 4 = {(a, b, c) : a + b + c 2Z}, are the fundamental groups of the 2-sheeted coverings f k : P k := T 2 S 1 / kerϕ k T 2 S 1, k = 1, 2, 3, 4, where T 2 S 1 denotes the universal covering T 2 S 1 = R R R T 2 S 1 given by (θ 1, θ 2, θ 3 ) (e 2πiθ 1, e 2πiθ 2, e 2πiθ 3 ). Now let e 1 and e 2 be a symplectic basis of the first homology group of T 2 (which also correspond to the two circles that form the torus) (see figure 3). e 2 e 1 Figure 3 For 1 i 4 and k = 1, 2 we have: ẽ k = [ e k ] + f, so If we denote the induced cohomology class corresponding to each Spin- structure by the same letter, q ϕi (e k ) = ϕ i, ẽ k = ϕ i, [ e k ] + f = ϕ i, [ e k ] + ϕ i,f = ϕ i ([ e k ]) + 1 ; thus the values of the quadratic forms associated to the four Spin-structures on the 2- dimensional torus and their Arf - invariants are as shown in table (1): 15

24 Table 1: Values of q ϕ and Arf (q ϕ ) for Spin -structures on T 2 q ϕ (e 1 ) q ϕ (e 2 ) Arf (q ϕ ) ϕ ϕ ϕ ϕ

25 3 Proof of the Lifting Theorem This section deals with the Haefliger s theorem, originally proved in [6], about the necessary and sufficient conditions for the existence of an immersion lift to R 3 over a map from a surface into R 2 via the projection map. Interested readers are referred to e.g. [11] or [9] for generalized results. Theorem 1. Let f : Σ R 2 be a stable map from a smooth and closed surface Σ into the plane and pr : R 3 R 2 be the projection map. Then there exists an immersion ˆf : Σ R 3 which satisfies f = pr ˆf, if and only if each connected component of its singular set S f, say C i, has an orientable (or non-orientable) neighborhood provided that the number of cusps on C i is even (resp. odd). The statement of the theorem is the direct consequence of the following lemmas: Lemma 3.1. Over a connected component C 0 of S f the number of cusps is even if a neighborhood of C 0 and the bundle of kernels of df (i.e. the line bundle whose fiber over any point p C 0 is the kernel of df p ) are both orientable or non-orientable, and odd if one is orientable and the other one is non-orientable. Proof. Take a neighborhood U 0 of C 0, which is either an annulus or a Möbius band immersed in Σ, based on U 0 is orientable or non-orientable. Thus we can define a diffeomorphism from U 0 onto the quotient of the plane R 2 obtained by identifying points whose x-coordinates differ by 1 and y-coordinates are the same or opposite, provided that U 0 is an annulus, or a Möbius strip respectively, in such a way that C 0 U 0 is mapped onto the quotient of the line y = 0. Note that the bundle of kernels of df along C 0 gives rise to a field of transverse vectors N 0 (x) along y = 0, which is invariant under the above identification. Define an orientation for N 0 (x) by choosing one of the two possible directions of N 0 (x) at the origin and propagating it continuously to all the vectors defined over the points (x, 0). Let n 2 (x) be the projection of the unit vector defining the direction of N 0 (x) on the y-axis, now if U 0 and the bundle of kernels of df C0 are both orientable or non-orientable then n 2 (0) = n 2 (1), and if one is orientable and the other one is non-orientable, then n 2 (0) = n 2 (1). Thus the number of cusp points on C 0, i.e. the number of points on the interval [0, 1], where n 2 (x) vanishes while dn 2 (x)/dx is nonzero 6, is even in the first case and odd in the second case. Lemma 3.2. The stable map f from Σ into R 2 can be factorized into an immersion ˆf : Σ R 3 and the projection map pr : R 3 R 2 as pr ˆf if and only if the bundle of kernels of df C is orientable. 6 By returning to the local form of the map, implied by theorem

