New York Journal of Mathematics
New York J. Math. 1 (1995) 130–148.
The Index of Discontinuous Vector Fields
Daniel H. Gottlieb and Geetha Samaranayake
Abstract. The concept of the index of a vector ?eld is one of the oldest in Algebraic Topology. First stated by Poincare and then perfected by Heinz Hopf and S. Lefschetz and Marston Morse, it is developed as the sum of local indices of the zeros of the vector ?eld, using the idea of degree of a map and initially isolated zeros. The vector ?eld must be de?ned everywhere and be continuous. A key property of the index is that it is invariant under proper homotopies. In this paper we extend this classical index to vector ?elds which are not required to be continuous and are not necessarily de?ned everywhere. In this more general situation, proper homotopy corresponds to a new concept which we call proper otopy. Not only is the index invariant under proper otopy, but the index classi?es the proper otopy classes. Thus two vector ?elds are properly otopic if and only if they have the same index. This allows us to go back to the continuous case and classify globally de?ned continuous vector ?elds up to proper homotopy classes. The concept of otopy and the classi?cation theorems allow us to de?ne the index for space-like vector ?elds on Lorentzian space-time where it becomes an invariant of general relativity.
Contents Introduction A. The Results B. Organization of the Paper C. De?nition of Otopy and Index D. Guides 1. The De?nition for One-dimensional Manifolds 2. The Index De?ned for Compact n-Manifolds 3. The Index for Locally De?ned Vector Fields 4. The Index of a Defect 5. Properties of the Index References 130 131 131 132 134 136 138 142 144 145 147
The authors would like to thank James C. Becker for many excellent suggestions which improved the de?nitions and eliminated subtle errors of exposition.
Received May 15, 1994. Mathematics Subject Classi?cation. 55M20, 55M25, 57R25. Key words and phrases. manifolds, homotopy, degree, ?xed point, critical point, space-time.
c 1995 State University of New York ISSN 1076-9803/95
A. The results. We generalize the notion of homotopy of vector ?elds to that of otopy of vector ?elds. Using otopy we can: 1. Classify the proper homotopy classes of vector ?elds. The index is a proper homotopy class invariant but two vector ?elds with the same index may not be proper homotopic. (See (4) in Section 5.) 2. Show that two vector ?elds are properly otopic if and only if they have the same index. (See (3).) 3. Extend the de?nition of index to any vector ?eld, without hypotheses. (See Subsection C in the Introduction.) 4. De?ne a local index for any connected set of defects for any vector ?eld instead of merely for isolated zeros. (See the penultimate paragraph of C or Section 4.) Under an otopy these defects move and interact. The following conservation law holds: The sum of the indices of the incoming defects is equal to the sum of the indices of the outgoing defects. (See (1).) 5. Demonstrate that the concept of the index of a vector ?eld depends only on elementary di?erential topology, the concept of pointing inside, and the EulerPoincare number. This is done in Sections 1, 2, 3, 4. The classical approach depends on the degree of a map. In this paper we show how the degree of a map might be de?ned via the index of a vector ?eld by using (11). 6. We can study vector ?elds along the ?bre on ?bre bundles. An otopy generalizes to a vector ?eld V along the ?bre restricted to an open set. For a proper V , only certain values of the index of V restricted to a ?bre are possible. (See (14).) For example, the Hopf ?brations of spheres admit only the index zero. 7. We can study space-like vector ?elds on a Lorentzian space-time, M . The concept of otopy generalizes to a space-like vector ?eld restricted to an open set of M . For a proper space-like vector ?eld V , the index of V restricted to a spacelike slice is independent of the slice. Thus, the index is an invariant of General Relativity. Hence it should be used to describe physical phenomena. For example, the Coulomb electric vector ?eld E of an electron or a proton has index ?1 or 1 respectively. This remains true no matter what what coordinate system is used to describe the ?eld. B. Organization of the paper. There are a few features to be explicitly noted. First, we are actually de?ning two types of indices. These are usually denoted IndU (V ), and ind(P ). The ?rst takes values in the integers and ∞ and the last takes on the value ?∞ as well. Second, there are two di?erent de?nitions of these indices. The advanced de?nition is based on the de?nition of index already de?ned for continuous vector ?elds and is found in Subsection C of the Introduction. The elementary de?nition is given inductively in Sections 1, 2, 3, 4. This de?nition is equivalent to the ?rst, and the proof that it is well-de?ned is completely self contained, using only pointset topological methods. The only algebraic topological notion is that of the Euler Poincare number. Subsection C of the Introduction establishes notation and the formal concept of proper otopy as well as the key example of otopy which forces the concept on us. If the reader draws a few pictures and understands what is to be formalized,
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the formal de?nitions will be obvious except for a few small details. The formal de?nition of otopy is in Subsection C along with the advanced de?nition of index. Subsection D of the Introduction contains guides for three di?erent ways for reading the paper. Especially contained in D is a simpli?ed description of the index which if combined with the list of key properties of Section 5 should give the reader the essence of the subject without the technicalities of the proof. The main burden of the paper is the development of the elementary de?nition of the index. It is here that the two di?erent types of index, IndU (V ), and ind(P ), are carefully de?ned. The de?nition is made inductively on the dimension of the manifolds and is shown to be well-de?ned. This takes up Sections 1, 2, 3, and 4. In Section 5 we write a list of 14 properties of the index. There are short proofs of them. It is hoped that this list will be easy to use for the mathematician or physicist who needs to apply the idea of index in their work. C. De?nition of otopy and index. The concept of otopy arises from homotopy in a natural way. Consider a smooth compact manifold M with boundary ?M . Let V be a continuous vector ?eld de?ned on M . There is an associated vector ?eld ?V on ?M given by projecting the vectors of V on ?M to vectors which are tangent to ?M . (See the paragraph above Lemma 2.1 for the de?nition of projection.) Denote by ?? M the open set of ?M where the vectors of V point inside. Let ?? V denote ?V restricted to ?? M . For outward pointing vectors we de?ne ?+ M and ?+ V . Let ?0 M denote the closed set of ?M where the vectors of V are tangent to ?M . Now consider a homotopy Vt . It induces a homotopy ?Vt on ?M . Now ?? Vt is varying with t, but it is not a homotopy. We say it is an otopy. It is the key example of an otopy. The key observation about otopies. Consider a zero of ?V which passes from ?? M to ?+ M in ?M . As it passes over ?0 M it coincides with a zero of V which is passing through ?M . Thus there is a connection between the zeros of Vt which pass inside and outside of M through ?M and the zeros of ?Vt which pass inside and outside of (?? M )t . The concepts of proper homotopies and proper otopies and proper vector ?elds are introduced so that no zeros appear on ?M or ?0 M . De?nition of continuous otopy. Let N be a manifold and let V be a continuous vector ?eld de?ned on N × I so that V is tangent to the slices N × t. Then we say that V is a continuous homotopy and that V0 = V (m, 0) and V1 = V (m, 1) are homotopic vector ?elds. Suppose that T is an open set on N × I and V is a continuous vector ?eld de?ned on T so that V is tangent to the slices N × t. Then we say that V is a continuous otopy and that V0 and V1 are otopic. Note that V0 or V1 are vector ?elds de?ned only on the open sets in M given by the intersection of T with M × i for i = 0 or 1. Thus V0 or V1 can be “empty” vector ?elds. Also note that “otopy” gives an equivalence relation on the set of vector ?elds de?ned on open sets in N . This follows just as in the homotopy case. But it is a trivial equivalence relation, since every vector ?eld is homotopic to the zero vector ?eld and every vector ?eld ?eld de?ned on an open set is otopic to the empty vector ?eld. De?nition of proper continuous otopy. If U is an open set in a manifold with boundary we will adopt the convention that the Frontier of U includes that part of
