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SINGULARITIES OF SMOOTH MAPS By RUSTAM SADYKOV A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005 Copyright 2005 by Rustam Sadykov ACKNOWLEDGMENTS The problem which I solve in the dissertation appeared in the early 90s in a paper by Osamu Saeki. I learned about the problem 4 years ago, before I become a graduate student. Since that time many people helped me. I wish to express my sincere gratitude to teacher and thesis adviser Alexander Nikolaevich Dranishnikov, to whom I owe thanks not only for valuable discussions but also for assisting me at every stage of the work. I am grateful to Peter Akhmetev for teaching me global singularity theory, and to Yuli Borisovich Rudyak for explaining ideas of algebraic topology. I am grateful to my friends, Sergey Melikhov and Yuri Turygin, for their interest in the work and for discussions of this and closely related topics. I gave the proofs of the main statements at the graduate topology seminars. I would like to thank Alexander Nikolaevichwho organized and led the seminars and the other participants for their help in checking the proofs. I am happy to thank Osamu Saeki whose papers motivated my interest in the problem and led to the dissertation. I also grateful to Osamu Saeki for the invitation and support of my trip to Japan, where I could present the results and learn about new developments in singularity theory. Finally, I appreciate the generous help of an .. iril: mous referee, who helped me to correct erroneous claims and whose numerous comments and recommenda tions considerably improved the presentation of results. TABLE OF CONTENTS page ACKNOWLEDGMENTS ................... ...... iii LIST OF FIGURES ................................ v ABSTRACT ...................... ............. vi 1 INTRODUCTION .................... ....... 1 1.1 Regular Points of Smooth Mappings ......... ...... .. 1 1.2 Singular Points of Smooth Mappings ...... ....... ... 2 1.3 Mappings of Surfaces into 3dimensional Spaces . . .. 2 1.4 Mappings Between Surfaces. .................. 3 2 THOMBOARDMAN SINGULARITIES ....... . .... 7 2.1 Naive Definition .................. ... 7 2.2 Finite Jet Space . . . . . . 8 2.3 Infinite Jet Space .................. ........ .. 10 2.4 ThomBoardman Singularities ............ .. .. .. 11 3 THOM POLYNOMIALS .................. ........ .. 14 4 SINGULARITIES OF MAPPINGS .31  N3 ............... 17 4.1 Normal Forms of Singularities ............ .. .. .. 17 4.2 Topological Description of Singularities ..... . . 17 4.3 Behavior of the Mapping in a Neighborhood of 7 . .... 19 4.4 Bifurcations of Singular Sets ................ .. .. 24 4.5 Surgery of 7(f) ................... ....... 26 5 THE INVARIANT e(f) .................. ........ .. 29 5.1 First Definition .................. ....... .. .. 29 5.2 Second Definition ..... . . ..... ........ 30 5.3 Invariance of the Secondary Obstruction Under Homotopy . 31 6 HPRINCIPLE .................. ............. .. 32 6.1 Differential Relations .................. ..... .. 32 6.2 HPrinciple .. .. ... .. .. .. .. ... .. .. .. ... .. .. 32 6.3 Morin Singularities. .................. ....... .. 33 6.4 AndoEliashberg Theorem .................. ..... 34 7 UNIVERSAL JET BUNDLES ................... .... 36 7.1 Construction 1 ............... .......... ..36 7.2 Construction 2 ............... .......... ..37 7.3 Construction 3 ............ . . ... 38 7.4 Corollary of AndoEliashberg Theorem ..... . . 39 8 SECONDARY OBSTRUCTION .................. ..... 40 8.1 Definition of the Secondary Obstruction . . ..... 40 8.2 Necessary Condition .................. ..... .. 40 8.3 Sufficient Condition .................. .. 41 9 COMPUTATION OF THE SECONDARY OBSTRUCTION ....... 48 10 SIMPLY CONNECTED CASE .................. .. 56 11 FINAL REMARKS .................. ......... .. 59 REFERENCE LIST .................. ............. .. 60 BIOGRAPHICAL SKETCH .................. ......... .. 63 LIST OF FIGURES Figure page 11 Whitney umbrella .................. ........ 3 12 Fold and cusp singular points ................ ...... 4 41 The image of a swallowtail singular point. .. . ..... 18 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SINGULARITIES OF SMOOTH MAPS By Rustam Sadykov August 2005 C'!h In: Alexander N. Dranishnikov Major Department: Mathematics A singular point of a smooth mapping f : M  N of manifolds is a point at which the rank of f is less than the minimum of dimensions of M and N. Singularities of smooth mappings have a nice classification, with respect to which for almost any smooth mapping f, the set of singular points of any type J forms a smooth submanifold Sj(f) c M. We study those topological properties of the set Sj(f) that does not change under homotopy of f. One of the first questions that arises in the singularity theory asks whether a singularity type J is inessential for a mapping f; in other words, does there exist a homotopy of f eliminating all the 3singular points? The primary obstruction is defined as the cohomology class [Sj(f)] E H*(M; Z2) dual to the closure of Sj(f). Remarkably, the class [S3(f)] is a polynomial, called Thom polynomial, in StiefelWhitney classes of the tangent bundle TM and the induced bundle f*TN. The Thom polynomial turns out not to be a complete obstruction; O. Saeki constructed an example of a mapping from a 4manifold into a 3manifold where the cohomology obstruction corresponding to certain singularities, cusps, is trivial though a homotopy to a general position mapping without cusp singular points does not exist. We consider smooth mappings of 4manifolds into 3manifolds, determine the secondary obstruction, prove its completeness and express it in terms of the cohomology ring of the source manifold. Definition A general position mapping of a 4manifold into 3manifold without cusp singular points is called a fold mapping. Theorem For a closed oriented 4u,,i,.I.: [ 1', the following conditions are equivalent: (1) .1' admits a fold i'".'l'.:. y into R3; (2) for every orientable 3 I,.;,:..' .1/ N3, every homot('i.,; class of I,,I,.'':,i. of im1 into N3 contains a fold i'l ''l.: l. (3) there exists a, 1. i,'i,,. ',i/;/ class x E H2(i 1; Z) such that x x x is the first Pontrjagin class of iF\. For a simply connected manifold 3 1T, we show that .[1 admits no fold mappings into N3 if and only if [1T is homotopy equivalent to CP2 or CP2# P2. CHAPTER 1 INTRODUCTION 1.1 Regular Points of Smooth Mappings Given a smooth mapping f of a manifold M of dimension m into a smooth manifold N of dimension n, the differential df(x) of the mapping f at a point x of M is a linear map from the tangent space T1M of M at x to the tangent space Tf(x)N of N at f(x), df(x) : TM Tf(,) N. We ,v that x E M is a ,.iJ.'; point of the in'rll.:.l f if the rank rkx(f) of the differential df(x) is exactly max(m, n). Otherwise we ~v that the point x is a S..:!,l.r, point of the iq. '.:,;j f. We observe that the set of regular points forms an open submanifold of the source manifold. Indeed, if a homomorphism h of vector spaces sends a set {ei} of independent vectors into a set of independent vectors, then every homomorphism sufficiently close to h also sends the vectors {ei} into independent ones. Consequently, if f is a smooth mapping and x is a point of the source manifold, then the rank of the differential df(x) at x is not greater than the rank of the differential df(y) at any point y sufficiently close to x. In particular, in a small neighborhood of a a regular point, the mapping f has no singular points. The regular points of a mapping have a simple description. In the case of a positive codimension, n m > 0, the regular points are precisely the points in a neighborhood of which the mapping f is an embedding. If the mapping f is of a nonpositive codimension, i.e., n m < 0, then the regular points are the points of the source manifold in a neighborhood of which the mapping f is a submersion. 1.2 Singular Points of Smooth Mappings We study singularities of smooth mappings up to an equivalence relation. Definition. Given two mappings fi : MI  Ni, i = 1, 2, we i that the points xl E M1 and x2 c 31 are of the same singularity type with respect to the right left equivalence if there are neighborhoods Ui containing xi, neighborhoods 1V containing fi(xi) and diffeomorphisms g : U  2 U2, h : V1 V2 that fit into the commutative diagram U1 U2 filUl f2U2 V, h V2, where the mappings fiUi are the restrictions of the mappings fi to Ui. It is convenient to describe a rightleft singularity type, 'r, by choosing a normal form, i.e., a mapping g : R' R' with singularity r at the origin. Once the normal form is chosen, we  that a mapping f : M N has rsingularity at a point x E M if in some local coordinate neighborhoods of x in M and f(x) in N, the mapping f has the form g. In the two subsequent sections we will consider examples of singularities in the cases of mappings of manifolds of small dimensions. 1.3 Mappings of Surfaces into 3dimensional Spaces One of the singularities of mappings from a surface into a 3manifold is the Whitney umbrella. In a neighborhood of a Whitney umbrella, in some coordinates, the mapping f has the form f(u, ) (uv, u, v2). Theorem 1.3.1 (Whitney) Every in'jrr,,:', of a ,'ifi..: into a 3ia,,,.',... 43 can be approximated by a i'.r'll'.,:. with singularities of only Wiili,. .1 umbrella /. The set of Whitney umbrellas is a discrete set. In particular, a mapping of a closed surface may have only finitely many Whitney umbrellas. Figure 11: Whitney umbrella. In fact, the number of Whitney umbrellas is even. To prove this, we describe the Whitney umbrellas as the end points of selfintersection curves. If the source surface is closed, then each connected component of selfintersection points is either a circle which has no end points or a closed interval which has two end points. Thus the number of Whitney umbrellas is twice the number of closed intervals of selfintersection points. If we consider mappings up to homotopy, then the Whitney umbrellas are no longer essential; every mapping of a surface into a 3manifold is homotopic to an immersion. This follows from the SmaleHirsch hprinciple for immersions, which we will discuss in later sections. 