26 Proof. Note that the existence of the immersion ˆf is equivalent to the existence of a height function h : Σ R for f such that ˆf : Σ R 3 ; (x, y) (f(x, y), h(x, y)) is an immersion. First assume that the immersion lift exists. Since the derivative of ˆf is d ˆf = f 1 x f 2 x h x then ker df ker dh = {(0, 0)} at any point p in a neighborhood U of S f. In other words, if we introduce a Riemannian metric on U, h at any point p of S f is not perpendicular to the fiber of the bundle of kernels of df over p, thus its orthogonal projection on the fiber is nonzero, and gives an orientation of the fiber. Conversely, suppose that the line bundle given by the kernel of df is orientable. We use the notation and presentation of the previous lemma. Let n 1 (x) be the projection on the x-axis of the unit vector defining the direction of the vector field N(x) at the point (x, 0). By our assumption of orientability, n 1 (x) is a function of period 1. It can be approximated by a differentiable function n 1 (x) of period 1 with a finite number of zeros on the interval [0, 1) and a non-zero derivative at these points, resulting in that n 1 (x) vanishes an even number of times on this interval; the positive number n 1 (x) can be assumed to be so close to n 1 (x) such that the vectors (n 1 (x), n 2 (x)) and (n 1 (x), n 2 (x)) are not orthogonal. Construct a differentiable function r(x) of period 1 as f 1 y f 2 y h y, dr(x) dx = a(x)n 1(x), where a(x) is a non-zero differentiable function, and define h(x, y) := a(x)n 2 (x)y + r(x), then h(x, 0) = (a(x)n 1 (x), a(x)n 2 (x)), which is not orthogonal to the fibers of the bundle of kernels of df As the function is invariant under the identification described in the proof of the above lemma, it defines, by passing to quotients, a differentiable function h 0 in the neighborhood U 0 of C 0 whose level lines are transverse to the field N 0 a any point of C 0. Repeating this construction for each component C i of S f and using the Whitney s extension theorem 7, we construct a differentiable function h on Σ which coincides with the functions h i built in a neighborhood of C i. 7 Whitney s Extension Theorem. Let M be a smooth manifold and let F : M R k be a continuous function. Given any positive continuous function δ : M R, there exists a smooth function F : M R k such that F(x) F(x) < δ(x) for all x M. If F is smooth on a closed subset A M,then F can be chosen to be equal to F on A. 18

27 4 Spin -structures and Classification of Immersions The aim of this section is to construct an example of two non regularly homotopic immersions from the torus into R 3 with identical projections on the plane. This is based on Pinkall s classification theorem; the main idea of the theorem is to correspond a mod 4 quadratic form to any immersion (see [15]). The desired quadratic form reduces to a mod 2 quadratic form for orientable surfaces. We present a method to obtain the latter based on Johnson s correspondence between the Spin- structure induced by the immersions and quadratic forms on the first homology group of the surface. In [14] Milnor proved the following lemma : Lemma 4.1. Given Spin-structures on two of the three bundles ξ, η, ξ η, there is a uniquely determined Spin-structure on the third. Let ξ : E B be a vector bundle, the determinant bundle, det ξ, of ξ is defined to be the line bundle over the same base space, whose fiber over any point in the base space is the top exterior product of the fiber of ξ over the point, or equivalently, the line bundle whose transition functions are determinants of the transition functions of ξ. A vector bundle ξ is orientable if and only if det ξ is the trivial bundle. Let Σ be an orientable surface (smooth, compact and without boundary. For simplicity, we also assume that the surface is connected), and f : Σ R 3 be an immersion. then the direct sum of the tangent bundle with the normal bundle τ Σ ν Σ is isomorphic to f τ R 3, the pullback of τ R 3 to Σ by df. Since R 3 is oriented, the normal bundle ν Σ of Σ in f τ R 3 is isomorphic to the determinant bundle detτ Σ, which is isomorphic to the trivial one dimensional vector bundle ε, due to the orientability of Σ. Thus we have f τ R 3 = τ Σ ε and by lemma 4.1 the unique Spin-structure on R 3 8, induces a Spin-structure on τ Σ and hence on Σ, called the Spin-structure induced by the immersion. More precisely, if at each point x Σ we complete the local oriented orthonormal frame SO x (Σ) by adding the outward unit normal vector, we get an element in SO(3); Then the pullback of the standard two sheeted covering λ : Spin(3) SO(3) via df gives P, the total space of the induced Spin-structure, and the covering map involved in its definition is the restriction of the projection map SO(Σ) Spin(3) SO(Σ) to P. Let α : S 1 Σ be a smooth simple closed curve on Σ and the representative of a homology class c in H 1 (Σ, Z 2 ), as we explained in Subsection 2.3 we can construct a framed loop α 8 Since H 1 (SO(R 3 ), Z 2 ) = H 1 (R 3, Z 2 ) = 0, there is only one Spin-structure on R 3 19