the boundary ?M of M which lies in U , as well as the usual frontier. The capital F will distinguish these two di?erent notions of Frontier and frontier. We say that V de?ned on an open set U in N is a proper vector ?eld on the domain U if the zeros of V form a compact set in U and if V extends continuously to a vector ?eld on U with no zeros on the Frontier of U . Thus if V is de?ned on a compact manifold M with boundary ?M , we say V is proper if there are no zeros on ?M . A proper otopy with domain T is an otopy V de?ned on the open set T with a compact set of zeros whose restriction to any slice is a proper vector ?eld. A proper homotopy is an proper otopy V de?ned on all of N × I. Now the index for proper continuous vector ?elds on a compact manifold with or without boundary, as well as the index for proper continuous vector ?elds de?ned on an open set U , were de?ned in [BG]. If V is a continuous proper vector ?eld de?ned on a compact manifold M with boundary ?M and Euler-Poincare number χ(M ), then ?? V is also proper and the index satis?es IndV = χ(M ) ? Ind(?? V ) Now we consider any arbitrary, possibly discontinuous, vector ?eld V on N . We assume we are in a smooth manifold N . A vector ?eld is an assignment of tangent vectors to some, not necessarily all, of the points of N . We make no assumptions about continuity. We consider the set of defects of a vector ?eld V in N , that is the set D which is the closure of the set of all zeros, discontinuities and unde?ned points of V . That is we consider a defect to be a point of N at which V is either not de?ned, or is discontinuous, or is the zero vector, or which contains one of those points in every neighborhood. We extend the notion of proper to arbitrary tangent vector ?elds by replacing the word zero by defect. De?nition of discontinuous proper otopy. We say that V is a proper vector ?eld on an open set U if the defects of V in U form a compact set and if V can be extended to U so that there are no defects on the Frontier of U . Thus for N a compact manifold with boundary we say that V is a proper vector ?eld if there are no defects on the boundary. A proper otopy W with domain T is an otopy in N × I whose defects form a compact set and whose restriction to every slice is a proper vector ?eld for that slice. A globally de?ned otopy is still called a homotopy. We will modify the word homotopy to discontinuous homotopy if needed. Remarks. 1. As before, the concept of proper discontinuous otopy is an equivalence relation on the locally de?ned vector ?elds of N . It is a simple exercise of pointset topology to show that every discontinuous vector ?eld is otopic to a continuous vector ?eld. Also, if two continuous locally de?ned vector ?elds are otopic, they are continuously otopic. So the extension of index theory from continuous to discontinuous vector ?elds is not mathematically challenging. But discontinuous vector ?elds arise very naturally in mathematics and physics and now the results of index theory can be applied to them without any mental anguish. 2. In order to avoid confusion between points at which the vector ?eld V is unde?ned inside the open set U and outside the open set U we can restrict our attention without loss of generality to vector ?elds which are de?ned everywhere on U , but
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are not necessarily continuous. Any vector ?eld on U which is not de?ned at some points in U can be replaced by the same vector ?eld returning the zero vector at those unde?ned places. In fact, for those readers who are uncomfortable with the notion of discontinuous vector ?elds, Remark 1 o?ers a way to proceed by thinking of only continuous vector ?elds. 3. Note that a defect of an otopy need not be a defect of the vector ?eld de?ned on the slice. For example, consider the unit vector ?eld pointing to the right on the real line and otopy it to the unit vector ?eld pointing to the left by letting the ?eld reverse direction when t = 1. The vector ?eld at t = 1 has no defects thought of as a vector ?eld on the real line, but the otopy defects are located at all the points of the t = 1 slice. Thus this otopy is not proper since the set of defects is not compact. Replacing the line by a closed interval, the above example has a compact set of defects, but it still is not a proper otopy because defects are on the Frontier of the t = 1 slice. If we replace the real line by a circle in our example, we again get defects on the top circle, but they form a compact set and there is no Frontier, so we consider this as a proper otopy, indeed a proper discontinuous homotopy. The advanced definition of index. If V is a proper discontinuous vector ?eld de?ned on a compact M with boundary, then ?? V is continuously de?ned on ?? M . So Ind(?? V ) is de?ned. Then Ind(V ) is de?ned by IndV = χ(M ) ? Ind(?? V ). If V is a proper discontinuous vector ?eld “de?ned” on the open domain U , we can ?nd a compact manifold M which contains the compact set of defects of V and has none on ?M . Then IndU V := IndM V where IndM (V ) means the index of V restricted to M . If V is not a proper vector ?eld on U , then we de?ne IndU (V ) := ∞ . We introduce ∞ to avoid saying that the index is unde?ned, since there is information when V is not proper. Now let P be a connected component of the set of defects D of a vector ?eld V on N . We will de?ne the local index of P , which we denote by ind(P ), as follows: Let U be an open set containing P and no other defects, so that V is proper on U . Then ind(P ) := IndU (V ). If there is no such U , but P is contained in an open set on which V is proper, then ind(P ) := ?∞. If there is no open set U containing P on which V is proper, then ind(P ) := ∞. The relationship between IndU (V ) and ind(P ) is very striking. If all the indices involved are ?nite, then IndU (V ) = D. Guides. 1. The intuitive picture. Consider the defects of a vector ?eld as “topological particles” Pi endowed with a “charge” denoted ind(Pi ). These Pi move and interact as the vector ?eld evolves in time (that is under otopy and homotopy). These “charges” are preserved under collisions just as electric charge is. See (1), the conservation law. Then for a region of space U or M , we have IndU (V ) = ind(Pi ) for Pi contained in U , ((8), the summation equation). The list of properties 1–14 in Section 5 can then be used to calculate the index. Particularly useful is the Law of Vector Fields, (2). The classi?cation of otopy by index means that any set of defects Pi can be transformed to any other set of Pj ’s if and only if the sum of ind(P ), where the sum is over all connected components P of D.
