The Index of Discontinuous Vector Fields by Daniel H. Gottlieb and Geetha Samaranayake Introduction At the frontier between the continuous and the discrete there is a naturall* *y occurring additive, integral "quantum number" which is preserved under "collisions" of di* *scontinu- ities. This quantum number depends only on the basic topological notions of com* *pactness, connectedness, dimension, and the concept of pointing inside. We assume we are in a smooth manifold N. A vector field is an assignment of* * tangent vectors to some, not necessarily all, of the points of N. We make no assumption* *s about continuity. We will call this N the arena for our vector fields. We consider * *the set of defects of a vector field V in N, that is the set D which is the closure of th* *e set of all zeros, discontinuities and undefined points of V . That is we consider a defect* * to be a point of N at which V is either not defined, or is discontinuous, or is the zero vect* *or, or which contains one of those points in every neighborhood. We are interested in the connected components of the defects and how they c* *hange in time. Those connected components of D which are compact we will call topolo* *gical particles. If we can find an open set about a particle which does not intersect* * any defect not in the particle itself, then we say the particle is isolated. If C is an is* *olated particle we can assign an integer which we call the index of C in V . We denote this by * *Ind(C). The key properties of Ind(C) are that it is nontrivial, additive over parti* *cles, easy to calculate and is conserved under interactions with proper components as V varie* *s under time. For example, let V be the electric vector field generated by one electr* *on in R3. Then the position of the electron e is the only defect and Ind(e) = -1. Now if * *V changes under time in such a way that there are only a finite number of particles at ea* *ch time, all contained in some large fixed sphere, then the sum of the indices of the partic* *les at each time t is equal to -1. Thus the electron vector field can change to the proton * *vector field only if the set of defects changing under time is unbounded, since the proton h* *as index +1 which is different from the index of the electron. In this case we will say* * that the transformation of the electron to the proton involves " topological radiation". Vector fields varying under time, and defect components interacting with ea* *ch other, can be made precise by introducing the concept of otopy, which is a generalizat* *ion of the concept of homotopy. An otopy is a vector field on N x I so that each vector is* * tangent to a slice N x t. Thus an otopy is a vector field W on N x I so that W (n; t) i* *s tangent to N xt. We say that V0 is otopic to V1 if V0(n) = W (n; 0) and V1(n) = W (n; 1). * *We say that a set of components Ci of defects on V0 transforms into a set of components of * *defects Dj of V1 if there is a connected component T of the defects of W so that T \ (N x * *0) = [Ci and T \ (N x 1) = [Dj. If T is a compact connected component of defects of W , * *which transforms a set of isolated particles Ci into isolated particles Dj, then we s* *ay there is no topological radiation and X X (1) Ind(Ci) = Ind(Dj): If T is not compact, we say there is topological radiation. We define Ind(C) as follows. Since C is an particle, there is an open set U* * containing C so that there are no defects in the closure of U except for C. We can define * *an index for any vector field defined on the closure of an open set so that the set of defec* *ts is compact and there is no defect on the frontier of the open set. We say such a vector fi* *eld is proper with domain the open set. In the case at hand, V restricted to cl(U) is proper * *with domain U. Hence we can define Ind(V |U). We set Ind(C) = Ind(V |U). Next we define Ind(V ) with domain U to be equal to the index of V |M where* * M U is a smooth compact manifold with boundary containing the defects of V in its i* *nterior. We can find such an M since the defects are a compact set in U. We call a vector field V defined on a compact manifold M proper if there ar* *e no defects on the boundary. Consider the open set of the boundary where V points inside. W* *e denote that set by @- M. We define the vector field @- V with domain @- M in the arena* * @M by letting @- V be the end product of first restricting V to the boundary and then* * projecting each vector so that it is tangent to @M which results in a vector field @V tang* *ent to @M, and then finally restricting @V to @- M to get @- V . Then we define Ind(V ) by* * the equation (*) Ind(V ) = O(M) - Ind(@- V ) where O(M) denotes the Euler-Poincare number of M. We know that @- V is a pro* *per vector field with domain @- M since the set of defects is compact unless there * *is a defect at the the frontier of @- M. If there were such a defect, it would be a zero of* * V tangent to @M and hence a zero of V on the boundary, so V would not have been proper. Now @- V is a proper vector field with domain the open set @- M which is on* *e dimension lower than M. Then Ind(@- V ) is defined in turn by finding a compact manifold * *containing the defects of @- V and using equation (*). We continue this process until eith* *er @- M is a zero dimensional manifold where every point is a defect and so Ind(@- V ) is * *simply the number of points, or where @- M empty in which case Ind(@- V ) = 0. To summarize, we define the index of a proper vector field V with domain U * *assuming that the index for vector fields is already defined for compact manifolds with * *boundary. Then the index of V is defined to be the index of V restricted to a compact s* *mooth manifold with boundary of codimension zero containing all the defects of V in U* *. We will show this definition is well-defined, that is it does not depend on the chosen * *manifold with boundary, by showing that a vector field with no defects defined on a compact m* *anifold with boundary has index zero. 2 The well-definedness of this definition will involve the first four section* *s of this paper. In section 5 we summarize the useful properties of the index which we have prov* *ed along the way, along with a few proved in other papers. The key property is that of * *a proper otopy described below. Suppose that V is a proper vector field with open domain U. A proper otop* *y is a proper vector field W defined on N x I with domain an open set where we require* * W to be tangent to the slices. Then we say W is a proper otopy of V if V is the rest* *riction of W to N x 0 and the domain of W intersects N x 0 in U. The key property of the i* *ndex of proper vector fields with open domains is that the index is invariant under pro* *per otopy. For connected manifolds the converse is true: Two proper vector fields are prop* *erly otopic if and only if they have the same index. We may generalize the concept of otopy in two ways. Recall an otopy is an o* *pen set T on N x I with a vector field W which is tangent to the slices. Now this can be * *generalized by considering a fibre bundle E ! B with fibre N and an open set T on E and a v* *ector field W whose vectors are tangent to the fibre. It is clear that if W is a prop* *er vector field, that is the defects form a compact set and there are no defects on the frontier* * of T , then W restricted to any fibre has an index. This index is the same for every fibre.* * In [B-G], for the case of continuous W , it is shown that there is an S-map which induces* * a transfer on homology with trace equal to this index. The second way to generalize an otopy is to note that N x I can be thought * *of as a manifold S with a natural non-zero vector field. Then W is a vector field wh* *ich is orthogonal to this vector field. In fact any vector field can be projected orth* *ogonal to the natural vector field. If S is a space-time, there is a field of light cones. If* * we consider a space-like vector field W on S, it is like an otopy. W restricts to any space-l* *ike slice and projects tangent to it. The index of the defects at any event is thus an invari* *ant of general relativity, it is invariant under any change of coordinate system. The defects* * propagate through space-time and the index satisfies a conservation law, just like the co* *nservation law of electric charges under particle collisions. It is very easy to believe t* *hat the index of a vector field, as here exposed, must lead to an explanation of the conservatio* *n of physical properties under collision based on the idea of connectivity and continuity and* * pointing inside. 