1.4 Mappings Between Surfaces. Singularities of mappings between surfaces were studied by Whitney who proved that every continuous mapping of surfaces can be approximated by a mapping with only regular points, fold singular points, and cusp singular points. A j. ,.n point of a mapping f, as it has been defined, is a point in a neighbor hood of which the mapping f is a diffeomorphism. The fold and cusp singular types are defined by normal forms. We v that a singular point p is of the fold type or the cusp type if in some neighborhoods of p \Z Figure 12: Fold and cusp singular points and f(p) there are coordinates in which the mapping f has the form f(x,y) = (x,y2) or fx, y) = 3 + xy,y) respectively (see figure 12). As it follows from the normal forms of singularities, the set of singular points S of a mapping f with only fold and cusp singular points forms a smooth curve in the source manifold. The set of cusp points is discrete, while the set of fold points is the 1dimensional complement to the cusp points in S. We note that the rank of the differential of the mapping f is 1 both at a point of the fold type and at a point of the cusp type. To distinguish a fold singular point from a cusp singular point, Whitney considered the restriction of the mapping f to the smooth curve of singular points S and observed that the cusp points of f are exactly the singular points of f S (see figure 12). The cusp singular points are essential even if we consider mappings up to homotopy. For example, the projective plane RP2 does not admit a mapping into R2 with only fold singular points [42]. Let us sketch a proof that motivates the notion I, .! .. ,gy obstruction." We note that any two mappings into R2 are homotopic. Thus to prove the claim it suffices to construct a mapping RP2 IR2 with fold and cusp singularities and then to show that the cusp singular points can not be eliminated by homotopy. Construction. Folding a triangle A along the three lines that join the three middle points of the sides of A defines a 4to1 mapping fo from A to the triangle A1/4 that is 4 times smaller than A. Let us z that two boundary points x, y of A are opposite if the segment [x, y] goes through the center of A. We observe that the fold mapping fo sends the opposite points x, y of the boundary to the same point fo(x) = fo(y). Thus, the mapping fo defines a mapping fo of the projective plane obtained from A by identifying the opposite boundary points. Slightly perturbing the mapping fo and composing it with an embedding of the triangle A1/4 into R2, we obtain a mapping f from RP2 into R2 with only fold and 3 cusp singular points. To prove that the cusp singular points of f can not be eliminated by homo topy, we need to consider the bifurcations of the set of singular points S that occur under homotopy of f. Every homotopy joining two mappings between surfaces can be approximated by a general position homotopy that changes the set of singular set by isotopy except for the finitely many points when one of the following bifurcations takes place [23]. Homotopy under which a circle of singular points with two cusp points appear: ft(x, y) = (x3 2 tx, ), t [ 1, ], Homotopy under which two new cusp singular points appear on a curve of fold singular points: ft(x, y) = (+ xy t2,Y), t E [1,], Homotopy under which a curve of singular points with two cusp points disappear: ft(x, y) = (x3 2 t ),t [1,l], 6 Homotopy under which two cusp singular points on a curve S disappear: ft (x, y) ( + xy + tx2), t [,]. Thus, under general position homotopy the number of cusp singular points changes by a multiple of two and therefore every mapping of RP2 into R2 with only fold and cusp singular points has an odd number of cusp points [42]. CHAPTER 2 THOMBOARDMAN SINGULARITIES The rightleft equivalence relation on singularities of smooth mappings is so fine that the number of different singularity types of a mapping is infinite in general. Besides, the behavior of the set of points of a rightleft equivalence class under homotopy of a mapping has no simple description. To overcome the difficulties arising here one may consider a coarser relation in which a class of equivalence is a union of some, perhaps infinitely many, rightleft equivalence classes of singularities. One of such relations pl wing a special role in singularity theory is the ThomBoardman classification. Every continuous mapping of smooth manifolds admits an approximation by a mapping f with singularities of only finitely many different ThomBoardman classes. Furthermore, the set of singular points of f of each ThomBoardman class is a submanifold of the source manifold. 2.1 Naive Definition Let TM and TN denote the tangent bundles of smooth manifolds M and N respectively and df the differential of a smooth mapping f : M  N. The set Si = S(f) is defined as the set of points x in M at which the kernel rank of f is krf i. Suppose that dim M = m > n = dim N. Suppose that for each i, the set Si is a submanifold of M, then we can consider the restriction f Si, of f to Si, and define the singular set Si,~, as the subset S2,(f Si,) of Si,. Again, if every set Sil,i is a submanifold of M, then the definition may be iterated. Thus, the set Sil,..., i is defined by induction as Si,(f lSi,.... ). The index J = (i, ...,ik) is called the symbol of the iin, Ii H;; We will write Sj for Sil,.... . For example the Whitney fold singular points of a mapping between surfaces and the Whitney umbrella of a mapping of a surface into a 3manifold are Thom Boardman singular points of the type Si,o. From the Whitney description of singular points of a mapping between surfaces, it follows that the cusp singular points are of the type Si,1,o. Certainly, this natural definition makes sense only under heavy restrictions; the singular set Si, .... i can be defined only if the singularity stratum Sil...,4_ is a submanifold of the source manifold. By passing to jet spaces Boardman was able to extend the definition over all singular sets. 2.2 Finite Jet Space A singularity type of a mapping f : M  N at a point x E M depends on the behavior of the mapping f only in a small neighborhood of x. So, we pass to germs. A germ at a point x E M is an equivalence relation on mappings under which two mappings f, i = 1, 2, defined on a neighborhood of x E M represent the same germ at x if there is a possibly smaller neighborhood of x where the mappings fl, f2 coincide. A kjet is, by definition, a class of ~kequivalence of germs. Two germs f and g at x are ~kequivalent if at the point x the mappings f and g have the same partial derivatives of order < k. As partial derivatives involved, our definition implicitly assumes coordinate systems in neighborhoods of x and f(x) = g(x). It is easy to verify, however, that if in some coordinate systems f and g have the same partial derivatives of order < k, then the same is true for any other choice of the coordinate systems. If a kjet ax is represented by a mapping f at x, then we also iv that ax is a kjet of the mapping f at x. The set of all kjets Jk(M, N) is called the kjet space of mappings of M into N. Let ax be a kjet at a point x E M represented by some mapping f. If a coordinate system in a neighborhood of x and a coordinate system in a neighborhood of f(x) are fixed, then the kjet ax is determined by the Taylor polynomial of f at x of order k. In its turn, the set of polynomials of order k is naturally isomorphic to the finite dimensional Euclidean space as each polynomial is characterized by the set of its coefficients. So, the kjet space has a natural structure of a smooth manifold. Formally, let U and V be coordinate covers of the manifolds M and N respectively. For each open set U E U and an open set V E V we define a subset Wuv of the kjet space as the set of the jets ax, x E U, represented by mappings sending x into V. Note that the subsets W = {Wuv}, where U and V range over elements of U and V, cover the space of kjets. Also, being isomorphic to the set of polynomials of order k, each of Wuv is isomorphic to a Euclidean space. These isomorphisms induce topologies, one for each Wuv, that coincide on intersections Wuv n Wuv,, U' E U, V' c V. Thus there is a natural topology on Jk (M, N). Moreover, the cover W together with homeomorphisms from Wuv into the Euclidean space, U E U, V E V, defines a smooth structure on Jk (M, N). In fact the cover W not only helps to introduce a smooth structure on Jk(M, N) but also allows us to introduce on Jk(M, N) a structure of a smooth locally trivial bundle over M x N. Indeed, Jk(M, N) is covered by the sets Wuv C W each of which is a trivial bundle over U x V. We note that the bundle projection Jk(M, N)  M x N sends a kjet ac represented by a mapping f into the point x x f(x). Example. If k = 1, then the kjet of a mapping f at x E M is the differential df : T1M Tf(x)N. Thus the space of 1jets is the locally trivial bundle HOM(M, N) over M x N the fiber over x x y of which consists of linear homomorphisms TM M TyN. For the definition of ThomBoardman singularities we need to pass to an infinite dimensional space J~ (M, N) which we define next. 2.3 Infinite Jet Space A kjet at a point x determines an ijet for each I < k. Thus for each pair k, I with I < k we have a natural projection S: Jk (M, N) J(M, N). The jet space J~ (M, N) is a topological space defined as the inverse limit of the system {(Jk(M, N), 7j}. Though the jet space is infinite dimensional and we cannot define a smooth structure on J"(M, N), still, using projections k7 : J' (M, N) Jk (M, N) we may define on J~"(M, N) smooth functions, tangent vectors and submanifolds. We v, that a function on the jet space is smooth if locally it is the compo sition of the projection onto some kjet space and a smooth function on the kjet space. Having defined smooth functions we may define a 'i',.. ,./ vector at a jet a. The set of germs F(a) of functions on J~(M, N) at the jet a is an algebra over R. A differential operator D, at a is a correspondence D, : F(a) > F(a) that is linear, i.e., D,(af + bg) =aD f(f) + bD,(g), f, g E F(a), a, b E R, and satisfy the Leibniz rule D f(fg) = D(f) + Dc(g), f, g E (a). We define a tangent vector at the jet a as a differential operator D,. We may view a vector D, as an infinite sequence of vectors D, k E N, respectively tangent to the jet spaces Jk(M, N) at (ca) such that (7k), : k D D k >1. (2.1) Indeed, let ak denote the kjet 7'"(). By definition of F(a), F(a) = UkENF(ak), where each F(cak) is identified with a subset of F(cak+1) under the mapping induced by the projection F(ak+l)  F(ak). Given a vector D,, its restrictions to F(ak) produce a sequence of operators D, On the other hand suppose a sequence of vectors D, satisfies (2.1). Then we define a differential operator D, as follows. Let f be a smooth function defined in a neighborhood of a. Then, by definition, for some k, f is lifted by 7r from a smooth function fk on Jk(M, N). We define D,(f) as the lift by 7r' of the smooth function Dl (fk). It is easily verified that so defined correspondence Do is a differential operator. The set of vectors at the jet a is a vector space called a ',",.,'. ,./ space at a. Finally, we i that a subset of the jet space is a subrni, .:[. ./.I if it is the inverse image of a submanifold of some kjet space. 