the indices of the Pi is equal to sum of the indices of the Pj . Note that ind(Pi ) in dimension one can only take on the values, ?1, 0, 1, ∞, ?∞. In higher dimensions ind(Pi ) can be any integer and ∞ and ?∞. The value ?∞ will most probably not appear in a physical application. 2. The elementary definition. This is the only modern complete account of indices that the authors are aware of. In Section 1 are listed 6 lemmas. The IndM (V ) and IndU (V ) are assumed to be de?ned in dimension n ? 1 and satisfy the 6 lemmas. Then for M a compact manifold we de?ne IndM (V ) := χ(M ) ? Ind?? M (?? V ). In Section 2 we show that this is well-de?ned. Then for U an open set, we de?ne in Section 3 IndU (V ) := IndM (V ) where M ? U contains the defects of V in U . In Section 2 and Section 3 we prove the lemmas in Section 1 for dimension n. The most subtle property to prove is the “existence of defects” (7). The lemmas for dimension 1 are proved in Section 1. After IndM (V ) and IndU (V ) are established, we de?ne in Section 4 the local index for a “topological particle”, that is a connected component P of the set of defects of V , then ind(P ) := IndU (V ) where U is an open set containing P and no other defects. The two cases where such a U cannot exist are given by ind(P ) = ±∞. In the rest of the paper IndU and IndM will frequently be shortened to Ind. The prerequisites for this development of index are elementary topology and di?erential topology. The only “sophisticated” results used are: The Tietze Extension Theorem; the existence of triangulations for smooth manifolds; transversality; smooth approximation to continuous cross-sections; the additivity of the EulerPoincare number. Most of these can be found in [GP]. In Section 5 all the key properties of the index are listed. Properties (1) to (8) are basically proved in the earlier sections. Properties (9) and (10), the product and sign rules, are proved as simple consequences of properties (1) to (8). Property (11) requires knowledge of the degree of a map. It is this result which shows that the index de?ned this way agrees with the other de?nitions as in [BG] or [M]. It should be mentioned that property (11) could stand as a de?nition of the degree, and presumably most of the properties of degree could be proved from properties (1) to (11). The main point is this: The index is independent of degree, and also intersection number, ?xed point index, and coincidence number. Properties (12), (13), (14) are proved elsewhere. The proofs employ the previous properties and sophisticated algebraic topology. Each one is a generalization of a famous theorem. 3. The advanced definition. For the Expert who knows homotopy theory and di?erential topology well, [BG] will be accessible. The concept of otopy was introduced in that paper, and the invariance of index under otopy was established. (Although otopy was ?rst published there, its actual discovery came from the underlying motivation of this paper: To de?ne the index by means of the Law of Vector Fields.) One should read the de?nition in Subsection C in the Introduction to extend the de?nition of the index for discontinuous vector ?elds. Then to prove the classi?cation theorems, (3) and (4), use the properties of [BG] where needed and the otopy extension property, which is proved in Section 2. 4. An advantage for the elementary definition. The elementary de?nition is based directly on the vector ?eld, unlike the other de?nitions. In [BG] a map is constructed and the degree of the map is the index. In Hopf’s de?nition the vector ?eld must be deformed until there are only a ?nite set of zeros. G. Samaranayake
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makes use of this advantage in her thesis [S]. She has a computer program which estimates the index of a zero using the Law of Vector Fields, (2). It works well because she does not need to prepare the vector ?eld in any substantial way. Using this program she can search for zeros of a static coulomb electric ?eld generated with a ?nite number of electrons and protons whose index is not ?1, 0, or 1. Placing protons at the vertices of the Platonic solids: tetrahedron, octahedron, cube, icosahedron, and dodecahedron, she estimates the index of the central zero to be –3, –5, 5, –11, and 11 respectively.
1. The De?nition for One-dimensional Manifolds
First we describe the organization of the de?nition of index and the way we will show it is well-de?ned. In most situations M will denote a compact smooth manifold with boundary ?M which is possibly empty. N will usually denote an arbitrary smooth manifold with or without boundary, and U will denote an open set in N or M . We usually consider vector ?elds V as globally de?ned over M or locally de?ned over N . If V is locally de?ned it is associated to an open set U in N on which it is globally de?ned. We say U is the domain of V . The inductive de?nition of index. Let φ denote the empty domain. De?ne Indφ (V ) := 0. If M is a compact connected manifold and V is globally de?ned on M with no zeros on ?M , then de?ne IndM (V ) by (?) IndM (V ) := χ(M ) ? IndU (?? V ) where U = ?? M.