1. The definition for one-dimensional manifolds The inductive definition begins with empty vector fields, that is domains w* *hich are empty. This could arise since @- M is empty if V never points inside from the b* *oundary. We define the index of an empty vector field to be equal to zero. Zero dimensio* *nal manifolds consist of discrete sets of points. The only vectors are zero vectors, so for a* * vector field to be proper it must consist of a finite number of zeros. One-dimensional compact * *manifolds with boundary consist of a finite disjoint union of compact components which ar* *e compact 3 intervals. We use the definition (*), that is Ind(V )= (number of components) - (number of boundary points where V is pointing inwards): In the case of components without boundaries, circles in this case, we define t* *he index to be O(circle) = 0. Lemma 1.1. Two vector fields V and V 0are properly otopic if and only if Ind(@- V ) = Ind(@- V 0) on each component of the boundary: Proof. Let W be a vector field so that W (m) = V (m)=kV (m)k for m on the b* *oundary 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 zer* *o vectors at the other end of the collar. Then we define the homotopy tV + (1 - t)W . This i* *s a proper homotopy, since at any point m on the boundary V (m) and W (m) both point eithe* *r inside or outside so no zero can arise on the boundary. If V should have a defect at s* *ome m in the interior, we may alter V by assigning V (m) = 0. Thus the homotopy is defin* *ed. Now both V and V 0are properly otopic to W , hence they are otopic to each other. Lemma 1.2. If M is a finite collection of manifolds with boundary and f is * *a diffeo- morphism so that the related vector field is denoted by V *, then Ind(V ) = Ind(V *): Proof. Pointing inside is preserved under diffeomorphism. Lemma 1.3. If V has no defects, then Ind(V ) = 0. Proof. Each connected component of M is 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, and thus on all the intervals. So Ind(V ) = 0 is true for M. Now suppose that the arena is a connected manifold N with no boundary and n* *ot compact. Thus an open interval. Then we define the index of V with open doma* *ins to be the index of V restricted to a union of compact intervals which contain the * *defects of V . This is well-defined. If M and M0 are two manifolds with boundary contain* *ing the defects, there is a compact manifold with boundary M00containing both M and M0.* * The vector field V restricted to M00- int(M) is a nowhere zero vector field, and th* *e previous lemma and the fact that the index is additive proves that the index is well-def* *ined. Next we deal with the case of the arena N being a closed manifold, in this * *case that is a finite set of circles. We will consider the case of a single circle, the genera* *l case will be given 4 by adding the indices for each connected component. The set of defects is close* *d. If the defects can be contained in a compact manifold with boundary, in this case diff* *eomorphic to a closed interval, we define the index of V to be the index of V restricted * *to the compact manifold. On the other hand, if the domain of V is the entire arena, then we de* *fine Ind(V ) = O(arena) - Ind(@- V ) = O(circle) - Ind(empty vector field) = * *0: These two definitions are consistent. If V has domain the entire circle, * *then it is properly homotopic to the zero vector field. Then we homotopic the zero vector * *field to V 0which is zero inside a large closed interval and not zero around a point wit* *h the vectors thus forced to point in the same sense around the circle. Then V 0restricted to* * the large closed interval has index zero which is just what the global definition gives. We make a few more observations before we finish with the one-dimensional c* *ase. Lemma 1.4. Given a connected arena N, two proper vector fields are properly* * otopic if and only if they have the same index. For every integer n there is a vector * *field whose index equals that integer. Proof. Suppose we have a proper otopy W with domain T on N x I. Let Vt deno* *te W restricted to N x t. We show that there is some interval about t such that Vs h* *as the same index for all s in the interval. Since the set of defects of the otopy is compa* *ct we can find a compact manifold M so that M x J, for some closed interval J, lies in T and con* *tains the defects inside @M x J. Thus the proper homotopy Vt on M x J preserves the index* * on M, and hence the proper