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THE INDEX OF DISCONTINUOUS VECTOR FIELDS:

arXiv:hep-th/9202088v1 26 Feb 1992THE INDEX OF DISCONTINUOUS VECTOR FIELDS:TOPOLOGICAL PARTICLES AND RADIATIONbyDaniel H. Gottlieb and Geetha SamaranayakeDepartment of MathematicsPurdue UniversityWest Lafayette, IndianaAbstractWe define the concepts of topological particles and topological radiation. Theseare nothing more than connected components of defects of a vector field.

To eachtopological particle we assign an index which is an integer which is conserved underinteractions with other particles much as electric charge is conserved. For space-likevector fields of space-times this index is invariant under all coordinate transforma-tions.

We propose the following physical principal: For physical vector fields theindex changes only when there is radiation. As an implication of this principal wepredict that any physical psuedo-vector field has index zero.1

2IntroductionAt the frontier between the continuous and the discrete there is a naturally oc-curring additive, integral “quantum number” which is preserved under “collisions”of discontinuities. This quantum number depends only on the basic topological no-tions of compactness, connectedness, dimension, and the concept of pointing inside.We assume we are in a smooth manifold N. A vector field is an assignmentof tangent vectors to some, not necessarily all, of the points of N. We make noassumptions 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 isthe closure of the set of all zeros, discontinuities and undefined points of V . Thatis we consider a defect to be a point of N at which V is either not defined, or isdiscontinuous, or is the zero vector, or which contains one of those points in everyneighborhood.We are interested in the connected components of the defects and how theychange in time.

Those connected components of D which are compact we will calltopological particles. If we can find an open set about a particle which does notintersect any defect not in the particle itself, then we say the particle is isolated.

IfC is an isolated particle we can assign an integer which we call the index of C inV . We denote this by Ind(C).The key properties of Ind(C) are that it is nontrivial, additive over particles,easy to calculate and is conserved under interactions with proper components asV varies under time.

For example, let V be the electric vector field generated byone electron in R3.Then the position of the electron e is the only defect andInd(e) = −1. Now if V changes under time in such a way that there are only afinite number of particles at each time, all contained in some large fixed sphere,then the sum of the indices of the particles at each time t is equal to −1.

Thus theelectron vector field can change to the proton vector field only if the set of defectschanging under time is unbounded, since the proton has index +1 which is differentfrom the index of the electron. In this case we will say that the transformation ofthe electron to the proton involves “ topological radiation”.Vector fields varying under time, and defect components interacting with each

3other, can be made precise by introducing the concept of otopy, which is a general-ization of the concept of homotopy. An otopy is a vector field on N ×I so that eachvector is tangent to a slice N ×t.

Thus an otopy is a vector field W on N ×I so thatW(n, t) is tangent to N × t. We say that V0 is otopic to V1 if V0(n) = W(n, 0) andV1(n) = W(n, 1). We say that a set of components Ci of defects on V0 transformsinto a set of components of defects Dj of V1 if there is a connected component Tof the defects of W so that T ∩(N × 0) = ∪Ci and T ∩(N × 1) = ∪Dj.

If T is acompact connected component of defects of W, which transforms a set of isolatedparticles Ci into isolated particles Dj, then we say there is no topological radiationand(1)XInd(Ci) =XInd(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 Ucontaining C so that there are no defects in the closure of U except for C. We candefine an index for any vector field defined on the closure of an open set so that theset of defects is compact and there is no defect on the frontier of the open set. Wesay such a vector field is proper with domain the open set.

In the case at hand, Vrestricted to cl(U) is proper with domain U. Hence we can define Ind(V |U).

Weset Ind(C) = Ind(V |U).Next we define Ind(V ) with domain U to be equal to the index of V |M whereM ⊂U is a smooth compact manifold with boundary containing the defects of Vin its interior. 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 are nodefects on the boundary.

Consider the open set of the boundary where V pointsinside. We 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 tothe boundary and then projecting each vector so that it is tangent to ∂M whichresults in a vector field ∂V tangent to ∂M, and then finally restricting ∂V to ∂−Mto get ∂−V . Then we define Ind(V ) by the equation(*)Ind(V ) = χ(M) −Ind(∂−V )

4where χ(M) denotes the Euler-Poincare number of M. We know that ∂−V is aproper vector field with domain ∂−M since the set of defects is compact unlessthere is a defect at the the frontier of ∂−M. If there were such a defect, it wouldbe a zero of V tangent to ∂M and hence a zero of V on the boundary, so V wouldnot have been proper.Now ∂−V is a proper vector field with domain the open set ∂−M which is onedimension lower than M. Then Ind(∂−V ) is defined in turn by finding a compactmanifold containing the defects of ∂−V and using equation (*).