Let M be a smooth manifold with a globally de?ned V . Then IndM (V ) := sum of indices on each path component. Let V be a proper vector ?eld on the open set U in N . De?ne IndU (V ) := IndM (V ) where M is a compact manifold in U containing the defects of V . remark. It will be clear that by Lemma 1.6, the equation (?) will hold for nonconnected manifolds also. We shall refer below to (?) without the connectedness hypothesis. We begin the induction at dimension ?1, the empty manifold. Here the index is zero. For dimension 0, the connected manifold is a point and the vector ?eld V consists of the zero vector. Applying (?) we see that IndM (V ) equals 1. Thus IndU (V ) equals the number of points in U . In dimension 1 there are two compact connected manifolds: The circle and the closed interval. Let V be a vector ?eld globally de?ned on a circle M . Then IndM (V ) = 0 follows from (?). Note that if V were the zero vector ?eld, it is proper when N is a circle. This contrasts to the fact that a zero vector ?eld can never be proper on an M or a U with non empty Frontier. Let M be a closed interval and let V be a proper vector ?eld. Then IndM (V ) = 1?(number of points on the boundary where V points inside). Thus IndM (V ) can take on the values 1, 0, ?1. Let M be a general compact 1-dimensional manifold with a globally de?ned V . Then M is a ?nite union of closed intervals and circles and IndM (V ) := sum of indices on each path component. So we have a de?nition for IndM (V ) which is obviously well-de?ned in one dimension. It will be necessary, however, to prove that IndM (V ) is well-de?ned beginning
with dimension 2 for each step of the induction. We must show in dimension 1 already that IndU is well-de?ned. We prove three lemmas about the M case and then after Lemma 1.3 we can show that IndU (V ) is well-de?ned. We state the lemmas in this Section for general manifolds. The proofs will be for dimension one. Frequently Ind will stand for either IndM or IndU . Lemma 1.1. Two vector ?elds V and V globally de?ned on M are properly homotopic if and only if Ind(?? V ) = Ind(?? V ) on each component of the boundary. Proof. We may assume that M is connected. If M is a circle, every globally de?ned V is properly otopic to any other globally de?ned V . On the other hand, there is no path component of the circle’s empty boundary. So the result is trivially true for the circle. Next assume that M is a closed interval. Let W be a vector ?eld so that W (m) = V (m)/ V (m) for m on the boundary of M . Assume that W (m) = 0 outside a collar of the boundary, and assume that W continuously decreases in size from the unit vectors on the boundary to the zero vectors at the other end of the collar. Then we de?ne the homotopy tV + (1 ? t)W . This is a proper homotopy, since at any point m on the boundary V (m) and W (m) both point either inside or outside so no zero can arise on the boundary. Now both V and V are properly homotopic to W , hence they are properly homotopic to each other. remark. Note that if V and V are continuous vector ?elds, there is a continuous proper homotopy between them. If they are smooth, then there is a smooth proper homotopy between them. Also note that for a vector ?eld V globally de?ned on an interval, there are only four proper homotopy classes. In higher dimensions there are in?nitely many proper homotopy classes. The corresponding result in higher dimensions is Theorem 2.2. Lemma 1.2. If M is a compact manifold di?eomorphic to M and the vector ?eld related to V by the di?eomorphism f is denoted by V ? , then IndM (V ) = IndM (V ? ) Proof. Pointing inside is preserved under di?eomorphism. Lemma 1.3. If V has no defects, then Ind(V ) = 0. Proof. We may assume that M is connected. Let M be an interval. Since V has no defects on this interval, V must point outside on one end and inside on the other. Thus Ind(V ) = 1 ? 1 = 0 on this interval. For M a circle the globally de?ned V must always have index zero. Now we can show that IndU (V ) is well-de?ned. If M and M are two compact manifolds containing the defects, and contained in U , there is a compact manifold M also contained in U and containing both M and M . The vector ?eld V restricted to M ? int(M ) is a nowhere zero vector ?eld, and the previous lemma and the fact that the index is additive proves that IndU (V ) is well-de?ned for those vector ?elds for which the defects sit inside a compact manifold with boundary.
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Lemma 1.4. Given a connected N , two proper locally de?ned (continuous) vector ?elds are properly otopic (by a continuous otopy) if and only if they have the same index. For every integer n there is a vector ?eld whose index equals that integer (provided N has positive dimension). Proof. Suppose we have a proper otopy W with domain T on N ×I. Let Vt denote W restricted to N × t. We show that there is some interval about t such that Vs has the same index for all s in the interval. Since the set of defects of the otopy is compact we can ?nd a compact manifold M so that M × J, for some closed interval J, lies in T and contains the defects inside ?M × J. Thus the proper homotopy Vt on M ×J preserves the index on M , and hence the proper otopy on N ×J preserves the index on N as t runs over J. Thus we have a ?nite sequence of vector ?elds each having the same index as the previous vector ?eld. Hence the ?rst and last vector ?elds have equal indices. Conversely, for any integer n, let Wn be the vector ?eld consisting of |n| vector ?elds de?ned on disjoint open intervals in N , each one of index 1 if n > 0 and of index ?1 if n < 0. Thus Ind(Wn ) = n. Now if V has index n, we must show that V is properly homotopic to Wn . Now the domain of V consists of open connected intervals, and only a ?nite number of them contain defects. Each of these intervals has index equal to 1, ?1, or 0. Now V is properly otopic to the same vector ?eld V whose domain is restricted to only those intervals which have nonzero indices. Now if two adjacent intervals have di?erent indices, there is a proper otopy which leaves the rest of the vector ?eld ?xed, and removes the two intervals of opposite indices. After a ?nite number of steps we are left with either an empty vector ?eld, if n = 0, or a Wn . The empty vector ?eld is W0 . Thus V is properly otopic to Wn . Lemma 1.5. IndU (V ) on N is invariant under di?eomorphism. Proof. Immediate from Lemma 1.2 and the de?nition of index for locally de?ned vector ?elds. Lemma 1.6. Let V be a vector ?eld over a domain U and suppose that U is the disjoint union of U1 and U2 . Then if V1 and V2 denote V restricted to U1 and U2 respectively, we have Ind(V ) = Ind(V1 ) + Ind(V2 ).
2. The Index De?ned for Compact n-Manifolds
The otopy extension property. Let V be a continuous vector ?eld on a closed manifold N . Let U be an open set in N . Any continuous proper otopy of V on the domain U can be extended to a continuous homotopy of V on all of N . In fact, if V and W are continuous vector ?elds with a proper continuous otopy between restrictions of them to open sets, then the otopy can be extended to a continuous homotopy of V to W .
Proof. The continuous proper otopy implies there is a continuous vector ?eld W on an open set T in N × I which extends to the closure of T with no zeros on the Frontier and which is V when restricted to N × 0. This vector ?eld W can be thought of as a cross-section to the tangent bundle over N × I de?ned over a closed subset. It is well known that cross-sections can be extended from closed sets to continuous cross-sections over the whole manifold. We assume that the index is de?ned for (n?1)-manifolds and that all the lemmas of Section 1 hold. First we consider the case of connected compact manifolds M We suppose that V is a globally de?ned proper vector ?eld on such a manifold M . We choose a vector ?eld N on the boundary ?M which points outside of M . Every vector v at a point m on ?M can be uniquely written as v = t + kN (m) where t is a vector tangent to ?M and k is some real number. We say t is the projection of v tangent to ?M . Then ?V is the vector ?eld obtained by projecting V tangent to ?M . Now we de?ne ?? V by restricting ?V to ?? M , the set of points such that V is pointing inward. Then we de?ne (?) IndM (V ) = χ(M ) ? IndU (?? V ) where U = ?? M.