2.4 ThomBoardman Singularities Given a smooth mapping f : M  N, at each point of the manifold M we have an infinite jet of f. The correspondence that takes a point x into the jet of f at x is a mapping jf: M J(M,N), called the jet extension of f. Similarly for each k, we define the kjet extension jkf of f. If v is a vector at a point x of M, then the sequence of vectors d(jkf)(v) satisfies the condition 2.1 and therefore defines a vector d(jf)(v) at jf(x). Note that the map d(jf) : TM TJ"(M, N) is injective. The union of images of d(jf) over all mappings M + N is called the total Ir,'.. ,./I bundle of the jet space and is denoted by D. Given a jet jf(x), we will use the injective homomorphism d(jf)lTx to identify the plane TxM with Dl\jf(x). Every 1jet at a point x E M determines a homomorphism TxM  Tf(x)N, where f is a germ at x representing the jet. Let y be a point of the jet bundle and Ky C Dy the kernel of the homomorphism defined by the 1jet component of y. Boardman proved that for every i1 the set i, = { ye Jcc(M, N)  dimK = ii } is a submanifold of J~(M, N). Let T' denote the set of r integers (i, ..., i,) such that i1 > .. > ir. Suppose that the submanifold Ejri has been already defined. Then define EZj { y E Ej1 I dim(Ky n TEj,) = ir}. Boardman proved that for every symbol Jr the set EZj is a submanifold of J (M, N). A mapping f is called a general position i,,r.lr.:l if the section jf is transver sal to every submanifold Ej. By the Thom Strong Transversality Theorem (see Arnold et al. [5] or Boardman [6]), every mapping can be approximated by a general position mapping. Given a mapping f : M + N, a point x E M is a singularity of type J if the image jf(x) is in EZ. As has been mentioned, for general position mappings, the definition of singularity types given by Boardman coincides with the naive definition given in section 2.1. Examples of general position mappings. For mappings from a surface into a 3manifold, general position mappings are those with singularities of only Whitney 13 umbrella type. For mappings between surfaces, general position mappings are exactly the mappings with only fold and cusp singular points. CHAPTER 3 THOM POLYNOMIALS Two of the main questions in the global singularity theory are Question 1. Given two i,1r,,..:[..1.14 M, N and a ;i,;.,l.; Ii; type r, does there exists a general position i.1i'1r.:', f : M N without 7 singularities? Question 2. Given a continuous i''rr'.:jj f : M N of smooth ,,<,;,4.:4..l., and a c.:,,il;.,~I:1, r,T, does there exists a homote(.i' of f to a general position In'1'.:i.j without 7 singularities. We already know the answers to these questions in two particular cases. Every mapping of a surface into R3 admits a homotopy to an immersion; thus the answers to the Questions 1,2 with 7 standing for Whitney umbrella are positive. The answers to the Questions 1,2 for mappings between surfaces without cusp singular points is negative in general. Every general position mapping of the projective plane into R2 has at least one cusp singular point. Lemma 3.0.1 (tliashberg [8]) A ,,,,'.:,.1 of a connected closed i, fi,.: into an orientable ,'f, ': is homotopic to a i.irr.':,.j with only fold .:,,g.l,.n points if and only if the Euler characteristic of the source ,,,,:f.;,l .11 is even. The obstruction to elimination of the cusp singular points of a mapping f of a closed surface into an orientable surface can be considered as a homology obstruction because it is represented by the fundamental class of the cusp singular points. In general, given a general position mapping f : M + N and a Thom Boardman singularity type 7, the closure of the set of rsingular points of f represents a homology class [r] E H,(M; Z2). Under general position homotopy the homology class [r] does not change and therefore defines a homology obstruction. It is remarkable that the cohomology class dual to the homology obstruction is .1. l a polynomial in terms of StiefelWhitney classes of the bundles TM and f*TN [14], [17]. For example in the case of cusp singularities of mappings from a surface into R2, the homology obstruction is dual to the StiefelWhitney class w2. The polynomial in terms of StiefelWhitney classes dual to the homology obstruction is called Thorn p i 'l ,.;;;,. Many papers are devoted to calculation of Thom polynomials of certain singularities. Some of the polynomials were already calculated by Thom [39]. For the list of results of 19701980 we refer to the survey [5] of V. Vasiliev. Recently, there was a significant progress in calculation of Thom polynomials by L. Feher, M. Kazarian and R. Rimanyi [11], [18], [26], [27]. The homology obstruction is not complete in general. For example, every mapping of a 2dimensional sphere into R2 has a curve of fold singular points, though the homology obstruction belongs to the trivial group H2(S2; Z). Except the trivial case of fold singular points that appear as the points corresponding to the boundary of the image, the question whether the homology obstruction provide a complete obstruction is not simple. Let us list some cases where the homology obstruction is known to be com plete: (1) cusp singularities for mappings from mmanifold into an orientable surface (H. Levine [20]; Y. Eliashberg [8]); (2) any stable singularity type except fold one for mappings between orientable 4manifolds (0. Saeki, K. Sakuma [35], Y. Ando [2], [3]); (3) nonfold singularities of mappings of nonpositive codimension between stably paralellizable manifolds (Y. Eliashberg [8], [9]); (4) zero dimensional singularities of mappings of nonpositive codimension between orentable manifolds of dimensions > 2 (Y. Ando [3]); (5) any Morin singularity except of fold and cusp types of mappings Mm ' N1 with m n > 0 odd (R. Sadykov [29]); In 1992 0. Saeki [30] presented an example of manifolds M, N and a singu larity type T such that the homology obstruction [r] to the existence of a rfree mapping M  N is trivial and nevertheless the manifold M does not admit a mapping into N without r singularities. In the example of O. Saeki M is a 4manifold homotopy equivalent to CP2, N is R3 and 7 is the cusp singularity. The closure of the cusp singularities of a mapping from a 4manifold into a 3manifold is a union of circles. Its homology class is trivial as the manifold M is simply connected. On the other hand O. Saeki proved that the manifold M does not admit a mapping into N without cusp singular points. In the latter sections we determine the secondary obstruction for the existence of mappings of 4manifolds into 3manifolds without cusp singularities and prove that the secondary obstruction is complete. CHAPTER 4 SINGULARITIES OF MAPPINGS M3[  N3 4.1 Normal Forms of Singularities In general position with respect to ThomBoardman singularities, a smooth mapping f of a 4manifold into a 3manifold has only fold, cusp and swallowtail singularities [5]. In fact, one may define a general position mapping of a 4manifold into a 3manifold as a mapping with singularities of only these types. In a neighborhood of any singular point of a mapping f, there are local coordinates, called special, in which the mapping f takes one of the normal forms: Definite fold singularity type S+ T = t1, T2 t2, Z q +q . Indefinite fold singularity type S_ Ti t1, T2 t2, Z= q q . Cusp singularity type 7 Ti tl, T2 t2, Z q + tx + x3 Swallowtail singularity type Ti tl, T2 t, Z = q + tlx + t2x2 + x4 4.2 Topological Description of Singularities Directly from the normal forms of singularities, it follows that the set of all singular points of a general position mapping f : M  N of a 4manifold into a 3manifold is a closed, perhaps nonorientable, 2dimensional submanifold S of M. However, the image of the singular set f(S) is not a submanifold of N in general; the restriction of the mapping f to S, may not be an immersion. The singular set of f S is exactly the set of cusp and swallowtail singular points of f. The image of the singular set under the mapping f S in a neighborhood of a swallowtail singular point is depicted on the figure below. Figure 41: The image of a swallowtail singular point. We see that the image of the swallowtail singular points can be identified with the end points of the selfintersection curve of f(S). In particular, the homology obstruction to eliminating swallowtail singular points by homotopy of f is ahli trivial. In fact, by the Ando theorem [3], if the manifolds M and N are orientable, then the homotopy eliminating the swallowtail singular points of f exists. We are interested in the question whether there exists a homotopy eliminating cusp singular points 7(f). The set of cusp singular points constitute a union of curves with boundary at the swallowtail singular points so that the closure {cusp points} {cusp points} U {swallowtail points} is a union of circles embedded into M. It is wellknown that if M and N are orientable, then the homology class [7(f)] E H2(M;Z ) is trivial. In fact a stronger assertion takes place. Lemma 4.2.1 Suppose that the I,,;.,[.,1 1N is orientable. Then the curve 7(f) bounds an orientable , .i f.. in M. Proof. Let us prove that for an orientable surface bounded by 7(f), we may chose the set of definite fold singular points S+(f). Firstly, we need to show that aS+(f) = 7(f). This follows directly from the normal forms of the cusp and swallowtail singular points; the curve 7(f) cuts the set of singular points S(f)= S+(f)U 7(f) US_(f) into the set of definite fold points S+(f) and indefinite fold points S_(f) so that each component of 7(f) belongs to the boundary of exactly one component of S+(f) and one component of S_(f). Secondly, we need to show that the set set S+(f) is orientable. This follows from the normal form of f in a neighborhood of a definite singular point. Indeed, in a neighborhood of a definite fold point, the mapping f can be written as the product idR2 x m of the identity mapping id : R2  R2 and the Morse function m : R2 R defined by m(qi, q2) = q2 + q The Morse function m gives rise to an orientation of R as the image of m in the set of nonnegative reals. Hence the image f(S+) also has a natural coorientation. Together with orientation of N, the coorientation of f(S+) defines an orientation of S+ 0. Remark. Note that in contrast with S+(f), the set S_(f) is nonorientable in general. Let us recall that a vector v(p) at a cusp point p is called a characteristic vector if there is a standard coordinate neighborhood of p such that v(p) = (p). A vector field on the curve 7(f) in .1 is called a characteristic vector field if it consists of characteristic vectors. The existence of a characteristic vector field for an arbitrary cusp mapping of an orientable 4manifold into an orientable 3manifold will follow from Lemma 4.3.2 bellow. 4.3 Behavior of the Mapping in a Neighborhood of 7 In this section we describe the behavior of the mapping f in a neighborhood of the curve of cusp singular points 7. For convenience of exposition, we assume that 7 is connected. Note that in a neighborhood of each cusp singular point, the mapping f can be written as the product of the identity mapping id[o,] : [0, 1] [0, 1] and the standard cusp i,'.':'hI g : D3 D2 which in special coordinates {t,q, x} of D3 and special coordinates of D2 has the form g(t, q,x) =(t,q2+ t +3). Our aim is to prove that in a neighborhood of 7, f has the product structure. As the first step of the proof, let us show that we can decompose a neighbor hood of the curve 7 into the product S1 x D3, = S1 x {0}, and a neighborhood of the curve f(7) into the product S1 x D2, f(7) S1 x {0}, with respect to which the mapping f is the product of the identity mapping idsl : S1 S1 and the standard cusp mapping g, i.e., for each point x E S1 the restriction f {x} x D3 maps the 3disc into the corresponding 2disc {f(x)} x D2 and, moreover, in some coordinates f {x} x D: {} x D x {x} x D2 is the standard cusp mapping. We start with an arbitrary tubular neighborhood of the curve f(7) in N and choose its decomposition into the product S1 x D2 so that, first, 7 coincides with S1 x {0} and, second, each disc {x} x D2, x e S1, intersects the curve 7 transversally. In a sufficiently small neighborhood of 7, the mapping f composed with the projection S1 x D2 S1 x {0} onto the curve 7 has rank 1. Hence, by the Inverse Function Theorem, there is a product neighborhood S1 x D3 of 7 each disc {x} x D3, x e S1, of which maps under f into the corresponding disc {f(x)} x D2 of the neighborhood of f(7). To complete the proof of the assertion that in a neighborhood of 7, the mapping f has a product structure, we need to show that for each x E S1 the mapping f {x} x D3 is the standard cusp mapping. For a point x E 7, let D' and D' denote the discs of the product structures of 7 and f(7) that go through the points x and f(x) respectively. Lemma 4.3.1 For every point x E 7, the restriction fx f D' is a general position