otopy on N x J preserves the index on N as t runs over J. * *Thus we have a finite sequence of vector fields each having the same index as the previ* *ous vector field. Hence the first and last vector fields have equal indices. Conversely, f* *or any integer n, let Wn be the vector field consisting of |n| vector fields defined on disjoi* *nt 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 finite number of them contai* *n defects. Each of these intervals has index equal to 1, -1, or 0. Now V is properly otopi* *c to the same vector field V whose domain is restricted to only those intervals which ha* *ve nonzero indices. Now if two adjacent intervals have different indices, there is a prope* *r otopy which leaves the rest of the vector field fixed, and removes the two intervals of opp* *osite indices. After a finite number of steps we are left with either an empty vector field, i* *f n = 0, or a Wn. The empty vector field is W0. Thus V is properly otopic to Wn. 5 Lemma 1.5. The index of a vector field on an open manifold is invariant und* *er diffeo- morphism. Proof. Immediate from Lemma 1.2 and the definition of index for open manifo* *lds. Lemma 1.6. Let V be a vector field over a domain U and suppose that U is th* *e disjoint union of U1 and U2. Then if V1 and V2 denote V restricted to U1 and U2 respecti* *vely, we have Ind(V ) = Ind(V1) + Ind(V2): 2. The index defined for compact n-manifolds The otopy extension property. Let V be a continuous vector field on a clos* *ed 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. Proof. The continuous proper otopy implies there is a continuous vector fie* *ld W on an open set T in N x I which extends to the closure of T with no zeros on the f* *rontier and which is V when restricted to N x 0. This vector field W can be thought o* *f as a cross-section to the tangent bundle over N x I defined over a closed subset. I* *t 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 defined for (n - 1)-manifolds in such a way tha* *t all the lemmas of section 1 hold. First we consider the case of compact manifolds such that every component i* *s a mani- fold with boundary. We suppose that V is a proper vector field on such a manifo* *ld M. We choose a vector field N on the boundary @M which points outside of M. Every vec* *tor 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 field obtained by projecting V tangent to @M. Now we define @-* * V by restricting @V to @- M, the set of points such that V is pointing inward. Then * *we define (*) Ind(V ) = O(M) - Ind(@- V ): Lemma 2.1. Ind(V ) is well-defined. Proof. We have already defined the index on (n - 1)-dimensional manifolds w* *ith open domains for proper vector fields. Note that @- V is proper since V is, since th* *e frontier of @- M is a subset of @0M where V is tangent to @M. So a defect of @- V on the fr* *ontier must come from a defect of V on @M. Hence Ind(@- V ) is defined. Now the vector* * field @- V obviously depends upon the outward pointing N. If we had another outward p* *ointing vector field N0 we would project down to a different @- V , call it W . Now the* * homotopy of 6 vector fields Nt = tN + (t - 1)N0 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 * *). We will also allow the case where N is not defined on a closed set of @M wh* *ich is disjoint from the frontier of @- M. Then @V has defects, but @- V is still p* *roper. A homotopy between N and N0, as in the lemma, still induces a proper otopy betwee* *n @- V and W , so the Ind(V ) is still well-defined in this case also. This case aris* *es when M is embedded as a co-dimension zero manifold in such a way that it has corners. Th* *en the natural outward pointing normal in this situation is not defined on the corners* *. But we still have the index defined if none of the corners is on the frontier of @- M. Now our goal is to prove that non-zero vector fields have index equal to ze* *ro on compact manifolds with boundary. Theorem 2.2. V is properly otopic 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 c* *onnected, we have that V is properly otopic to W if and only if Ind(V ) = Ind(W ). If V a* *nd W are both continuous, then "otopic" can be replaced by "homotopic" in the above stat* *ements. Proof. The theorem is true for manifolds one dimension lower by lemma 1.1. * *A proper otopy of V to W induces a proper otopy from @- V to @- W