Lemma 2.1. IndM (V ) is well-de?ned. Proof. We assume already de?ned the index on (n ? 1)-dimensional manifolds with open domains for proper vector ?elds. Note that ?? V is proper on ?M if V is proper on M , because the Frontier of ?? M is a subset of ?0 M , the subset where V is tangent to ?M . So a defect of ?? V on the Frontier must come from a defect of V on ?M . Hence IndU (?? V ) is de?ned. Now the vector ?eld ?? V obviously depends upon the outward pointing N . If we had another outward pointing vector ?eld N we would project down to a di?erent ?? V , call it W . Now the homotopy of vector ?elds Nt = tN + (t ? 1)N always points outside of M for every t. Hence it induces a homotopy from ?? V to W and this homotopy is proper. Thus Ind(?? V ) = Ind(W ) by Lemma 1.1. Hence IndM (V ) is well-de?ned for connected manifolds with boundary. If M has empty boundary, then IndM (V ) = χ(M ) by (?). Hence IndM (V ) is well-de?ned for all connected manifolds, and hence is well-de?ned for all N -manifolds. remark. The above lemma is also true in the case where the normal vector ?eld N is not de?ned on a closed set of ?M which is disjoint from the Frontier of ?? M . Then ?V is not everywhere de?ned, but ?? V is still proper. A homotopy between N and N , as in the lemma, still induces a proper otopy between ?? V and W , so the Ind(V ) is still well-de?ned in this case also. This case arises when M is embedded as a co-dimension zero manifold in such a way that it has corners. Then the natural outward pointing normal in this situation is not de?ned on the corners. But we still have the index de?ned if none of the corners is on the Frontier of ?? M . This point arises in Theorem 2.6. Now our goal is to prove that non-zero vector ?elds have index equal to zero on compact manifolds with boundary.
Gottlieb and Samaranayake
Theorem 2.2. On M the globally de?ned vector ?eld V is properly homotopic to W if and only if Ind(?? V ) = Ind(?? W ) for every connected component of ?M . So as a corollary in the case that ?M is connected, we have that V is properly homotopic to W if and only if Ind(V ) = Ind(W ). If V and W are both continuous, then “homotopic” can be replaced by “continuously homotopic” in the statements above. Proof. We may assume that M is connected. If M has empty boundary, the theorem is true since every globally de?ned vector ?eld is properly otopic to any other globally de?ned vector ?eld. So assume that M has non-empty boundary. The theorem is true for manifolds one dimension lower by Lemma 1.1. A proper homotopy of V to W induces a proper otopy from ?? V to ?? W in the manifold ?M . Hence Ind(?? V ) = Ind(?? W ). Hence Ind(V ) = Ind(W ) from (?). Conversely, we can ?nd a smooth collar ?M × I of the boundary so that V restricted to this collar has no defects. Then we homotopy V to V where V is de?ned by V (m, t) = tV (m) for a point in the collar and V = 0 outside the collar. Now since Ind(?? V ) = Ind(?? W ) for each connected component of the boundary, we can ?nd a proper otopy from ?? V to ?? W . Now this otopy can be extended to a homotopy of ?V to ?W by the otopy extension property. This homotopy in turn can be used to de?ne a proper homotopy from V to W . Here we assume W has the same de?nition relative to W as V has to V . Thus W is properly homotopic to V . Lemma 2.3. Suppose V is a proper vector ?eld on a compact manifold M . Let ?M × I be a collar of the boundary so small so that V has no defects on the collar. Then V restricted to M minus the open collar ?M × (0, 1] has the same index as V. Proof. Let ?Vt denote the projection of V tangent to the submanifold ?M × t for every t in I. Let W be the vector ?eld on the collar de?ned by W (m, t) = ?? Vt if (m, t) is a point in ?? M ×t. Then W can be regarded as a proper otopy, proper since V has no defects on the collar. Thus Ind(?? V ) = Ind(?? V0 ) and hence Ind(V ) = χ(M )?Ind(?? V ) equals the index of V restricted to M = M ?open collar, because the indices of the ?? vector ?elds are the same on their respective boundaries and χ(M ) = χ(M ). Lemma 2.4. Let V be a proper continuous vector ?eld on M . Suppose that ?? V is properly otopic to some locally de?ned vector ?eld W on ?M . Then there is a proper homotopy of V to a proper continuous vector ?eld X so that ?? X = W and the zeros of each stage of the homotopy Vt are equal. Proof. Use the otopy extension property to ?nd a homotopy Ht from ?V to a vector ?eld on ?M , which we shall call ?X. Let n(m, t) be a continuous real valued function on ?M × I which is positive on the open set T of the otopy between ?? V and W , zero on the Frontier of T , and negative in the complement of the closure of T , and so that n(m, 1) = n(m) where V (m) = n(m)N (m) + ?V (m) de?nes n(m). Such a function exists by the Tietze extension theorem. Using n(m, t), we de?ne
a vector ?eld X on ?M × I by X (m, t) = n(m, t)N (m) + Ht (m). We adjoin the collar to M as an external collar and extend the vector ?eld V by X to get the continuous vector ?eld X. Now M with the external collar is di?eomorphic to M . Under this di?eomorphism X becomes a vector ?eld which we still denote by X. We may assume this di?eomorphism was so chosen that X = V outside of a small internal collar. Then the homotopy tX + (1 ? t)V is the required homotopy which does not change the zeros of V . Lemma 2.5. If V is a vector ?eld with no defects on an n-ball B, then IndB (V ) = 0. Proof. For the standard n-ball of radius 1 and center at the origin, we de?ne the homotopy Wt (r) = V (tr). This homotopy introduces no zeros and shows that V is homotopic to the constant vector ?eld. The constant vector ?eld has index equal to zero, as can be seen by using (?). If we have a ball di?eomorphic to the standard ball, then the index of the vector ?eld under the di?eomorphism is preserved by Lemma 1.2, and hence it has the zero index. If the ball is embedded with corners so that the corners are not on the Frontier of the set of inward pointing vectors of V , then the index is de?ned and by Lemma 2.3 it is equal to the index of V restricted to a smooth ball slightly inside the original ball. This index is zero. Theorem 2.6. If V is a vector ?eld with no defects on a compact manifold M , then IndM (V ) = 0. Proof. Now M can be triangulated and suppose we have proved the theorem for manifolds triangulated by k ? 1 n-simplices. The previous lemma proves the case k = 1. We divide M by a manifold L of one lower dimension into manifolds M1 and M2 each covered by fewer than k n-simplices so that the theorem holds for them. We arrange it so that L is orthogonal to ?M . We use Lemma 2.4 to homotopy V to a vector ?eld with no defects so that the new V is pointing outside orthogonally to ?M at L ∩ ?M . Then a simple counting argument shows that IndM (V ) = 0 since the restrictions of V to M1 and M2 have index zero. This argument works if M has no corners. If M has corners we ?nd a collar of M which gives a smooth embedding of ?M × t for all t but the last t = 1. Then by Lemma 2.3 above, we ?nd that V , restricted to the manifold bounded by ?M × t for t close enough to 1, has the same index as V . That is zero. The counting argument follows. By induction, Ind(V |M1 ) = Ind(V |M2 ) = 0. Thus Ind(?? V1 ) = χ(M1 ) and Ind(?? V2 ) = χ(M2 ). Now we have the following equation Ind(?? V ) = Ind(?? V1 ) + Ind(?? V2 ) ? Ind(W ) where W is the projection of V on the common part of the boundary of M1 and M2 , that is L. This follows from repeated applications of Lemma 1.6. Now Ind(W ) = χ(L) since W points outwards at the boundary of L. Hence Ind(?? V ) = Ind(?? V1 ) + Ind(?? V2 ) ? Ind(W ) = χ(M1 ) + χ(M2 ) ? χ(L) = χ(M ). Hence IndM (V ) = 0 from (?).