ll l I ": "l,!l Proof of Lemma 4.3.1. In terms of jet spaces, it suffices to show that under the assumptions of the lemma, first, the jet extension j"f, of the mapping fx sends Dj to the set Q2 C J (D', D ) and, second, that the mapping j fx is transversal to the submanifolds of E2 and E2,1points. We will write J] for J([1, 1] x D [1, 1] x D ) and Jf for the restriction of J,, ] to {0} x D Let s: J (D, n3 D J,,p be the embedding that relates the jet section of a mapping g:D D3 D2 with the jet section of the mapping id x g : [1, 1] x D [1, ] x D2 restricted to {0} x D where id stands for the identity mapping of [1, 1]. More precisely, by definition, s is a unique embedding that makes the diagram J g (Dj, (D) J 39 Ij(gx id) D [t, t] x D commutative (see also the definition before Lemma 9.0.9). Here the bottom mapping is the embedding identifying D' with the disc {0} x Du . Let us write E' for the set of 3points in J~,i] and Ej for the set of 3points in J~(DD, D'). We claim that s'(E',) = Ej. We will prove the claim by induction over the length of the symbol J. The mapping s defines a homomorphism s, of the tangent bundles of J({D, Dx) and J[,71]. Note that for every y e J<(D, Dj), the homomorphism s, bijectively sends the kernel Ky defined in section 2.4 into the kernel K,(y). This implies the claim for symbols J of length 1. If the claim holds for all symbols of length k, then for each jk of length k, s,(K n TYZk) =K,(y) n T',. Hence each set Ejme+ with symbol of length k + 1 maps under s into the correspond ing set E',+i. This completes the induction. In particular, the inverse images of the sets of E2, Z2,1 and 22points of J[1,,] under the mapping s are the corresponding sets in J%(D D ). By [28, Lemma 4.3], the mapping s is transversal to E2 n J' and to E2,1 J in J By definitions (7.1) and (7.2), the sets E2 and E2,1 are transversal to the fiber J" in J",,]. Therefore the mapping s is transversal to E2 and E2,1 in J . Identifying D$ with {0} x D$ C [1,1] x D3 C S1 x D3, we obtain (jf)D3 s o jf,. Thus, to prove Lemma 4.3.1 it suffices to show that (C1) the image of (jf)uD is in 2 C [J1,1] and (C2) (jf)lD3 is transversal to the sets of E2 and E2,1points. The condition (C1) holds since Im(if) C Q2. Let us prove (C2). The differential of the jet section jf at x splits into the sum of homomorphisms d(jf)= d(f)\lTD + d(jf)lT(y, (4.1) where TDD' and T17 are the tangent spaces of D' and 7 at x respectively. The dif ferential d(jf)lT., sends Tx7 into the tangent space of the E2,1points. Now, since f is in general position, the equation (4.1) implies that d(jf)lTD3 is transversal to the sets of E2,1 and E2points. This completes the proof of Lemma 4.3.1. U Lemma 4.3.2 There are product neighborhoods S1 x D3 of / and S1 x D2 of f(7) such that f restricted to S1 x D3 is the product of the :/. 1,'/:/.;/ i'"ll':.'j of S1 and the standard cusp i'.',:,,:'.' g. Proof. In view of Lemma 4.3.1, the collection of the mappings {fx} indexed by the points of an interval of 7 can be viewed as a homotopy of the standard cusp mapping, which is known to be homotopically stable (for example see Theorem 7.1 in [12]). Therefore, there is a cover {Ia} of the curve 7 by intervals such that for each interval l, of the cover, the mapping f restricted to a, x D3 is equivalent to the product ida x g of the identity mapping of I, and the standard cusp mapping 9. The trivializations {I1 x D3} and {I x D2} lead to bundle structures of S x D3 and S1 x D2 over S1 with common cover {I,} of S1 and with transition mappings consistent with f. The latter means that each pair (i, Q) of the corresponding transition mappings belongs to the group stabilizing the standard cusp mapping g. Since the normal bundles of 7 in .31 and of f(7) in N3 are orientable, the transition mappings are elements of the group Aut(g) = {(, 9) e Diff (D3, 0) x Diff+(D2, 0) 0 o go 01 g}, where Diff+(D2, 0) and Diff+(D3, 0) stand for the groups of orientation preserving autodiffeomorphisms of (D2, 0) and (D3, 0) respectively. The group Aut(g) reduces to a maximal subgroup MCAut(g) conjugate to a linear compact subgroup [16], [40]. To prove Lemma 4.3.2, it remains to show that the group MCAut(g) is trivial. Let CK denote the orientable version of the contact group, i.e. /K is a semiprod uct of Diff+(D2, 0) and of the group of germs (D2, 0)  Diff+(D1, 0). The group Autkc(h) of a germ h : (D2, 0)  (D1, 0) is defined by A11/. (h)={ cEIC I(h)= h }. The standard cusp mapping g is a miniversal unfolding of the germ go : (D2, 0) (D1, 0), defined by go(q, x) = q2 + x3. Hence by [40, Proposition 3.2], the group MCAut(g) is isomorphic to MC1A,1. (go). The latter group is isomorphic to a compact subgroup H of the group Aut QgO of automorphisms of the local algebra of go [25, Theorem 1.4.6]. Let b E GL+(2) be a linear automorphism of R2 that preserves orientation. It defines an action on germs (R2, 0)  (R, 0) by sending a germ h into ho Suppose that this action factors through an action on local algebra Aut QgO. Then b defines an element in Aut QgO. By the proof of [25, Theorem 1.4.6], we may assume that each element of H < Aut QgO is induced by a linear map of GL+(2). The linear maps b E GL+(2) corresponding to elements of H leave invariant the principal ideal P generated by a germ that in some coordinates has the form (q, x) v> q2 + 3. Note that up to a scalar multiple the vector is determined by the property that for every germ h c P, the partial derivative (0, 0) = 0. Consequently, if a map ib E GL+(2) corresponds to an element in H, then b leaves the direction of the vector a invariant, i.e. &b(a) C= a with a : 0. Moreover, since ib leaves the ideal P invariant, we conclude that a > 0. Hence, the group H is contractible. This completes the proof of Lemma 4.3.2. U 4.4 Bifurcations of Singular Sets We w that a homotopy of a mapping f is in general position if it is a general position mapping. Lemma 4.4.1 Every homot(ei', joining two general position in.ry'r,:isl can be approximated by a general position homot(e'i', Proof. Let fi : M  N, i = 0, 1, be two general position mappings, and F: M x [0,1] N x [0,1] be a homotopy from fo x {0} to fl x {1}. We may assume that for a small number E, the homotopy F does not change the mapping in intervals [0, E) C [0, 1] and (1 1] [0, 1]. As a n lppli.' the homotopy F admits a C1close approximation by a general position mapping. Moreover, we may choose an approximation F that coincides with F on the E intervals [0, E) and (1 E, 1] Since F is C'close to F, the composition p o F : Mx [0, 1] [0, 1], where p : N x [0, 1] [0, 1] is the projection onto the second factor, has no singular points. Therefore, for every moment to E [0, 1], the inverse image F1(N x {to}) is diffeomorphic to M. Thus, F can be considered as a new homotopy joining fo and fl, which as a mapping is in general position. U Lemma 4.4.1 guarantees that any two homotopic general position mappings can be joined by a homotopy in general position. That is why we restrict our attention to bifurcations that occur only under general position homotopy. In the rest of the section we describe bifurcations of homotopies of mappings from a 4manifold into a 3manifold. By dimensional reasoning (see, for example, Boardman [6] and Ando [4]) a general position mapping f from a 5manifold into a 4manifold has only Morin singularities, and D4 singularities with symbol 3 = (2, 2, 0). In a neighborhood of a Morin singular point, in some local coordinates, the mapping f can be written as k1 f(ti,t2,3,q,x) (tl 2, 3, q2 + tii +xk+l'), k 1,2, 3,4. (4.2) i= 1 For a D4 point of f, there are local coordinates in which the mapping f takes the form f(to, tl, t2, v) = ,t, t2, 2 v3 + tU + t V + t 2). (4.3) Suppose that F : V1 T x [0, 1]  N3 x [0,1] is a general position homotopy joining two general position mappings fo and fl. From the normal forms (4.2) and (4.3), it is easy to verify that the singular set of a general position homotopy F : .1T x [0, 1]  N3 x [0,1] is a submanifold of .1 T x [0, 1]. Therefore, S(F) defines an embedded bordism between the singular sets of fo and fl. 4.5 Surgery of 7(f) In this section we study a surgery of the singular set of a cusp mapping f : 1 T  N3 and find a sufficient condition for the existence of a homotopy of f realizing a given surgery. Let f : 1 T  N3 be a cusp mapping of an orientable 4manifold into an orientable 3manifold. In general, the restriction of f to the curve of cusp singular points 7 is an immersion. To simplify arguments, we make f\ an embedding by a slight perturbation of f in a neighborhood of 7. Let v be a field of characteristic vectors on 7. We ? that an orientable surface H is a basis of surgery (see Eliashberg [9]), if 1. H = 7, 2. the vector field v is tangent to H and has an inward direction, 3. H \ OH does not intersect S(f), and 4. the restriction f H is an immersion. We will show that if a basis of surgery H exists, then we can reduce 7 by modification of f in a neighborhood of H. We assume that 7 is connected and f has no other cusp singular points. The proof in the general case is similar. We need one more preliminary observation. Let I denote the closed interval [1, 1] and g : I x D2 I x I be the standard cusp mapping defined by g(t, q, x) = (t, q2 + tx + X3), where t and q, x are the coordinates of the first and the second factors of the domain I x D2 respectively. Then g can be considered as a homotopy gt : D2 I,t c I, defined by gt(q, x) = q2 + tx + 3. Note that g1 : D2  I is a Morse function. Since Morse functions are homotopically stable, there are coordinates in which g8 = g1 for each s E [1, 1 + E). Lemma 4.5.1 Suppose that there is a basis of surgery Hi. Then there is a fold "''. ':'r.j f : 31 N3, which differs from f only in a neighborhood of Hi. Proof. Let U(7) = S x D3 be a neighborhood of 7 in 1 given by Lemma 4.3.2. Let t2 be a cyclic coordinate on the circle S1, (t, q, x) be coordinates on D3 and (T1, T2, Z) be coordinates in a neighborhood of f(7) with T2 cyclic such that f u(y) is given by Ti ti, T2 t2, Z = q2 + t + 3. We may assume that H, n U() = { (t,t2,q,x) x q =0, ti c [0,1] }. We define Ho = { (ti,t2,q,x) x =q=0, tic [1,0] } and set H = Ho U H1. We regard a tubular neighborhood of a submanifold as a disc bundle. The properties (3) and (4) of the definition of a basis of surgery guaranties that the submanifold H has a tubular neighborhood A such that the restriction of A to H n U(7) is in U(7), the intersection S(f) n OA is in the restriction of the bundle A to OH and the set A \ U(7) contains no singular points of the mapping f. To simplify explanations we assume that f H is an embedding. Then the image of A, which we denote by B, is a line bundle over f(H). In the following, for a manifold X with boundary aX, let CX denote a collar neighborhood of aX in X, and let I denote [1, 1]. First, by the remark preceding the lemma, the manifolds B1 = B f(CH) and A1 = f1(BI)nA have product structures A1 = CHxIxI and B1 = f(CH)xI such that f A1 is a product of a diffeomorphism CH f(CH) and a Morse function. We may assume that the restriction of this Morse function to CI x I U I x CI is the projection onto the factor corresponding to the coordinate x, C I x  I, I x CI CI, and that A1 = ACH. Next, we extend the product structure of B1 to a product structure f(H) x I of B. Then we restrict this product structure to B2 = f(H) x CI and define A2 f (B2) n A. The mapping f A, is regular and therefore we may assume that A2 = H x I x CI is a trivial line bundle over B2 with projection f A2 along the second factor. Finally, we can find A3 C A and a product structure H x CI x I of A3 such that f A3 is a trivial CIbundle over f(H) x I and A1 U A2 U A3 is a collar neighborhood of OA. The connected components of A2 and A3 are orientable 1dimensional bundles with bundle mappings given by the restrictions of f. Since the structure group of orientable line bundles reduces to the trivial group, we can make the third coordinates of A1, A2 and A3 agree on intersections. We fix an extension of the product structures of A1, A2 and A3 to a product structure H x I x I of A. Let p E OH and fp denote the restriction of f to the fiber of the bundle A over p. We let f(x) = f(x) for x e [1 \ A and f(u, v, w) f(u) x fp(v, w) for x = (u, v, w) E A = H x I x I. It is easily verified that f is a smooth mapping and f satisfies the requirements of the lemma. U CHAPTER 5 THE INVARIANT e(f) In this section we give several definitions of a secondary obstruction to eliminating the cusp singular points of a general position mapping of a closed orientable 4manifold into an orientable 3manifold; and discuss some of its properties. 