in the arena @M. He* *nce Ind(@- V ) = Ind(@- W ). Hence Ind(V ) = Ind(W ) from (*). Conversely, we can* * find a smooth collar @M x I of the boundary so that V restricted to this collar has no* * defects. Then we otopy V to V 0where V 0is defined by V 0(m; t) = tV (m) for a point in * *the collar and V 0= 0 outside the collar. Now since Ind(@- V ) = Ind(@- W ) for each conn* *ected component of the boundary, we can find 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 define a proper homotopy from V 0to W 0. Here w* *e assume W 0has the same definition relative to W as V 0has to V . Thus W is properly ot* *opic to V . Lemma 2.3. Suppose V is a proper vector field on a compact manifold M each* * of whose components has a non-empty boundary. Let @M x I be a collar of the bounda* *ry so small so that V has no defects on the collar. Then V restricted to M minus t* *he open collar @M x (0; 1] has the same index as V . Proof. Let @Vt denote the projection of V tangent to the submanifold @M xt * *for every t in I. Let W be the vector field on the collar defined by W (m; t) = @- Vt if* * (m; t) is a point in @- M x t. Then W is a proper otopy, proper since V has no defects on t* *he collar. Thus Ind(@- V ) = Ind(@- V0) and hence Ind(V ) = O(M) - Ind(@- V ) equals the i* *ndex of V restricted to M0 = M - open collar, because the indices of the @- vector fiel* *ds are the same on their respective boundaries and O(M) = O(M0). 7 Lemma 2.4. Let V be a proper continuous vector field on M. Suppose that @- * *V is properly otopic to some vector field W on @M. Then there is a proper homotopy o* *f V to a proper continuous vector field X so that @- X = W and the zeros of each stage* * of the homotopy Vt are not changed. Proof. Use the otopy extension property to find a homotopy Ht from @V to a * *vector field on @M which we shall call @X. Let n(m; t) be a continuous real valued fu* *nction on @M x 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; 0) = n(m) where V (m) = n(m)N(m) + @V (m) defines n(m). Such a function ex* *ists by the Tietze extension theorem. Using n(m; t), we define a vector field X0 on* * @M x I by X0(m; t) = n(m; t)N(m) + Ht(m). We adjoin the collar to M as an external co* *llar and extend the vector field V by X0 to get the continuous vector field X. Now M* * with the external collar is diffeomorphic to M. Under this diffeomorphism X becomes * *a vector field which we still denote by X. We may assume this diffeomorphism was so chos* *en that X = V outside of a small internal collar. Then the homotopy tX + (1 - t)V is th* *e required homotopy which does not change the zeros of V . Lemma 2.5. If V is a vector field with no defects on an n-ball, then Ind(V * *) = 0. Proof. For the standard n-ball of radius 1 and center at the origin, we def* *ine the homo- topy Wt(r) = V (tr). This homotopy introduces no zeros and shows that V is homo* *topic to the constant vector field. The constant vector field has index equal to zero* *, as can be seen by using (*). If we have a ball diffeomorphic to the standard ball, then t* *he index of the vector field under the diffeomorphism is preserved, and hence it has the ze* *ro index. If the ball is embedded with corners so that the corners are not on the frontier o* *f the set of inward pointing vectors of V , then the index is defined 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 field with no defects on a compact manifold s* *uch that all the components have non-empty boundary, then Ind(V ) = 0. Proof. Now M can be triangulated and suppose we have proved the theorem for* * man- ifolds triangulated by k - 1 n-simplicies. 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 c* *overed by fewer than k n-simplicies 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 field with no defects so that the new V is pointing outside orthogonally* * to @M at L \ @M. Then a simple counting argument shows that Ind(V ) = 0 since the restri* *ctions of V to M1 and M2 have index zero. This argument works if M has no corners. If * *M has corners we find a collar of M which is a smooth embedding of @M x t for all t b* *ut the last t = 1. Then by lemma 2.3 above, we find that V , restricted to the manifold bou* *nded by @M x t for t close enough to 1, has the same index as V . That is zero. 8 The counting argument goes as follows. By induction, Ind(V |M1) = Ind(V |M* *2) = 0. Thus Ind(@- V1) = O(M1) and Ind(@- V2) = O(M2). Now Ind(@- V ) = Ind(@- V1* *) + Ind(@- V2) - Ind(W ) where W is the projection of V on the common part of the b* *oundary of M1 and M2, that is L. This follows from repeated applications of lemma 1.6.