Gottlieb and Samaranayake
3. The Index for Locally De?ned Vector Fields
Let N be an n-manifold and let V be a proper vector ?eld on N with domain U . Then the set of defects of V in U is compact. Thus we can ?nd a compact manifold M which contains the defects of V . We de?ne (??) IndU (V ) := IndM (V ).
Lemma 3.1. IndU (V ) is well-de?ned. Proof. If M and M are two compact manifolds containing the defects, there is a compact manifold M containing both M and M . The vector ?eld V restricted to M ? int(M ) is a nowhere zero vector ?eld. Then Theorem 2.6 implies that the index of V restricted to M ? int(M ) is zero. Now the index of V restricted to M equals the index of V restricted to M by the following lemma. Lemma 3.2. Suppose M is the union of two manifolds M1 and M2 where the three manifolds are compact manifolds so that the intersection of M1 and M2 consist of part of the boundary of M1 and is disjoint from the boundary of M . Suppose that V is a proper vector ?eld de?ned on M which has no defects on the boundaries of M1 and M2 . Then IndM (V ) = IndM1 (V1 ) + IndM 2 (V2 ) where Vi = V |Mi . Proof. Ind(V ) = χ(M ) ? Ind(?? V ) = χ(M ) ? (Ind(?? V1 ) + Ind(?? V2 ) ? Ind(?? V1 |L) ? Ind(?? V2 |L)) by Lemma 1.6 where L = M1 ∩ M2 . Now Ind(?? V1 |L) + Ind(?? V2 |L) = Ind(?? V1 |L) + Ind(?+ V1 ) = χ(L). Thus Ind(V ) = χ(M1 ) + χ(M2 ) ? Ind(?? V1 ) ? Ind(?? V2 ) = Ind(V1 ) + Ind(V2 ), as was to be proved. Lemma 3.3. Let V be a proper vector ?eld with domain U . Suppose U is the union of two open sets U1 and U2 such that the restriction of V to each of them and to U1 ∩ U2 is a proper vector ?eld denoted V1 and V2 and V12 respectively. Then (???) IndU (V ) = IndU1 (V1 ) + IndU2 (V2 ) ? IndU12 (V12 ).
Proof. We choose disjoint compact manifolds M1 , M2 , and M12 containing the zeros of V which lie in U1 ? U12 and U2 ? U12 and U12 respectively. Then the index of V is equal to the index of V restricted to the union of M1 , M2 , and M12 . But the index of V1 is the index of V restricted to M1 and M12 , and the index of V2 is the index of V restricted to M2 and M12 , and the index of V12 is the index of V restricted to M12 . Hence counting the index gives the equation (???).
Corollary 3.4. The index of a vector ?eld V on a closed manifold M whose domain is the whole of M is equal to χ(M ). Proof. This is true by (?) a priori. We note that Lemma 3.1 implies that any other way to calculate the index of V will give the same answer. We illustrate, using Lemma 3.2 twice: Let V be a vector ?eld which is non-zero on a small n-ball B about a point. Now let V1 be V on the n-ball and let V2 be V on the complement. Then Ind(V1 ) = 0, so Ind(?? V1 ) = 1. Now Ind(?? V2 ) = (?1)n?1 . So Ind(V2 ) = χ(M ? B) ? (?1)n?1 = χ(M ) ? (?1)n ? (?1)n?1 = χ(M ). Hence Ind(V ) = Ind(V1 ) + Ind(V2 ) = 0 + χ(M ). Theorem 3.5. Given a connected manifold N , two locally de?ned (continuous) proper vector ?elds are properly otopic (by a continuous otopy) if and only if they have the same index. For every integer n there is a vector ?eld whose index equals that integer (provided N has positive dimension). Proof. Suppose we have a proper otopy W with domain T on N ×I. Let Vt denote W restricted to N × t. We show that there is some interval about t such that Vs has the same index for all s in the interval. Since the set of defects of the otopy is compact we can ?nd a compact manifold M so that M × J, for some closed interval J, lies in T and contains the defects so that the defects avoid ?M × J. Thus by Theorem 2.2, the proper homotopy Vt on M × J preserves the index on M , and hence the proper otopy on N × J preserves the index on N as t runs over J. Thus we have a ?nite sequence of vector ?elds each having the same index as the previous vector ?eld. Hence the ?rst and last vector ?elds have equal indices. Conversely, for any integer k, let Wk be the locally de?ned vector ?eld consisting of |k| vector ?elds de?ned on disjoint open balls in N , each one of index 1 if k > 0 or of index ?1 if k < 0. Thus Ind(Wk ) = k. Now if V has index k, we must show that V is properly otopic to Wk . Now the defects of V form a compact set which is contained in a compact manifold with boundary M so that V is proper and has no defects on the boundary. We may proper otopy V ?rst to a continuous vector ?eld, and then to a smooth vector ?eld. Then we consider V as a cross-section to the tangent bundle of M . Using the transversality theorem, we can smoothly homotopy the cross-section so that it is transversal to the zero section of the tangent bundle keeping the cross-section ?xed over the boundary. The dimensions are such that the intersection consists of a ?nite number of points. Thus we proper otopy V to a vector ?eld with only a ?nite number of zeros. Now we put small open balls around each of these zeros. The index of the vector ?eld on the ball around each of these zeros is either 1 or ?1. Classically this follows from transversality, but we do not need that fact. We may ?nd a di?eomorphic n-ball which contains exactly |k| zeros so that around these zeros the vector ?eld restricts to Wk . The two vector ?elds have the same index on the n-ball and thus are properly homotopic, since from (?) the index on the boundary of the inward pointing ?? vector ?elds is the same, and so by induction they are properly otopic, hence by the otopy extension property the ? vector ?elds are homotopic. This homotopy can be extended to a homotopy of the two vector ?elds originally on the n-ball. Then using the sequence of homotopies and otopies, we can piece together a proper otopy of V to Wk .