28 in the principal S 1 bundle SO(Σ). Following Johnson s method, if α : S 1 SO(Σ) lifts to a path with distinct endpoints in P then we define q f (c) = 1, otherwise we define q f (c) = 0. Here by subscript f we indicate that the Spin-structure is induced by the immersion f. In what follows, we shall associate a loop in SO(3) to a simple closed curve on a surface immersed in the space, using its plane projection. Then the above mentioned lifting occurs if and only if the associated loop lifts to a path in Spin(3) connecting two distinct points. 9 Let g : Σ R 2 be a stable map, that lifts to an immersion f : Σ R 3 via the projection map pr : R 3 R 2. Thus there exists a height function h : Σ R with respect to the axis of the projection, such that f : Σ R 3 ; (x, y) (g(x, y), h(x, y)). If we view Σ in the direction of the projection, the local structure of the surface in a neighborhood of the general fold, looks like the surface is folded over on itself. α p α Figure 4 Let α be an oriented curve in Σ equipped with its standard (Frenet) frame, write (e 1,e 2 ) for this frame where e 1 is a tangent vector of α and e 2 a normal vector and assume that α is transverse to the fold curve. We will associate a loop in SO(3) to α using the projection. Let α denote the projection of α to the plane: α = pr α. Let P denote the finite set of points where α intersects the fold curve. First change the frame (e 1,e 2 ) by homotopy so that e 1 (p) lies in ker(dg) at all p P. For points q α P we define the frame (ê 1 (q),ê 2 (q), ± x3 ), where (ê 1,ê 2 ) is the frame which results by applying the Gram-Schmidt orthogonalization procedure to the frame (pr(e 1 (q)), pr(e 2 (q))) and where the sign of x3 equals the sign of det(dg(q)). In order to connect the framed arcs α P to a loop in SO(3) we need only to describe how to close the loop up at intersection points p. Here we use the height function h. Let p P. Note that the normal to the fold and x3 determines a plane at p. The limiting frame vector ê 1 (q), q p, where the last limit means that we approach p from the left, lies in this plane and determines a half plane. Let A be the half circle centered at the origin and lying in this half plane oriented from the point with x 3 coordinate larger than 0 to the point with x 3 coordinate less than 0 if d (h α)(p) < 0 and oriented in the opposite direction dt 9 π 1 (SO(3)) = Z 2, generated by any loop describing a 2π-rotation about a fixed axis. Any contractible loop in SO(3) lifts to a loop in Spin(3), while a noncontractible loop lifts to a path joining antipodal points in Spin(3) = SU(2) = S 3. 20