* * Now Ind(W ) = O(L) since W points outwards at the boundary of L. Hence Ind(@- V ) = Ind(@- V1) + Ind(@- V2) - Ind(W ) = O(M1) + O(M2) - O(L) = O(M* *): Hence Ind(V ) = 0 from (*). 3. The index for open n-manifolds Let N be an n-manifold and let V be a proper vector field on N with domain * *U. Then the set of defects of V in U is compact. Thus we can find a compact manifold M * *which contains the defects of V . We define (**) Ind(V ) = Ind(V |M): Lemma 3.1. Ind(V ) is well-defined. Proof. If M and M0 are two manifolds with boundary containing the defects, * *there is a compact manifold with boundary M00containing both M and M0. The vector field V restricted to M00- int(M) is a nowhere zero vector field. Then Theorem 2.6 impl* *ies that the index of V restricted to M00- int(M) is zero. Now the index of V restricted* * to M00 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 with boundary so that the intersection of M1 an* *d M2 consist of part of the boundary of M1 and is disjoint from the boundary of M. S* *uppose that V is a proper vector field defined on M which has no defects on the bounda* *ries of M1 and M2. Then Ind(V ) = Ind(V1) + Ind(V2) where Vi= V |Mi. Proof. Ind(V )= O(M) - Ind(@- V ) = O(M) - (Ind(@- V1) + Ind(@- V2) - Ind(@- V1|L) - Ind(@- V2|L)) where L = M1 \ M2. Now Ind(@- V1|L) + Ind(@- V2|L) = Ind(@- V1|L) + Ind(@+ V1) = O(L): Thus Ind(V ) = O(M1) + O(M2) - Ind(@- V1) - Ind(@- V2) = Ind(V1) + Ind(V2); as was to be proved. 9 Lemma 3.3. Let V be a proper vector field with domain U. Suppose U is the u* *nion of two open sets U1 and U2 such that the restriction of V to each of them and t* *o U1 \ U2 is a proper vector field denoted V1 and V2 and V12 respectively. Then (***) Ind(V ) = Ind(V1) + Ind(V2) - Ind(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 rest* *ricted to M2 and M12, and the index of V12 is the index of V restricted to M12. Hence cou* *nting the index gives the equation (* * *). Theorem 3.4. Given a connected arena N, two proper vector fields are propel* *y otopic if and only if they have the same index. For every integer n there is a vector * *field whose index equals that integer. Proof. Suppose we have a proper otopy W with domain T on N x I. Let Vt deno* *te W restricted to N x t. We show that there is some interval about t such that Vs h* *as the same index for all s in the interval. Since the set of defects of the otopy is compa* *ct we can find a compact manifold M so that M x J, for some closed interval J, lies in T and c* *ontains the defects so that the defects avoid @M x J. Thus the proper homotopy Vt on M* * x J preserves the index on M, and hence the proper otopy on N x J preserves the ind* *ex on N as t runs over J. Thus we have a finite sequence of vector fields each having* * the same index as the previous vector field. Hence the first and last vector fields have* * equal indices. Conversely, for any integer k, let Wk be the vector field consisting of |k|* * vector fields defined 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 homotop* *ic to Wk. Now the defects of V form a compact set which are contained in a compact ma* *nifold with boundary M so that V is defined and has no defects on the boundary. We m* *ay proper otopy V first to a continuous vector field, and then to a smooth vector * *field. Then we consider V as a cross-section to the tangent bundle of M. Using the transv* *ersality theorem, we can smoothly homotopy the cross-section so that it is transversal t* *o the zero section of the tangent bundle keeping the cross-section fixed over the boundary* *. The dimensions are such that the intersection consists of a finite number of points* *. Thus we proper otopy V to a vector field with only a finite number of zeros. Now we put* * small open balls around each of these zeros. The index of the vector field on the ball aro* *und each of these zeros is either 1 or -1. This follows from transversality, but we do not* * need that fact. We may find a diffeomorphic n-ball which contains exactly |k| zeros so th* *at around these zeros the vector field restricts to Wk. The two vector fields have the