Gottlieb and Samaranayake
remark. Note that this proof is more complicated than it need be because it does not use the concept of degree of a map or of intersection number. Corollary 3.6. The proper homotopy classes of continuous proper vector ?elds on a compact manifold with connected non-empty boundary is in one-to-one correspondence with the integers via the index. Of course, the manifold must have dimension greater than one for this to hold. Lemma 3.7. The index of a locally de?ned vector ?eld on a manifold N is invariant under di?eomorphism.
4. The Index of a Defect
Let V be a vector ?eld on an manifold N . Let D be the set of defects of V . Then D breaks up into a set of connected components Di . If a component Di is compact and is an open set in the subspace topology of D, we can de?ne an index denoted ind(Di ). Note the lower case ‘i’ here as opposed to the upper case ‘I’ in the de?nition of the global and local indices. We call ind(Di ) the index of the defect (or zero) Di . De?nition. If the defect set D is connected, compact and isolated, then we can ?nd a open set U of N containing D and no other defects of V . Then we de?ne the index of D by (????) ind(D) := IndU (V ).
If D is not isolated, then every open set containing D must contain another defect of V . In this case we say ind(D) := ?∞. If D is not compact, we say ind(D) := ∞. Now if the set of defects of V on N consists of a ?nite number of compact Di , then IndN (V ) = i ind(Di ). However it is possible that V is a proper vector ?eld and there are an in?nite number of Di . Then at least one of the Di is not isolated in D. But the index of V is still de?ned. A one dimensional example occurs when M is the interval [?1, 1] and the vector ?eld V is de?ned by V (x) = x sin(1/x) for x = 0 and V (0) = 0. Then 0 is a connected component of the defects which is not open in the set of defects of V . Thus ind(0) := ?∞, whereas IndM (V ) = 1. If we have an otopy Vt , we imagine the components of the defects Dt as changing under time. We can say that Dti at time t transforms without topological radiation into Dsj at time s if there is a compact connected component T of the defects of the otopy from time t to time s so that T intersects N × t in exactly Dti and T intersects N × s exactly at Dsi . The index of Dti is the same as the index of Dsj if T is compact. In other words if a ?nite number of “particles” Di at time t are transformed into a ?nite number of particles Cj at time s by a compact T , the sum of the indices are conserved. (1) Conservation Law. ind(Ci ) = ind(Dj ).
Thus the idea of otopy allows us to make precise the concept of defects moving with time and changing with time and undergoing collisions. The index is conserved
under these collisions as long as the “world line” T of the component is compact. That is, as long as there are is no “topological radiation”, that is as long as the relevant component in the otopy is compact. As we mentioned in Subsection A of the Introduction, the concept of otopy can be thought of as a space-like vector ?eld in a space time. So if a physicist wants to model something by defects of a vector ?eld, there is a conservation law preserving an invariant of General Relativity which automatically comes along with the model.
5. Properties of the Index
(2) Law of Vector Fields. Ind(V ) + Ind ?? V = χ(M ). This is in fact the equation (?) which de?nes the index. We remark here that any theory of index in which the Law of Vector Fields holds must agree with our de?nition. (3) Classification of proper otopy by the index. Let N be a connected manifold. V is properly otopic to W if and only if Ind V = Ind W . If V and W are continuous vector ?elds, then the otopy can be continuous. For any integer n there is a continuous vector ?eld W so that n = Ind W . (4) Classification of proper homotopy. Suppose M is a compact connected manifold with non-empty connected boundary ?M , and suppose V and W are continuous globally de?ned proper vector ?elds on M . Then V is properly homotopic to W if and only if Ind V = Ind W . For any integer n there is a continuous proper vector ?eld W so that n = Ind W , provided the dimension of M is greater than one. In general for M compact, V is proper homotopic to W if and only if Ind (?? V ) = Ind (?? W ) on every connected component of ?M . (5) Poincare-Hopf Theorem. If M is a closed compact manifold and V is a vector ?eld whose domain is all of M , then Ind V = χ(M ). Proofs. Property (3) is Theorem 3.5. Property (5) is Corollary 3.6. Property (4) follows from Theorem 2.2 and Lemma 1.4, along with the Otopy Extension Property. (6) Additivity. Let A and B be open sets and let V be a proper vector ?eld on A ∪ B so that V |A and V |B are also proper. Then Ind(V |A ∪ B) = Ind(V |A) + Ind(V |B) ? Ind(V |A ∩ B). Proof. Lemma 3.3. (7) Existence of defects. If Ind V = 0 then V has a defect. Proof. Theorem 2.6 for compact manifolds with boundary. (8) Summation equation. Suppose V is a proper vector ?eld and the set of defects consists of a ?nite number of connected components Di . Then Ind V = ind(Di ).