We continue thisprocess until either ∂−M is a zero dimensional manifold where every point is adefect and so Ind(∂−V ) is simply the number of points, or where ∂−M empty inwhich case Ind(∂−V ) = 0.To summarize, we define the index of a proper vector field V with domain Uassuming that the index for vector fields is already defined for compact manifoldswith boundary. Then the index of V is defined to be the index of V restricted toa compact smooth manifold with boundary of codimension zero containing all thedefects of V in U.

We will show this definition is well-defined, that is it does notdepend on the chosen manifold with boundary, by showing that a vector field withno defects defined on a compact manifold with boundary has index zero.The well-definedness of this definition will involve the first four sections of thispaper. In section 5 we summarize the useful properties of the index which we haveproved along the way, along with a few proved in other papers.

The key propertyis that of a proper otopy described below.Suppose that V is a proper vector field with open domain U. A proper otopy is aproper vector field W defined on N × I with domain an open set where we requireW to be tangent to the slices.

Then we say W is a proper otopy of V if V is therestriction of W to N × 0 and the domain of W intersects N × 0 in U. The keyproperty of the index of proper vector fields with open domains is that the index isinvariant under proper otopy.

For connected manifolds the converse is true: Twoproper vector fields are properly 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 openset T on N × I with a vector field W which is tangent to the slices.

Now this can

5be generalized by considering a fibre bundle E →B with fibre N and an open setT on E and a vector field W whose vectors are tangent to the fibre. It is clear thatif W is a proper vector field, that is the defects form a compact set and there areno defects on the frontier of T , then W restricted to any fibre has an index.

Thisindex is the same for every fibre. In [B-G], for the case of continuous W, it is shownthat there is an S-map which induces a transfer on homology with trace equal tothis index.The second way to generalize an otopy is to note that N × I can be thoughtof as a manifold S with a natural non-zero vector field.

Then W is a vector fieldwhich is orthogonal to this vector field. In fact any vector field can be projectedorthogonal to the natural vector field.

If S is a space-time, there is a field of lightcones.If we consider a space-like vector field W on S, it is like an otopy.Wrestricts to any space-like slice and projects tangent to it. The index of the defectsat any event is thus an invariant of general relativity, it is invariant under anychange of coordinate system.

The defects propagate through space-time and theindex satisfies a conservation law, just like the conservation law of electric chargesunder particle collisions. It is very easy to believe that the index of a vector field, ashere exposed, must lead to an explanation of the conservation of physical propertiesunder collision based on the idea of connectivity and continuity and pointing inside.As a first step in this direction we make the following proposal.

Every physicalvector field for which the index is defined, has the same index under any choiceof coordinates and orientation. Hence we conjecture that any psuedo vector fieldmust have either the index equal to zero or the index undefined.

Also we proposethat whenever a physical vector field has a change in its index, then there musthave been radiation.1. The definition for one-dimensional manifoldsThe inductive definition begins with empty vector fields, that is domains whichare empty.

This could arise since ∂−M is empty if V never points inside from theboundary. We define the index of an empty vector field to be equal to zero.

Zerodimensional manifolds consist of discrete sets of points. The only vectors are zerovectors, so for a vector field to be proper it must consist of a finite number of zeros.

6One-dimensional compact manifolds with boundary consist of a finite disjoint unionof compact components which are compact intervals. We use the definition (*), thatisInd(V ) = (number of components) – (number of boundarypoints where V is pointing inwards).In the case of components without boundaries, circles in this case, we define theindex to be χ(circle) = 0.Lemma 1.1.

Two vector fields V and V ′ are properly otopic if and only ifInd(∂−V ) = Ind(∂−V ′) on each component of the boundary.Proof. Let W be a vector field so that W(m) = V (m)/∥V (m)∥for m on theboundary of M. Assume that W(m) = 0 outside a collar of the boundary, andassume that W continuously decreases in size from the unit vectors on the boundaryto the zero vectors at the other end of the collar.

Then we define the homotopytV + (1 −t)W. This is a proper homotopy, since at any point m on the boundaryV (m) and W(m) both point either inside or outside so no zero can arise on theboundary. If V should have a defect at some m in the interior, we may alter Vby assigning V (m) = 0.

Thus the homotopy is defined. Now both V and V ′ areproperly 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 adiffeomorphism so that the related vector field is denoted by V ∗, thenInd(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 defectson this interval, V must point outside on one end and inside on the other.

ThusInd(V ) = 1 −1 = 0 on this interval, and thus on all the intervals. So Ind(V ) = 0is true for M.

7Now suppose that the arena is a connected manifold N with no boundary andnot compact. Thus an open interval.

Then we define the index of V with opendomains to be the index of V restricted to a union of compact intervals whichcontain the defects of V . This is well-defined.

If M and M ′ are two manifoldswith boundary containing the defects, there is a compact manifold with boundaryM ′′ containing both M and M ′. The vector field V restricted to M ′′ −int(M) isa nowhere zero vector field, and the previous lemma and the fact that the index isadditive proves that the index is well-defined.Next we deal with the case of the arena N being a closed manifold, in this casethat is a finite set of circles.