5.1 First Definition Given a general position mapping f of a 4manifold M into an orientable 3manifold, to define the homology obsturction, i.e., the primary obsruction, we consider the closure of the cusp singular points and take its fundamental homology class in the group Hi(M; Z2). To define the secondary obstruction, consider the set of all singular points S(f) of the mapping f. This is a closed 2dimensional submanifold of M. We may slightly perturb S(f) so that the new surface S'(f) intersects the original surface S(f) at finitely many points. With each intersection point we associate a number 1, the sign of the intersection point; and then define the secondary obstruction as the sum of the associated numbers. To define the sign of the intersection point p n p' e S(f) n S'(f) we choose an orientation of the manifold M and an orientation of a neighborhood of p in S(f). The latter orientation determines an orientation of a neighborhood of p' in S'(f). Indeed, for any choice of a Riemannian metric on M the projection of the tangent plane of S'(f) at p' into the tangent plane of S(f) at p is an isomorphism provided that the perturbation S'(f) is C1close to S(f). The orientation of S(f) at p and the orientation of S'(f) at p' determine an orientation of M at p = p'. If the latter orientation coincides with one chosen for M, then the intersection point p is assigned +1, otherwise 1. Note that to define the secondary obstruction we choose an orientation of M and after perturbing S(f) choose local orientations of S(f). The choice of the orientation of M is essential; the change of the orientation of M leads to the change of signs of each intersection point and thus the secondary obstruction of the same mapping f corresponding to different orientations on M differ by sign. Let us show that the secondary obstruction does not depend on our choice of the orientations of the singular set in neighborhoods of the intersection points. Let p n p' be an intersection point, (e1, e2) be two independent vectors tangent to S(f) and (e3, 4) the corresponding vectors tangent to S'(f). Then the orientation el A e2 of S(f) at p leads to the orientation el A e2 A e3 A e4 of M. The other orientation e2 A el of S(f) leads to the same orientation e2 A el A e4 A e3 = el A e2 A e3 A e4. It is easy to show that the secondary obstruction is independent from the choice of the perturbation S'(f). We omit the proof since it is wellknown and will also follow from the definition of the secondary obstruction which we give below. In terms of homology groups, the secondary obstruction can be defined as the homology class of the signed set of intersection points, i.e., as an element of Ho(M; Z), which under assumption that M is connected can be identified with Z. 5.2 Second Definition Let K be a 2dimensional submanifold of an oriented 4manifold .3', and F be an orientation system of local coefficients over K. We refer to the paper [21] for the definition of the normal class or the Euler class of the normal bundle over K in [1', which is a cohomology class e e H2({K; ). The number (e, [K]), where [K] E H2(K; .) is the fundamental class of K, is an integer called the normal Euler number of the embedded manifold K. We denote this number by e(K). Note that the sign of the normal Euler number depends on the orientation of 31 (see Massey [21]). For a general position mapping f from an oriented closed 4manifold [T into a 3manifold N3, we define the secondary obstruction e(f) as the normal Euler number of the embedded 2dimensional submanifold S(f). This definition is consistent with the first definition of the secondary obstruc tion. 5.3 Invariance of the Secondary Obstruction Under Homotopy Let us show that the secondary obstruction e(f) is invariant under homotopy of the mapping f : 1  N3. Lemma 5.3.1 Suppose that fo and fl are two homotopic general position map pings. Then e(fl) = e(f2). Proof. Let F : 1' x [0, 1] N3 x [0, 1] be a general position homotopy joining fo : .1T x {0} N3 x {0} and fl : 11 x {1}  N3 x {1}. The boundary of the singular set B of F is the union of the singular sets Bo of fo and B1 of fi. Let it : Bt B, t = 0, 1, denote the inclusion, F the orientation system of local coefficients on B, and let e E H2(B; F) be the Euler class of the normal bundle of B in .[1T x [0, 1]. Then e(fo) e(fi) = (ie, [Bo]) (i*e, [Bi]) (e, io [Bo] i [Bi]) = 0, since io* [Bo] i*, [B] corresponds to the boundary of B and vanishes in H2(B; T). * CHAPTER 6 HPRINCIPLE The Hprinciple applies to differential relations over a fiber bundle and if holds, it reduces the question of the existence of a solution to the question of the existence of a section of a certain map. 6.1 Differential Relations Given a fiber bundle 7 : E  M, the jet space Jk(w), k = 1, 2,..., oo, associated with 7 is the space of germs of sections of the bundle 7. We will be interested in a particular case where the space E is the product M x N and the bundle map 7 is the projection onto the first factor. The sections of such a bundle are naturally identified with mappings M  N into the fiber and the jet space Jk (() is isomorphic to the jet space Jk(M, N). A differential relation R of order k over a bundle 7 is an arbitrary subset of the jet space Jk(r). As in section 2.4, a section f of the bundle 7 leads to a jet extension jf. We ;? that f is a solution of the differential relation R if the image of the jet extension jf is in 7. C Jk(T). Example. If the manifolds M is R" and the manifolds N is R", then a mapping M > N is a set of n smooth functions {fi} and a differential relation R of order k is an arbitrary relation on the values and the partial derivatives of {fi} of order order < k. A mapping {fi} is a solution of the differential relation R if its values and partial derivatives satisfy the relation. 6.2 HPrinciple We ?v that a mapping f : Y X is a section of a mapping p : X Y if the composition p o f is the identity mapping of X. If f is a solution of a differential relation 7, then its jet extension jf is a section of the composition 7rM : 7Z J(r)  E M of the inclusion, projection map of the jet bundle and r. The Hprinciple ii, 1 the statement converse to the given one up to homotopy: HPrinciple. Every section of the mapping 7M : 7  M is homotopic through sections to a section that arises as the jet extension of a section of r. We refer to [13], [10] and [38] for numerous examples of differential relations for which the hprinciple holds. Let us conclude the section with illustration of the hprinciple by interpreting the SmaleHirsh theorem [15] in terms of jet spaces. The SmaleHirsch theorem asserts that a smooth mapping f of a manifold M of dimension m into a manifold N of dimension n, n < m is homotopic to an immersion if and only if the differential df : TM TN is homotopic to an injective homomorphism, i.e., to a homomorphism of tangent bundles that is injective on each fiber. To interpret the SmaleHirsch theorem in terms of jet spaces, let us recall that the space of linear homomorphisms T1M TyN, where x ranges over the points of M and y ranges over the points of N, is isomorphic to the 1jet space J1(M, N). We define the differential relation R as the set of those points in J1(M, N) that correspond to injective homomorphisms. Then the SmaleHirsch theorem asserts that the differential relation R abides by the hprinciple. 6.3 Morin Singularities. Let M be a manifold of dimension m and N a manifold of dimension n < m. The mildest singularity of a mapping f : M  N of order 1 is the fold singularity in some coordinate neighborhood of which the mapping f has the form P(,. 11,a X') = (Xi, ...,a' X _1, *** x ). We will denote its ThomBoardman symbol J = (m n + 1,0) by J1. The mildest singularities of f of order k are called Morin singularities. The normal form of f in a neighborhood of a Morin singularity of order k is Ti = ti, i l1,2,...,n r, Li = li, i 2,3,...,r, (6.1) r Z = Q+ ltkt+k Q k ...k 2 t=2 Its ThomBoardman symbol is J = (m n + 1,t1,..., 1, 0) of length r + 1 and will be denoted by U,. We note that the Morin singularity with symbol J2 is the cusp singularity and the Morin singularity with symbol J3 is the swallowtail singularity. Returning to the jet spaces, let 0o = k C Jk(M, N) be the subset corre sponding to the set of regular points and for r > 0, Q, r= k denotes the subset corresponding to the ThomBoardman singularities with symbols Jt, t < r. Theorem 6.3.1 (Morin [23]) Suppose that the jet extension of a general position ,'rl. ':,jl f M  N takes M into the set Q". Then each .:,.i1,.lI, point of the i,'I; ':, f is Morin. We emphasize that in the Morin theorem the requirement that the mapping f is in general position, i.e., that the jet extension jf is transversal to Thom Boardman stratum, is essential. 6.4 AndoEliashberg Theorem The AndoEliashberg theorem asserts that if dim M > dim N > 2, then the differential relations Q, abide by the hprinciple. The Morin theorem allows to formulate the AndoEliashberg theorem as follows. Theorem 6.4.1 (AndoEliashberg [3]) Let M and N be orientable z,,<,;.: ..],] Let dimN > 2. Then for i,;, continuous section s : M + Q there exists a Morin map g : M  N such that jg : M  Q becomes a section fiberwise homotopic to s in Qr. The proof of Theorem 6.4.1 in [3] shows that the relative version of Theorem 6.4.1 is valid as well. In other words, suppose that U is an open set in M and s : M + Q, is a section such that the restriction of s to a neighborhood of M \ U is the jet section jg induced by a Morin mapping g : M \ U + N. Then g admits an extension to a Morin mapping g : M + N whose jet section jg is fiberwise homotopic in Qr to s by homotopy constant over M \ U. In particular the AndoEliashberg theorem reduces the question of the existence of a fold mapping of an orientable 4manifold M into an orientable 3manifold N to the problem of finding a continuous section of the mapping 7m/r: Q, ) j2 (M, NV) ) M x N M. CHAPTER 7 UNIVERSAL JET BUNDLES Let Jok' ,IR) be the vector space of kjets of germs (.' ,0) ( (R", 0), i.e., the fiber of the jet bundle Jk([. R") over the point (0, 0) e R" x R". A universal kjet bundle J(m, n) is a bundle with fiber Jk(P.' R") over a universal space BQ such that for any manifolds M and N, the kjet bundle Jk(M, N) can be induced from J(m, n) by an appropriate mapping of the base space M x N  BQ. The universal kjet bundles turns out to be very powerful in study of singulari ties of smooth mappings. Using the universal kjet bundles, Haefliger and Kosinski proved that the homology obstructions are Poincare dual to polynomials in terms of StiefelWhitney classes. Recently, Kazarian constructed and described spectral sequences associated with classifications of singularities which encode not only the fundamental classes of singularities but also the .,li i:ency of classes of different singularities. 