Gottlieb and Samaranayake
Proof. This follows from the de?nition of Ind(Di ) and (3). (9) Product rule. Let V and W be proper vector ?elds on A and B respectively. Let V × W be a vector ?eld on A × B de?ned by V × W (s, t) = (V (s), W (t)). Then Ind(V × W ) = (Ind V ) · (Ind W ). Proof. We can assume that A and B are open sets in their respective manifolds. Then V is otopic to Vn where Vn is restricted to a ?nite set of open sets in A homeomorphic to the interior of J k when k = dim A and J = [?1, 1], so that Vn (t1 , . . . , tk ) = (±t1 , t2 , . . . , tk ) where the +t1 is taken if Ind V is positive and ?t1 is taken if Ind V is negative. The index of the Vn |J k is ±1 respectively by (2). So Ind(Vn |Jik ) × (Wn |Jj ). Now it is easy to see Ind (V × W ) = Ind(Vn × Wn ) =
k that Ind((Vn |Jik ) × (Wn |Jj )) = Ind(Vn |Jik ) · Ind(Wn |Jj ). i,j
(10) Sign rule. (?1)n Ind(V ) = Ind(?V ) where n = dim M .
Proof. The theorem is true for n = 1. Assume it is true for (n ? 1)-manifolds. Now using (2) we have Ind(?V ) = χ(M ) ? Ind(?? (?V )) = χ(M ) ? Ind(??+ V ) = χ(M ) ? (?1)
by (2) by de?nition of ?? V and ?+ V by induction
Ind(?+ (V ))
= χ(M ) + (?1) (χ(?M ) ? Ind(?? V )) since χ(?M ) = Ind(?? V ) + Ind(?+ V ). If n is even then Ind(?V ) = χ(M ) + (0 ? Ind(?? V )) = Ind V If n is odd then Ind(?V ) = χ(M ) ? (2χ(M ) ? Ind(?? V )) = ?(χ(M ) ? Ind(?? V )) = ?Ind V by (2). by (2).
(11) Index defines degree. Suppose M is a compact sub-manifold of Rn of codimension 0. Let f : M → Rn be a map so that f (?M ) does not contain the origin. De?ne a proper vector ?eld V f on M by V f (m) = f (m). Then Ind V f = f (m) deg f , where f : ?M → S n?1 is given by f (m) = f (m) .
Proof. We homotopy f if necessary so that 0 is a regular value. Then f ?1 (0) is a ?nite set of points. There is a neighborhood of f ?1 (0) of small balls so that f : ?(ball) → Rn ? 0 ? S n?1 . Now, in each of these small balls, f has either degree = 1 or ?1. If degree equals 1, then f |?(ball) is homotopic to the identity. If degree = ?1, then f |?(ball) is homotopic to re?ection about the equator. In these cases Ind(V f |ball) = ±1 = deg f |?(ball). Now Ind(V f ) = = Ind V f |(balls) by proper otopy
deg f |?(balls) = deg f .
(12) Brouwer Fixed Point Theorem. Suppose f : M → Rn where M ? Rn is a codimension zero compact manifold. De?ne Vf (m) = m ? f (m). Then Ind Vf = ?xed point index of f (assuming no ?xed points on ?M ). Proof. The ?xed point index is de?ned to be the degree of the map m → from ?M → S n?1 . Hence by (11) we have the result.
m?f (m) m?f (m)
(13) Gauss-Bonnet Theorem. Let f : M → N where M and N are Riemannian manifolds and f is a smooth map. Let V be a vector ?eld on M . De?ne the pullback vector ?eld f ? (V ) by f ? V (m), vm = V (f (m)), f? (vm ) . Then if f : M n → Rn so that f? |?M has maximal rank and f (?M ) contains no zeros of V , then ? Ind f ? V = vi wi + (χ(M ) ? deg N ) where vi = Ind(xi ) where xi is the ith zero of V , wi is the winding number of f |?M ? about xi , and N : ?M → S n?1 is the normal (or Gauss) map. Proof. In paper [G5 ]. (14) Transfer Theorem. Let F @ > i >> E@ > p >> B be a smooth ?bre bundle with F a compact manifold and B a closed manifold. Let V be a proper vector ?eld on E with vectors tangent to the ?bres. Then there is an S-map τ : B + → E + so that in ordinary homology p? ?τ? (cohomology τ ? ?p? ) is multiplication by the index of V restricted to a ?bre, Ind(V |F ). Proof. In paper [BG].
[BG] James C. Becker, Daniel H. Gottlieb, Vector?elds and transfers, Manuscripta Mathematica 72 (1991), 111–130. [G1 ] Daniel H. Gottlieb, A certain subgroup of the fundamental group, Amer. J. Math. 87 (1966), 1233–1237. [G2 ] , A de Moivre formula for ?xed point theory, ATAS do 5? Encontro Brasiliero de Topologia, Universidade de S?o Paulo, S?o Carlos, S.P. Brasil 31 (1988), 59–67. a a
[G3 ] [G4 ] [G5 ] [G6 ] [GP] [M] [P] [S]
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, A de Moivre like formula for ?xed point theory, Proceedings of the Fixed Point Theory Seminar at the 1986 International Congress of Mathematicians (R. F. Brown, ed.), Contemporary Mathematics, no. 72, AMS, Providence, Rhode Island, 1986, pp. 99–106. , On the index of pullback vector ?elds, Proc. of the 2nd Siegen Topology Symposium, August 1987 (Ulrich Koschorke, ed.), Lecture Notes in Math., no. 1350, Springer Verlag, New York, 1988, pp. 167–170. , Zeroes of pullback vector ?elds and ?xed point theory for bodies, Algebraic topology, Proc. of Intl. Conference March 21–24, 1988,, Contemporary Mathematics, no. 96, AMS, Providence, Rhode Island, 1988, pp. 168–180. , Vector ?elds and classical theorems of topology, Renconti del Seminario Matematico e Fisico di Milano 60 (1990), 193–203. V. Guillemin and A. Pollack, Di?erential Topology, Prentice-Hall, Englewood Cli?s, New Jersey, 1974. Marston Morse, Singular points of vector ?elds under general boundary conditions, Amer. J. Math 51 (1929), 165–178. Charles C. Pugh, A generalized Poincare index formula, Topology 7 (1968), 217–226. Geetha Samaranayake, Calculating the Indices of Vector Fields on 2 and 3 Dimensional Euclidean Space, Thesis, Purdue University, 1993.
Department of Mathematics, Purdue University, West Lafayette Indiana, 47907 firstname.lastname@example.org Department of Mathematics, Ohio State University, Newark, Ohio email@example.com Typeset by AMS-TEX
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