29 if d dt (h α)(p) > 0. Connect the frames at opposite side of the fold by letting (ê 1, N) rotate according to the Frenet frame of the half circle and keeping ê 2 constant. Here N denotes the third vector of the frame and the plane is oriented by ê 2. Let λ(α) denote the loop in SO(3) obtained in this way. Noting that f is regularly homotopic to the map given by (g(x, y), ɛh(x, y)) for arbitrary small ɛ > 0 it is straightforward to check that q f (c) = [λ(α)] + 1, (4.1) where [λ(α)] denotes the homotopy class of λ(α) in π 1 (SO(3)) = Z 2. Example 4.1. Consider the following projection of a torus on the plane: Figure 5 By the lifting theorem the figure can be considered as the projection of an immersed torus in R 3, that has only fold singularities on two curves. Note that the theorem asserts such an immersion exists and says nothing about the uniqueness, in this simple case we can find at least two such immersion lifts : (i) The immersion that maps the both circles e 1 and e 2 (see Figure 3 on page 15) into figure eight shaped loops ; (ii) The immersion, by which e 1 is mapped into a figure eight loop and e 2 remains unchanged. Let s name the immersions f 1 and f 2 respectively. By equation (4.1), q f1 (e 1 ) = 1, q f1 (e 2 ) = 1 q f2 (e 1 ) = 1, q f2 (e 2 ) = 0 e 1 e 1 e 2 e 2 Figure 6 The projection of f 1 (T 2 ) and f 2 (T 2 ) onto the plane So we have: Arf (q f1 ) = q f1 (e 1 )q f1 (e 2 ) = 1 Arf (q f2 ) = q f2 (e 1 )q f2 (e 2 ) = 0 21

30 Thus by Pinkall s theorem f 1 (T 2 ) and f 2 (T 2 ) are not regularly homotopic. 22

31 A Quadratic Forms and the Arf -invariant A Z 2 - vector space V is called a symplectic vector space, if it is endowed with a symplectic form, that is a non-degenerate symmetric bilinear form V V Z 2 ; (x, y) x y, such that x x = 0 for all x V. A symplectic basis of V is a basis a 1,...a g, b 1,...b g such that a i a j = b i b j = 0 and a i b j = δ ij for 1 i, j g. Each symplectic vector space has a symplectic basis and is always even dimensional. A mod 2 quadratic form on V is a map q : V Z 2 satisfying q(x + y) = q(x) + q(y) + x y, for all x, y V. Definition. Let V be a symplectic Z 2 - vector space, and q be a mod 2 quadratic form. Let a 1,...a g, b 1,...b g be a symplectic basis for V. Then the Arf-invariant of q is defined to be Arf (q) := g q(a i )q(b i ). i=1 One can show Arf (q) is independent of the choice of the symplectic basis. 23

32 B Spin Groups Let V be an n-dimensional real vector space endowed with a non-degenerate symmetric bilinear form V V R ; (x, y) x y and denote its corresponding norm by.. Then the Clifford algebra associated to V, denoted by Cliff ± (V ), is the associative linear algebra with identity generated by the elements of V by the relations vw + wv = ± 2 (v w). Let e 1,...e n be an orthogonal basis for V, then the above relations imply that e i e i = ±1, for 1 i n, and e i e j = e j e i, for i j. For each naturally ordered subset A = {i 1,...i k } of {1,...n}, put e A = e i1 e i2 e ik, while for A =, e = 1 (the identity element in Cliff ± (V )). Then {e A : A {1,...n}} form a basis for Cliff ± (V ) regarded as a vector space over R. The compact Lie group Pin ± (n), named Pin group, is defined as the subset of Cliff ± (V ) consisting of products of all finite sequences of unit vectors in V, i.e. Pin ± (n) = {v 1 v 2 v k : v i V, v i = 1, 1 i k}. The connected component of the identity in both cases is isomorphic to Spin group: Spin(n) = {v 1 v 2 v k : v i V, v i = 1, 1 i k, k is even}, which is a compact, simply connected Lie group for n 3. Define a transposition e t A := e i k e i1 = ( 1) k 1 e A and an algebra homomorphism α(e A ) := ( 1) k 1 e A = ( 1) A e A and extend them linearly to Cliff ± (V ); then there are representations ρ ± : Pin ± (n) O(n) given by: ρ (w)(v) = wvw t ρ + (w)(v) = α(w)vw t. In this way, Pin ± (n) defines a 2-sheeted covering of the orthogonal group O(n), and the restriction of the representations to Spin(n) gives the 2-sheeted covering map of SO(n). 24

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