7.1 Construction 1 Let ESOk > BSOk be a universal vector bundle classifying the orientable vector bundles of dimension k. For each manifold X of dimension i, let 'x : X  BSOi denote the mapping classifying the tangent bundle of X. Let xx y be a point of BSOm x BSO,, F?" be the fiber of ESOm over x, T. be the fiber of ESO, over y. Then the fiber of the bundle ESOm x ESO, over xx y is " x R". We define the set Jf~x(m, n) as the set of kjets of mappings (.' 0) (R, 0) and then define a universal kjet space Jk(m, n) as the union UJfx(mn, n) that ranges over the pairs x x y of BSOm x BSO,. Note that the universal kjet space has a natural structure of a bundle over BSOm x BSO,. Let us prove that for any mmanifold M and nmanifold N the bundle induced from the universal kjet bundle by the mapping TM X TN : M N  BSOm x BSO, is isomorphic to the kjet bundle Jk(M, N). Indeed, let Jk(M, N) be the space of kjets of mappings TzM  TyN, where x x y ranges over the points of M x N. By construction, the space Jk(M, N) is a bundle over M x N isomorphic to the bundle induced from the universal kjet bundle by TM x TN. Lemma 7.1.1 Let M and N be Riemannian ,,,r,:... 4. Then the kjet bundle Jk(M, N) is cano,.:. /ll/; isomorphic to Jk(M, N). Proof. For each point x of M there is a neighborhood Ux of the origin in TxM diffeomorphic under the exponential mapping to a neighborhood V, of x in M, ExpM : UX V,. Let {Exp } be the set of exponential mappings for the manifold N. Then the mapping Jk(M, N) Jk(M, N) that takes the kjet over x x y represented by a mapping f : TxM TN into the kjet of the mapping ExpN o f o (ExpfM)1 is a canonical isomorphism 0. 7.2 Construction 2 In this section we sketch the construction by Kazarjan. Let k be a positive integer. The group G JOkDiffR x JkDiI!. , where Dil!F.' and DiffR" stand for the orientation preserving autodiffeomorphism groups of (F.' 0) and (R", 0) respectively, is a finite dimensional Lie group with a natural action on the vector space JOk(.' I R). The natural action of g is defined so that an element of g represented by a pair of autodiffeomorphisms dm x dn takes the kjet represented by a germ f into the kjet of the germ d, o f o d,. Let E  BQ be the universal gbundle over BQ. Then the space Jk(m, n) is defined as the bundle Jk(mn, E xg ( E xJ ), over Bg associated with the universal bundle E. Since the group g is homotopy equivalent to SO, x SO,, the space BQ is homotopy equivalent to BSO, x BSO,. It is easily verified that the bundle Jk(m, n) is the universal jet bundle. 7.3 Construction 3 If ( is a vector space, then (or = ( o ( o ... o ( denotes the vector space defined as the vector space (0r factored by the relation of equivalence: vl 0 v2 & "." r v w 1 w2 wr if and only if there is a permutation of r elements a such that I = ,,., for i 1, ..., r. The space (o" is called the symmetric rtensor product of (. As in the interpretation of SmaleHirsch theorem in terms of jet spaces, for every r, the bundles ( and T give rise to the bundle 'OM/(r, TI). The fiber of 'OM(0', Tr) over a point x E M is the set of homomorphisms Hom(r'", rx) between the fibers r" and q, of the bundles (or and r respectively over x. The space 'OM ((, 7) in the formulation of the SmaleHirsch Theorem is generalized by the vector bundle S' (, T) = HOM (, T) ED OM.( o Tr) E... E H OM (, r) over M (see paper [28] of Ronga). As above we define S"(0 Tr) as the inverse limit lim S(, ). Let kr(g) denote the rank of the kernel of a linear function g. A point of S"((, T) over a point x E M is a set g = {g} that consists of homomorphisms gi E Hom(o', ,i). We set i, = UE1M{(g S(>z, rI)l kr(gl) ii}. (7.1) Let Kh and Ch respectively denote the kernel and cokernel of a homomorphism h E Hom(, r). The composition of natural homomorphisms Hom( o TI) ) Hom(, Hom(, T7)) Hom(Kgl, Hom(Kgl, Cgl)) takes the homomorphism g2 E Hom( o x, qTp) into some homomorphism g2. We define Ei',iz UEM{g E S'(, T,) Ig E Ei, and kr(g2) 2}. (7.2) This construction proceeds by induction. We refer the reader to [6], [29]. The bundle J'(M, N) is isomorphic to the bundle S'(, TI), where = TM and ] = f*TN. Moreover, there is an isomorphism of bundles S"(', rI) and J~(M, N) that takes each Ej isomorphically onto Ej [29]. 7.4 Corollary of AndoEliashberg Theorem As a corollary of the AndoEliashberg theorem, we obtain that the existence of a homotopy eliminating the cusp singular points of a mapping f into an orientable 3dimensional manifold is independent of f. Corollary 7.4.1 The homote(.i' class of f : M"  N3,m > 3, contains a fold I,'.rl'.:.j if and only if there is a fold in':rl'.:.'j g : Mm R3 Proof. By AndoEliashberg theorem, the homotopy class of f contains a fold mapping if and only if there is a section Mm  1 C J2(TMm, f*TN3). The latter does not depend on f or TN3 since the tangent bundle of an orientable 3manifold is trivial. U CHAPTER 8 SECONDARY OBSTRUCTION 8.1 Definition of the Secondary Obstruction Let M be a closed connected oriented 4manifold. The intersection form defined on the free part of H2(3'; Z) is a quadratic form. Let Q(3[1) denote the set of integers taken on by this quadratic form. We note that the set Q(.1') and the normal Euler number of a general position mapping depend on the choice of orientation of M ['. It is easily verified, however, that for a given mapping f : [1P  N3, the condition e(f) E Q([1) does not depend on the orientation of M\. In sections 8.2 and 8.3 we will prove the main theorem . Theorem 8.1.1 Let f : [1P  N3 be a general position inr.,':..j from an orientable closed connected 4,,n1.,.:f./.I into an orientable 3,,,in.;,.: .I. Then the homote'i./; class of f contains a fold I'"'l'l.:I if and only if e(f) E Q(i1). 8.2 Necessary Condition The invariant e(f) allows us to give a necessary condition for the existence of a fold mapping into N3. In the later sections we will prove that this condition is also sufficient. With every oriented closed connected 4dimensional manifold 3.1' we associate the set Q(.1) of integers each of each is the normal Euler number of an orientable surface in 1\. Lemma 8.2.1 If f : 1M' N3 is a fold I'""''r.,:. then e(f) E Q(i[). 1 After the paper was written, O. Saeki informed the author that he obtained similar results using a different approach [32]. Proof. The singular set of a fold mapping consists of the surfaces S_ (f) of indef inite fold singular points and S+(f) of definite fold singular points. Therefore, e(f) = e(S_(f)) + e(S+(f)). In [30], [1] it is proved that e(S_(f)) = 0. Hence, e(f) = e(S(f)). Since S+(f) is orientable (see Lemma 4.2.1), we conclude that e(f) e Q(3'). U Corollary 8.2.1 Suppose that the homote(i'., class of a general position I''rr'.:" f : M' N3 contains a fold inqr. Then e(f) E Q9(1'). U 8.3 Sufficient Condition The objective of this section is to prove that the condition e(f) E Q(01) is sufficient for the existence of a fold mapping homotopic to a general position mapping f : [1  IR3. In view of Corollaries 4.4.1 and 7.4.1 this completes the proof of Theorem 8.1.1. Lemma 8.3.1 Let f be a general position ii,,l'l'.'from a connected closed oriented 4,,,,r,:f.1,* I[T1 intoR3. Suppose e(f) E Q(3i1). Then .i1 admits a fold in'"r'. :' into R3. Proof. The condition e(f) e Q(Pi1) guarantees the existence of an orientable 2submanifold S of .1 with normal Euler number e(f). Let us prove that in the complement .1 \ S, there is an orientable possibly disconnected embedded surface S such that (P) every orientable surface embedded in 1 1 \ S with nontrivial normal bundle intersects S. If l[1 \ S admits no orientable embedded surface with nontrivial normal bundle, then the property (P) holds for any orientable embedded surface S. Suppose that in .1 [ \ S there is an orientable embedded surface with nontrivial normal bundle and that a surface with property (P) does not exist. Then for any positive integer k there is a family of oriented embedded surfaces {Fi}i, 1...,k such that each of the surfaces has a nontrivial normal bundle and does not intersect the other surfaces of the family. Let Tor H2(3 PT \ S; Z) denote the subgroup of H2(3[ P \ S; Z) that consists of all elements of finite order. The group H2(3 T \ S; Z)/Tor H2(3 \ S; Z) is finitely generated. Fix a set of generators el, ..., es. Every surface Fi represents a class [Fi] in H2 (3 \ S; Z)/Tor H2([ \ S; ), which is not trivial since F, has a nontrivial normal bundle. Moreover, [Fi] [Fi] / 0 and [Fi] [Fj] = 0 for i / j. If the number k of the surfaces is greater than the number s of the generators, then there is a combination al[FI] + a2[F] + + ak[Fk] = 0 with ac + + ac / 0. Multiplication of both sides by [Fi], i = 1,..., k, gives aci[Fi] [F] = 0. Therefore, ai = 0 for every i = 1, ..., k. Contradiction. Thus a surface S with property (P) exists. Let us construct a mapping for which the set S U S is the part of the singular set. We recall that we identify the base of a vector bundle with the zero section. Lemma 8.3.2 There is a general position Ini'l'.:.j h : NS R3 from the normal bundle NS of S in 1' such that the set S is the set of /. I;,'.:/. fold .:,,i.,lr, points of h and h has no other : ..:l.,, points. Proof. The fiber of the bundle NS is diffeomorphic to the standard disc D2 { (x, y) E R2 x2 + y2 < 1}. Let m : D2 [2, 2] be the mapping defined by the formula m(x, y) = x2 + 2. Then m is a Morse function on D2 with one singular point. For every open disc U, in S, the restriction of the normal bundle NS to U, is a trivial bundle U, x D2 + U,. Let 13 denote the segment (3, 3). We define the mapping g_ : U x D2 U x I3 by ga(u, z) = (u, m(z)), where u E Ua, and z E D2. Note that rotations of D2 do not change the function m. We may assume that the fiber bundle NS  S is an SO2bundle. Then the mappings g9 give rise to a mapping g : NS  S x I3. The open oriented manifold S x I3 admits an immersion into IR3. We define h : NS  IR as the composition of g and this immersion. U Lemma 8.3.3 Let NS be the normal bundle of S in M1'. Then there is a general position i,.r'l'':', h : NS  R3 such that the set S is the :u.gqlar set of h and every component of S has at least one cusp .:,u.;,1.i, point of h. Proof. For a closed disc D C S, the restriction of the bundle NS S to S \ D is a trivial bundle (S \ D) x D2 (S \ D), where D2 is the disc as in Lemma 8.3.2. Let 13 denote the segment (3, 3). The function mi : D2  3 defined by mi (x, y) = x2 y2 is a Morse function with one singular point at the origin. Let idl : S \ D , S \ D be the identity mapping. Put El = (S \ D) x D2 and BI = (S \ D) x Is. Then idi x mi : El B is a fold mapping. Set E2 = S1 x (1, 1) x D2 and B2 = S1 x (1, 1) x I3. There is a mapping g : E2 B2 such that 1) the set S1 x (1,0) x {(0, 0)} is the set of all indefinite fold singular points of g, 2) the set S1 x (0, 1) x {(0, 0)} is the set of all definite fold singular points of g, 3) the curve S1 x {0} x {(0, 0)} is the set of all cusp singular points of g. Let U denote