sa* *me index on the n-ball and thus are properly homotopic, since from (*) the index on the bou* *ndary of 10 the inward pointing @- vector fields is the same, and so by induction they are * *properly otopic, hence by the otopy extension property the @ vector fields are homotopic* *. This homotopy can be extended to a homotopy of the two vector fields originally on t* *he n-ball. Then using the sequence of homotopies and otopies, we can piece together a prop* *er otopy of V to Wk. Corollary 3.5. The proper homotopy classes of continuous proper vector fiel* *ds on a compact manifold with connected boundary is in one-to-one correspondence with* * the integers via the index. Lemma 3.8. The index of a vector field on an open manifold is invariant und* *er diffeo- morphism. Lemma 3.9. The index of a vector field V on a closed manifold M whose domai* *n is the whole of M is equal to O(M). Proof. First otopy V to the zero vector field. Then homotopy the zero vecto* *r field to a vector field V 0so that it is a non-zero vector field on a small n-ball B abo* *ut a point. Now let V1 be V 0on the n-ball and let V2 be V 0on the complement. Then Ind(V1)* * = 0, so Ind(@- V1) = 1. Now Ind(@- V2) = (-1)n-1. So Ind(V2) = O(M - B) - (-1)n-1 = O(M) - (-1)n - (-1)n-1 = O(M): Hence Ind(V ) = Ind(V1) + Ind(V2) = 0 + O(M). 4. The Index of particles Let V be a vector field on an arena N. Let D be the set of defects of V .* * Then D breaks up into a set of connected components Di. We define an index for each co* *mponent Di which is compact and is an open set in the subspace topology of D. That is,* * in the terminology of the Introduction, we define the index of an isolated particle. F* *or isolated particles we can find a compact manifold M containing Di and no other defects. * *Then we define (****) Ind(Di) = Ind(V |M): P Now if we have a finite number of particles Di in the domain of V , then In* *d(V ) = iInd(Di). However it is possible that V is a proper vector field and there ar* *e an infinite number of Di. Then at least one of the Di is not isolated in D. But the index o* *f V is still defined. This event is very rare in practical situations. A one dimensional exa* *mple occurs when M is the interval [-1; 1] and the vector field V is defined by V (x) = x s* *in(1=x) for x 6= 0 and V (0) = 0. Then 0 is a connected component of the defects which is n* *ot open in the set of zeros of V . 11 If we have an otopy Vt, we imagine the components of the defects Dt as chan* *ging under time. We can say that Dtiat time t transforms without radiation into Dsj at ti* *me 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 x t in exactly Dtiand T intersects N x s exactly at Ds* *i. The index of Dtiis the same as the index of Dsj if T is compact. In other words if* * a finite number of particles Di at time t are transformed into a finite number of partic* *les Cj at time s by a compact T , the sum of the indices are conserved. That is X X (1) Ind(Ci) = Ind(Dj): Thus the idea of otopy allows us to make precise the concept of defects mov* *ing with time and changing with time and undergoing collisions. The index is conserved u* *nder these collisions as long as the "world line" T of the component is compact. That is, * *as long as there are is no radiation. 5. Properties of the Index (2) Ind(V ) + Ind@- V = O(M) This is in fact the equation (*) which defines the index. (3) Let N be a connected arena. V is a properly otopic to W if and only if Ind* *V = IndW . For any integer n there is a vector field W so that n = Ind W . (4) Suppose M is a compact manifold so that @M is connected, and suppose V and* * W are continuous proper vector fields 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 field * *W so that n = Ind W . (5) If M is a closed compact manifold and V is a vector field whose domain is * *all of M, then Ind V = O(M). Proof. Property (3) and (4) are Theorem 3.4 and Corollary 3.5 respectively * *for the homotopy part. For the fact that n = Ind W for some vector field W , we apply (* *2) and induction starting with Lemma 1.4. The proof of (5) is Lemma 3.9. (6) Let A and B be open sets and let V be a proper vector field on A [ B so th* *at V |A and V |B are also proper. Then Ind(V |A [ B) = Ind(V |A) + Ind(V |B) - Ind(V |A* * \ B). Proof of (6). Lemma 3.3 (7) Suppose V us a vector field with no defects. Then Ind V = 0. Proof. Theorem 2.6 for compact manifolds with boundary. 