the intersection of S \ D and a collar neighborhood of 9(S \ D) in S \ D. Then U is diffeomorphic to S1 x (1, 1/2). We can identify the subset U x D2 of E1 with the subset S1 x (1, 1/2) x D2 of E2 and the subset U x I3 of B1 with the subset S1 x (1, 1/2) x I3 of B2 so that the obtained sets El U E2 and B1 U B2 are manifolds and the mapping idl x mi coincides with the mapping g on the common part of the domains El n E2 C El U E2. Thus, idl x m1 and g define a cusp mapping c : E1 U E2 B1 U B2. Note that E1 U E2 is diffeomorphic to (S \ D) x D2 and B1 U B2 is diffeomorphic to (S \ D) x 13. Let m3 : D2  I be the Morse function, defined by m3(x, y) = 2 + y2, 0 0 0 and id3 : D be the identity mapping of the open 2disc D D \ OD. Then id3 x m3 : D x D2 D x I3 is a fold mapping. Let V be the intersection of D and a tubular neighborhood of OD in S. Then V is diffeomorphic to S1 x (1/2, 1). We identify the part V x D2 of E3 = Dx D2 with the part S' x (1/2, 1) x D2 of E2 C El U E2 and the part V x I3 of B3 = Dx Is with the part S1 x (1/2, 1) x Is of B2 C B1 U B2 so that 1) the obtained sets E = El U E2 U E3 and B = B1 U B2 U B3 are manifolds, 2) the mapping id3 x m3 coincides with c on the common part of the domains, 3) the manifold E is diffeomorphic to NS. The condition (3) can be achieved since the mapping m3 does not change under rotations of the fiber D2 Then id3 x m3 and c define a cusp mapping NS  B. Note that B M S x 3I is an open orientable 3manifold. Thus, it admits an immersion into R The composition of NS > B and the immersion B > R3 is a cusp mapping satisfying the conditions of the lemma. U We identify NS and NS with open tubular neighborhoods of S and S in PT respectively. There is a general position mapping g : 1P  IR3 which extends h : NS IR3 and h : NS  R. In general the extension g has some swallowtail singular points. Let us prove that we may choose g to be a cusp mapping. Ando (see Section 5 in [3]) showed that the obstruction to the existence of a section of the bundle Q2(Tl 1 TR3) over the orientable closed 4manifold .1 coincides with the number of the swallowtail singular points of a general position mapping .1  R3 modulo 2. Also Ando calculated that this obstruction is trivial. Since the mapping h U h does not have swallowtail singular points, the obstruction to the existence of an extension of the section j3(h U h), defined over NS U NS, to a section of Q2 over 1 P is trivial. Therefore, the relative version of the Ando Eliashberg theorem (see chapter 7) implies the existence of an extension to a cusp mapping g : 3[1  R3. The singular set S(g) consists of S U S and probably of some other connected submanifolds A, ..., Ak of 1\. We have e(f) = e(S(g)) = e(S) + e(S) + e(Ai) + e(A2) + + e(Ak). (8.1) The normal Euler number of the submanifold S equals e(f). Hence the sum of the normal Euler numbers e(Ai) + + e(Ak) equals e(S). Let At be a component of UAj. Suppose At is a surface of definite fold singular points. The surface At is orientable (see Lemma 4.2.1) and does not intersect S U S. By definition of S, this implies e(At) = 0. Suppose At is a surface of indefinite fold singular points. Then again e(At) = 0 [30], [1]. Therefore, e(At) is nontrivial only if the surface At contains cusp singular points. Let us recall that the union of those components of the singular submanifold S(g) that contain cusp singular points is denoted by C = C(g). The equation (8.1) implies that e(C) e(S) + e(Ai) +... + e(Ak) 0. It remains to prove the following lemma. Lemma 8.3.4 If g : [1V  R3 is a cusp ina'l.':hg and e(C) = 0, then there exists a homote'l,, of g eliminating all cusp /.:,tuli r points. Proof. If the curve of cusp singular points is not connected, then there exists a homotopy of g to a mapping with one component of the curve of cusp singular points. We may require that the homotopy preserves the number e(C). We omit the proof of these facts since the reasoning are similar to those in section 4.5. We will assume that the curve of cusp singular points 7(g) is connected and hence so is C(g). Lemma 8.3.5 Let v(x) be a characteristic vector field on 7(g). If e(C) = 0, then v(x) can be extended on C(g) as a normal vector field. Proof. For a general position mapping g : 3i1 R3, the set F =f (f(C (g))) 9 1/' is an immersed 3manifold. The selfintersection points of F correspond to the points of the surface C_(g). We v that two vectors vl and v2 of a vector space have the same direction if vl = Av2 for some scalar A / 0. There is an unordered pair of directions (I1(p), 12(P)) over C_(g) [1] with the following property. For every point p of C_(g) there are a neighborhood U about p with coordinates (xl, X2, x3, x4) and a coordinate neighborhood about g(p) such that the restriction glu has the form (xi, x2, x3 x4) and the directions of the vectors 8/8x3 and 8/8x4 coincide with l1(p) and 12(p) respectively. An Lpair is a pair (/i(p),12(p)) that satisfies this property. Let FI C C_(g) denote the complement of a regular neighborhood of the curve 7(g) in C_(g). The proof of Lemma 3 in [1] shows that there is a vector field v(p) in the normal bundle over Fi with directions li(p) + 12(p) or 11(p) 1(p) over the boundary OF, for some pair (li(p), 2(p)). We v that a direction at a cusp singular point is an xdirection if it is tangent to the surface S(f) and transversal to the curve 7(g). Note that for a special coordinate neighborhood about a cusp singular point the direction of the vector 8/8x has an xdirection. It is easily verified that for an pair (II(p),12(p)), the directions lI(p) 2(p) are tangent to F at every point p in C_(f). Furthermore the directions li(p) + 2(p) and 11(p) 2(p) approach the same xdirection as p approaches 7(g). It implies that the vector field v(p) over Fi has an extension to C_(g) such that v(p) is transversal to C (g) at every point of C (g) and has an xdirection at every point of 7(g). If necessary, we multiply the vector field v(p) by 1 to get a vector field which points toward C (g) over 7(g). Now the vector field v(p) can be modified in a neighborhood of 7(g) so that a new v(p) is normal to C_ (g) at every point of C_(g) and the restriction of v(p) to 7(g) is the characteristic vector field v(p). 47 The obstruction to the existence of an extension of v(p) to a vector field over C(g) is the normal Euler number e(C). Since e(C) = 0, such an extension exists.E The ends of the vectors v(p), p c C+(g), define an embedding of an orientable surface H diffeomorphic to C+(g) into M '. We modify the embedding in a neigh borhood of the boundary OH so that the new embedding defines a basis of surgery. Now Lemma 8.3.4 follows from Lemma 4.5.1. U The proof of Lemma 8.3.1 is complete. U CHAPTER 9 COMPUTATION OF THE SECONDARY OBSTRUCTION In this chapter we will express the secondary obstruction in terms of the Pontrjagin class pi(C1V) of the tangent bundle of Mi'. Namely, we will prove the formula1 e(f) = (pi(i/C), [M4]), where [M4] is the fundamental class of the manifold 1'. Lemma 9.0.6 Let f be a general position i'.ri.':.j from a closed oriented connected 4 i,n,,:f.4.1 11' into an orientable 3if,,,.',.....1N3. Let pi(M ) denote the first Pontrjagin class of 1 and [M4] the fundamental class of I 1I. Then e(f) (pi( ), [M4]). It allows us to formulate the theorem 8.1.1 in terms of the cohomology ring of m\. Theorem 9.0.1 Let f : [1P  N3 be a continuous i'n'r'.:',j from an orientable closed connected 4,,,,/,..:.. into an orientable 3n,,.n,..'.11 Then the homotel,'q class of f contains a fold il.ni1'. i if and only if there is a 4'l..1,. /.i./; class x E H2(l ;Z) such that pi(C) =x2. A smooth mapping f : P1 N3 induces a section j2f of the 2jet bundle J2(T 1, f*TN3) over .M. To calculate the invariant e(f) we consider sections Ai1 [ J2( l), where ( is an arbitrary orientable 4vector bundle over [11 and Tr is an arbitrary orientable 3vector bundle over 11. 1 After the paper was written, the author learned that this equality is a special case of a result obtained in [24]. The singular set E in the bundle J2( rT) over .31 is a manifold with sin gularities. By dimensional reasoning, the image of a general position section j : ll  J2((, T) does not contain singular points of the manifold with singular ities E. Consequently, the :,i.l;,jri set j1(E) of the section j is a submanifold of i[1. We define the normal Euler number e(j) of the section j as the normal Euler number e(j1(E)). A regular neighborhood E of E in J = J2(, r( ) is an open manifold. There is a system of local coefficients F over E, the restriction FlT of which gives a Zorientation of E. The Poincar6 homomorphism for cohomologies and homologies with twisted coefficients takes the fundamental class [E] onto some class Tr H2(E, E \ E; ). Note that TT is in H4(E, E \ E; Z). Let i be the composition H4(E, E \ E; Z) H4(J, J \ E; Z) H4(J, 0; Z) of the excision isomorphism and the homomorphism induced by the inclusion. We define h((, Tr) = i(rr). Then we claim that e(j) (j*h((, f ), 1']). (9.1) Lemma 9.0.7 For every general position section j : ll J2 (, I), the normal Euler class of the ;n, i'.,. j'() is given by (9.1). In particular, for every i,,j'~r.,:, f : N3, we have e(f) (( 2f)*h(TM, f*TN3), [M4]). Proof. Let A C [ll denote the singular set j1(E) and B denote a tubular neighborhood of A. The tubular neighborhoods E of E and B of A may be viewed as vector bundles. Since j : B > E is transversal to E, there is a commutative diagram of vector bundles B E A 1 A S . The Thorn class 7 possesses the property of naturality. Hence the Thorn class of the bundle B  A is j*(r). We have a commutative diagram i: H4(E, E \ ; Z) H (JJ E;Z) H4(J;Z) H4(B,B \ A;Z) H4( F1', [1 \ A;Z) H4 (I;Z), which completes the proof. U Lemma 9.0.7 shows that the number e(j) depends only on the bundles ( and T1. That is why we will denote this number by e( T). In the following, for an arbitrary manifold V, we denote the trivial line bundle over V by r(V) or simply by 7. Lemma 9.0.8 There is an integer k / 0 such that for i,:, orientable 4vector bundle ( over wiq:; closed oriented 4il., .f.'41./ 1\', the c(,;,;/I (pi(0, [M41) ke(, 3r) holds. Proof. We recall (see chapter 7) that the bundle J2( 3r) over P1I is induced by an appropriate mapping p : 3[1 BS04 x BSO3 from some bun dle J2(E4, E3) over BS04 x BSOs. As above we define a cohomology class h(E4, E3) E H4(j2(E4, E3); Z). Let a be an arbitrary section of the bundle J2(E4,E3). Together with p, the section a defines a section j : Pi  J2(, 3r) such that the diagram j2 2 (E4, E3 31 BSO4 x BSO3 commutes. We have j*h(, 3r) = j*p*h(E4, E3) = p*a*h(E4, E3), where p denotes the upper horizontal homomorphism of the diagram. Con sequently, the class j*h(3, 37) is induced by p from some class a*h(E4, E3) in H4(BS04 x BSO3; Z). Moreover, since 3, is a trivial bundle, the mapping p is homotopic to a mapping .1 P BS04 x pt c BS04 x BSO3 and therefore j*h(, 37) is induced from some class in H4(BSO4; Z). Modulo torsion the group H4(BSO4; Z) is isomorphic to Z E Z and is generated by the first Pontrjagin class pi and the Euler class W4. Since H4(3F; Z) is torsion free, for some integers k and 1, we have j*h(r, 37) =kpi( ) + lW4(). (9.2) Let us apply (9.2) to the 4sphere S4 with = TS4. The singular set of the standard projection f : S4 R4 + R3 is a 2sphere with trivial normal bundle in S4. Hence (j*h((, 37), [S4]) e(f) = 0. Since pi(TS4) = 0 and W4(TS4) = 2, we conclude that 1 0. Finally, k / 0 follows from pi(0) / 0 for some U To find the