12 (8) Suppose V is a proper vector field andPthe set of defects consists of a fi* *nite number of connected components Di. Then Ind V = Ind(Di). i Proof. This follows from the definition of Ind(Di) and (3). (9) Let V and W be proper vector fields on A and B respectively. Let V x W be * *a vector field on AxB defined by V xW (s; t) = (V (s); W (t)). Then Ind(V xW ) = (Ind V * *).(Ind W ). Proof. We can assume that A and B are open sets in their arenas. Then V is * *otopic to Vn where Vn is restricted to a finite set of open sets in A homeomorphic to the* * interior of Ik when k = dim A and so that Vn(t1; : :;:tk) = (t1; t2; : :;:tk) where the +t* *1 is taken if Ind V is positive and -t1 is taken if Ind V is negative. The index ofPthe Vn* *|Ik is 1 respectively (by induction on (9)). So Ind (V x W ) = (Ind Vn x Wn) = Ind(Vn* *|Iki) x i;j (Wn|I`j). Now it is easy to see that Ind(Vn|Iki) x (Wn|I`j)) = Ind(Vn|Iki) . In* *d(Wn|Ikj)). (10) (-1)nInd(V ) = Ind(-V ) where n = dim M. Proof. The theorem is true for n = 1. Assume it is true for (n - 1)-manifol* *ds. Now using (2) we have Ind(-V )= O(M) - Ind(@- (-V )) by (2) = O(M) - Ind(-@+ V ) by definition of @- V and @+ V = O(M) - (-1)n-1Ind(@+ (V )) by induction = O(M) + (-1)n(O(@M) - Ind(@- V )) since O(@M) = Ind(@- V ) + Ind(@+ V ): If n is even then Ind(-V ) = O(M) + (0 - Ind(@- V )) = Ind V by (2): If n is odd then Ind (-V )= O(M) - (2O(M) - Ind(@- V )) = -(O(M) - Ind(@- V )) = -Ind V by (2) (11) Suppose M is a compact sub-manifold of Rn of 0-codimension. Let f : M ! R* *n be a map so that f(@M) does not contain the origin. Define a proper vector field V* * fon M by V f(m) = f(m). Then Ind V f= deg f0 where f0 : @M ! Sn-1 by f0(m) = _f(m)_kf* *(m)k. Proof. We homotopy f if necessary so that "0is a regular value. Then f-1 ("* *0) is a finite set of points. There is a neighborhood of f-1 (0) of small balls so that f : @(* *ball) ! Rn-0 ~= 13 Sn-1 . Now, in each of these small balls, f has either degree 1 or -1. If degre* *e equals 1, then f|@(ball) is homotopic to the identity. If degree = -1, then f|@(ball) is * *homotopic to reflection about the equator. In these cases Ind(V f|ball) = 1 = deg f|@(bal* *l). Now X Ind(V f)= IndV f|(ball) by proper otopy X = deg f|@(balls) = deg f0: (12) Suppose f : M ! Rn where M Rn is a codimension zero compact manifold. Define Vf(m) = m - f(m). Then Ind Vf = fixed point index of f (assuming no fix* *ed points on @M) Proof. The fixed point index is defined to be the degree of the map m ! _m-* *f(m)__km-f(m)k from @M ! Sn-1 . Hence by (11) we have the result (13) Let f : M ! N where M and N are Riemannian manifolds and f is a smooth ma* *p. Let V be a vector field on M. Define the pullback vector field f*(V ) by = : Then if f : Mm ! Rn so that f*|@M has maximal rank and f(@M) contains no zeros* * of V , then X Ind f*V = viwi+ (O(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 ! Sn-1 is the normal (or Gauss) map. Proof. In paper [G5]. 14 REFERENCES [G1] Daniel H. Gottlieb, A certain subgroup of the fundamental group, Amer. J.* * Math., 87 (1966), pp. 1233-1237. [G2] _______________, A de Moivre formula for fixed point theory, ATAS do 5O E* *ncontro Brasiliero de Topologia, Universidade de S"ao Paulo, S"ao Carlos, S.P. Brasil, 31 (1* *988), pp. 59-67. [G3] _______________, A de Moivre like formula for fixed point theory, Proceed* *ings of the Fixed Point Theory Seminar at the 1986 International Congress of Mathematicians* *, R. F. Brown (editor), Contemporary Mathematics, AMS Providence, Rhode Island, 72, pp. 99-106. [G4] _______________, On the index of pullback vector fields, Proc. of the 2nd* * Siegen Topology Sym- posium, August 1987, Ulrich Koschorke (editor), Lecture Notes of Mathemat* *ics, Springer Verlag, New York. [G5] _______________, Zeroes of pullback vector fields and fixed point theory * *for bodies, Algebraic topology, Proc. of Intl. Conference March 21-24, 1988, Contemporary Mathe* *matics, 96, pp. 168-180. [G6] _______________, Vector fields and classical theorems of topology, Rencon* *ti del Seminario Matem- atico e Fisico, Milano. [M] Marston Morse, Singular points of vector fields under general boundary co* *nditions, Amer. J. Math, 51 (1929), pp. 165-178. [P] Charles C. Pugh, A generalized Poincare index formula, Topology, 7 (1968)* *, pp. 217-226. Purdue University 15