number k of Lemma 9.0.8 we need another description of the invariant e(j). Let Tr and ( be vector bundles over a manifold .1'. There are natural projections pr : ( E ()or _+ or and inclusion i : TI  TI (. A point of J"'(, rl) is a set of n homomorphisms {gi}i 1,...,, (see chapter 7). Define the embedding by sn(gl,,gk) =(gl id, io g2 op2, ... io gn opn). The homomorphism s,, n > 1, is called the stabilization homomorphism afforded by (. Lemma 9.0.9 (Ronga 7']) 2. E2 1(( TI E D )) i(,) and 2. s21(Yi,j( E(,TI(D)) Ei^j(,TI), and 3. the embedding s2 is transversal to the ,D',l,,,i.:l..1 j (, () and Let ( be an orientable 4vector bundle, T the trivial 3vector bundle over . ', and S2 the stabilization homomorphism afforded by the trivial line bundle 7 over .i. Lemma 9.0.9 allows us to give a definition of e(f) in terms of some cohomology class of H4(J2( T Z; Z). Lemma 9.0.10 There is a /. ,./.,i/; class h E H4(J2( T, T); Z) such that for a section j : 31 J2(, l), we have e(j) ((2 o j)*(h), [M4]). The proof of Lemma 9.0.10 is the same as that of Lemma 9.0.7. Let ( and rl be the vector bundles of dimensions 4 and 3 respectively over the standard 4disc D4. The set of regular points in J2( E Tr, rl E r) is homotopy equivalent to SO5. Therefore, each section j : S3 j 2( Tr TI r) that sends S3 into the set of regular points defines an element j in the set of homotopy classes [S3, Os]. The space SOs is an Hspace; hence j is an element of w3(S05) =Z. Since J2( TE, 'E ) is contractible, the section j admits the extension to a section over D4. We obtain a mapping e : 73(S05) Z that sends the homotopy class ) of a section j to the normal Euler number of the singular set of the section j extended over D4 Lemma 9.0.11 The i''r!.:.! e : 73(SO5) Z is a well /. J,... homomorphism. Proof. Let jl and j2 be two sections of the bundle J2 ( T', T E r) over D4 whose restrictions ji aD4 and j2 aD4 map S3 = OD4 into the set of regular points of J2( T~ T E r) and represent the same homotopy class j e [S3, SOs]. The arguments similar to those in the proof of Lemma 5.3.1 show that the normal Euler numbers of the submanifolds ji l() and j2 1(), where Z is the singular set of J2( T, T r), are equal. Therefore, the number e(j) does not depend on the choice of representative of the homotopy class j. We need to verify that the equality e(j + j2) = e(I) +e(2) (9.3) holds for every pair of elements 1, j2 of r3(S05). For i = 1, 2, let ji : OD' J2( D, T E r) be a section that leads to Ji. We can modify jl by homotopy that does not intersect the singular set of J2( E T, rl D T) so that the sections ji and j2 agree on some open subset of OD = OD. Then jl and j2 determine a section j3 : aOD#OD/ J2( T, y T), which leads to an element ji + J2 of 73(S05). Extensions of ji and j2 to D' and D' respectively give rise to an extension of j3 to DI#D' the singular set of which is the union of the singular sets of the extensions of jl and j2. Therefore, (9.3) holds. U We have defined the normal Euler number of a general position section of the bundle J2((, r) in the case where ( is an orientable 4vector bundle and Tl is an orientable 3vector bundle. If ( is an orientable 5vector bundle and Tl is an orientable 4vector bundle, then again the singular set of a general position section j of J2( q) is a 2submanifold of .31 and therefore we can define the normal Euler number e(j) and the number e(, Tr) in the same way as above. Let g : S3  SOs be a generator of 73(S0O) and d the number e(g). Let us calculate e(6 E T, 4r), where 6 is the 4vector bundle associated with the Hopf fibration S7 S4. Lemma 9.0.12 e(6, 37) = e(6 E T, 47) = d, up to sign. Proof. The sphere S4 is a union of two discs D1 and D2 with OD1 = OD2. A choice of trivializations of 6 over D1 and D2 defines a gluing homomorphism a: 6 E rlaDi 6 T aD2, (9.4) which being identified with a mapping S3 SOs represents a generator [a] E 73(SO5). Let J1 and J0 respectively denote the space J1(6 E r, 4r) and the complement to the singular set E in J1. To prove Lemma 9.0.12, it suffices to determine the normal Euler number of j1(E) for a particular section j : S4  J1. We regard a section of J1 as a bundle homomorphism 6 T  4r. If j is given over D1, then the diagram STrlaD1, E TrlaD2 t JD1 ido(jlD1)o0 ) 1 4lT I idD2 shows that in the trivialization of 6 over D2 the section jlo D is id o (jl aD) o a1, where id is the identity mapping. If we choose j to be constant over D1, then in the trivialization of 6 over D2 the section jlo D induces a mapping S3 J0 M S05 representing the homotopy class [a] E 73(SOs). Thus the normal Euler number of j extended over D2 is 0 up to sign. U Lemma 9.0.13 There is an integer q such that e(TCP2, 3) 1 + Id. Proof. There is a mapping f of a regular neighborhood E of CP1 C CP2 into R3 such that the singular set of f is CP1 (see Lemma 8.3.2). Let f be a general position extension of f on CP2. The number e(f) is the sum of the normal Euler number of CP1 and the normal Euler number of the surface of singular points that lies in the disc D4 = P2 \ E. The latter number is a multiple of 0. Hence for some q, e(f) 1 + '. U To calculate the exact value of e(TCP2, 3r) we use the notion of the connected sum of two bundles. For i = 1, 2, let 31 T be a closed oriented 4manifold and (i an orientable 4vector bundle over 31T'. Identifying the fiber of (1 over some point in 3 [f with the fiber of 12 over some point in 3i\ we obtain a bundle over 3 / V i\1 which is transferred to a bundle over .3 #3 1 IT by a natural mapping .3 T # 3 J 3 [ V 3 f. We denote the resulting bundle over .3[/ #3 IT by (1#2. It follows that the additivity properties (Pi(I #62), : (pi(I), [M4]) + (pi (2), !L and e(1#2, 3wT/: V) e(I, 3,T +)) +e(2, 3T(1j)) take place. Lemma 9.0.14 e(TCP2, 3) 3. Proof. Let 6 be the 4vector bundle over S4 with pl(6) = 2. Lemma 9.0.12 implies that e(6, 37) 0. For K = 2CP2, we have pi(TK#6#3) = 0, where 6#3 stands for 6#6#6. Lemma 9.0.8 shows that e(TK#6#3, 3T) is also zero. By additivity, 0 = e(TK#6#3, 3) = 2(1 + qd) + 30. Since q and 0 are integers, we conclude that e(TK, 3r) = 6. This implies that e(TCP2, 3r) = 3. On the other hand it is known (see Sakuma [36] and [37]) that e(TCP2, 3) 3 (mod 4). Therefore, e(TCP2,23T) 3. U Lemma 9.0.14 shows that the integer k in Lemma 9.0.8 equals 1. Thus, for every oriented 4manifold .1' and a general position mapping f : I'  R3, we have e(f) = (pi(3 ), [M4]). This completes the proof of Lemma 9.0.6. U Theorem 9.0.1 is proved. U CHAPTER 10 SIMPLY CONNECTED CASE This chapter is devoted to the case where the manifold .31 is simply con nected. We examine the equation pi(3 [) = x2 and determine when it has a solution. Theorem 10.0.2 Suppose that .1' is an orientable closed connected .:i,,'lp connected 4,m,,. ,: f. /.I and N3 is an orientable 3 n,,',:. f./I Then a homote('i.; class of a f.'' ':j f 3: I  N3 has no fold ii l'l!"''! if and only if 3i1 is homot(.ji'/ equivalent to CP2 or CP2# CP2. Here homote(.c' equivalence is not supposed to be orientation preserving. As has been shown, a homotopy class of a general position mapping f from a connected closed oriented 4manifold .3 into an orientable 3manifold N3 has a fold mapping if and only if e(f) = (pi(.1), [M4]) E Q(1). That is the number (pi(,3), [M4]) is a value of the intersection form of .31. First, let us consider the case where the intersection form of .31 is indefinite. If pi( ) = 0, then for every f, e(f) = 0 E Q(3P). Suppose pi(3 ) / 0. Lemma 10.0.15 If the intersection form of a closed .:,,1m; connected m,,;,,...f..1.1 3 with pl(M ) / 0 is .,/. I,./: odd, then Q(3 ) = Z. In particular (pi(1), [41) e Proof. Since the intersection form of .1I is odd, in H2(3P; Z) there exists a basis g,9g2, ...,g s *..., k such that the value of the intersection form at ale + + aes + 2 2 2 2 S+ akek is C+ +1 ... 2. Therefore, the number e(f) is in Q(.1) if and only if e(f) can be represented in the form ca + + ca ac2+ a for some integers ca, i 1, ..., k. Since the intersection form is indefinite, this sum has at least one positive square and at least one negative square. Since the signature a(o(1) = pi(P\') / 0, the number k of squares is at least 3. Suppose that the number e(f) is odd. Then it can be represented as the difference of two squares. Suppose that e(f) is even. Then the odd number e(f) 1 can be represented as the difference of two squares and the third square of the sum can be used to add T1 to the difference to get e(f). Hence Q(lP) Z. U Suppose that the intersection form of .3 [ is indefinite even. Being even, it is isomorphic to a direct sum of some copies of Esform and some copies of the form 0 1 with matrix Consequently, the number (pi(\l ), [M4]) = 3o(31) is 1 0) even. Every even indefinite intersection form contains a subform isomorphic to Since this subform takes every even value, we have (pi(\l ), [M4]) 1 0 Thus, every closed simply connected 4manifold with indefinite or trivial intersection form admits a fold mapping into R3. To treat the case where the intersection form of .3 1 is definite, we need the Donaldson Theorem. Let kJ, k / 0, denote the form of rank Ikl given by the diagonal matrix with eigenvalues 1 if k > 0 and 1 if k < 0. Theorem 10.0.3 (Donaldson [71) Every 1 /7i,,'/. intersection form of a closed oriented smooth 411,,:ni f./. is isomorphic to the form kJ for some integer k / 0. Lemma 10.0.16 Suppose that the intersection form of a connected closed .: ,, '1 connected innf../,1 i3 with pji(f) / 0 is 1, [f,,/, Then (pi(p'), [14]) E Q(i[1) if and only if the intersection form is isomorphic to kJ, Ik > 3. Proof. It suffices to consider only the case where k > 0. If k = 1, then the inter section form is isomorphic to that of CP2 and (pi(i ), [M41) = 3 1(1) is not in Q(.1l). For k = 2, the set Q(3.1) consists only of integers that can be represented as the sum of at most two squares. Hence the number (pi(i. ), [M4) = 6 is not in Q(lM'). If k = 3, then (pi(fl1), [M4]) = 9 E Q([1). Finally, by the Lagrange 58 theorem, every positive integer can be represented as a sum of four squares. Thus for k > 4, we have (pl(3 i), [M4]) C Q(i1). * In view of Theorem 8.1.1, Lemmas 10.0.15 and 10.0.16 imply that .3l1 admits a fold mapping into R3 if and only if the intersection form of 31 T is different from J and 2J. By the J. H. C. Whitehead Theorem about the oriented homotopy type of a simply connected 4manifold [22], [41], this completes the proof of Theorem 10.0.2. U CHAPTER 11 FINAL REMARKS Remark 1. If two manifolds 3M/ and .3j admit a fold mapping into R3, then the connected sum M,#M2 also admits a fold mapping into R3. In [34] the authors conjectured that the obstruction to the existence of a fold mapping into R3 is additive with respect to connected sum, and the manifold kCP2#lP2 admits a fold mapping into R3 if and only if k + 1 is odd. Theorem 10.0.2 solves the conjecture in the negative. Remark 2. Sakuma conjectured (see Remark 2.3 in [19]) that a closed orientable manifold with odd Euler characteristic does not admit a fold mapping into R" for n = 3, 7. Saeki [33] presented an explicit counterexample to this conjecture. Theorem 10.0.2 shows that there are many manifolds with odd Euler characteristic admitting fold mappings into R3. However, it should be mentioned that Theorem 10.0.2 does not i. 1 a method of an explicit construction and the question of an explicit construction of a fold mapping for a given manifold seems difficult. Remark 3. 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