We will consider the case of a single circle, the generalcase will be given by adding the indices for each connected component. The setof defects is closed.

If the defects can be contained in a compact manifold withboundary, in this case diffeomorphic to a closed interval, we define the index of Vto be the index of V restricted to the compact manifold. On the other hand, if thedomain of V is the entire arena, then we defineInd(V ) = χ(arena) −Ind(∂−V ) = χ(circle) −Ind(empty vector field) = 0.These two definitions are consistent.

If V has domain the entire circle, then itis properly homotopic to the zero vector field. Then we homotopic the zero vectorfield to V ′ which is zero inside a large closed interval and not zero around a pointwith the vectors thus forced to point in the same sense around the circle.

Then V ′restricted to the large closed interval has index zero which is just what the globaldefinition gives.We make a few more observations before we finish with the one-dimensional case.Lemma 1.4. Given a connected arena N, two proper vector fields are properlyotopic if and only if they have the same index.

For every integer n there is a vectorfield whose index equals that integer.Proof. Suppose we have a proper otopy W with domain T on N × I.

Let Vt denoteW restricted to N × t. We show that there is some interval about t such that Vshas the same index for all s in the interval. Since the set of defects of the otopy iscompact we can find a compact manifold M so that M ×J, for some closed interval

8J, lies in T and contains the defects inside ∂M × J. Thus the proper homotopy Vton M ×J preserves the index on M, and hence the proper otopy on N ×J preservesthe index on N as t runs over J.

Thus we have a finite sequence of vector fieldseach having the same index as the previous vector field. Hence the first and lastvector fields have equal indices.

Conversely, for any integer n, let Wn be the vectorfield consisting of |n| vector fields defined on disjoint open intervals in N, each oneof index 1 if n > 0 and of index −1 if n < 0. Thus Ind(Wn) = n. Now if V hasindex n, we must show that V is properly homotopic to Wn.

Now the domain ofV consists of open connected intervals, and only a finite number of them containdefects. Each of these intervals has index equal to 1, −1, or 0.

Now V is properlyotopic to the same vector field V whose domain is restricted to only those intervalswhich have nonzero indices. Now if two adjacent intervals have different indices,there is a proper otopy which leaves the rest of the vector field fixed, and removesthe two intervals of opposite indices.

After a finite number of steps we are left witheither an empty vector field, if n = 0, or a Wn. The empty vector field is W0.

ThusV is properly otopic to Wn.Lemma 1.5. The index of a vector field on an open manifold is invariant underdiffeomorphism.Proof.

Immediate from Lemma 1.2 and the definition of index for open manifolds.Lemma 1.6. Let V be a vector field over a domain U and suppose that U is thedisjoint union of U1 and U2.

Then if V1 and V2 denote V restricted to U1 and U2respectively, we haveInd(V ) = Ind(V1) + Ind(V2).2. The index defined for compact n-manifoldsThe otopy extension property.

Let V be a continuous vector field on a closedmanifold N. Let U be an open set in N. Any continuous proper otopy of V on thedomain 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 field Won an open set T in N × I which extends to the closure of T with no zeros on

9the frontier and which is V when restricted to N × 0. This vector field W can bethought of as a cross-section to the tangent bundle over N ×I defined over a closedsubset.

It is well known that cross-sections can be extended from closed sets tocontinuous cross-sections over the whole manifold.We assume that the index is defined for (n −1)-manifolds in such a way that allthe lemmas of section 1 hold.First we consider the case of compact manifolds such that every component isa manifold with boundary. We suppose that V is a proper vector field on sucha manifold M.We choose a vector field N on the boundary ∂M which pointsoutside of M. Every vector v at a point m on ∂M can be uniquely written asv = t + kN(m) where t is a vector tangent to ∂M and k is some real number.

Wesay t is the projection of v tangent to ∂M. Then ∂V is the vector field obtained byprojecting V tangent to ∂M.

Now we define ∂−V by restricting ∂V to ∂−M, theset of points such that V is pointing inward. Then we define(*)Ind(V ) = χ(M) −Ind(∂−V ).Lemma 2.1.

Ind(V ) is well-defined.Proof. We have already defined the index on (n −1)-dimensional manifolds withopen domains for proper vector fields.

Note that ∂−V is proper since V is, sincethe frontier of ∂−M is a subset of ∂0M where V is tangent to ∂M. So a defectof ∂−V on the frontier must come from a defect of V on ∂M.

Hence Ind(∂−V ) isdefined. Now the vector field ∂−V obviously depends upon the outward pointingN.

If we had another outward pointing vector field N ′ we would project down toa different ∂−V , call it W. Now the homotopy of vector fields Nt = tN + (t −1)N ′always points outside of M for every t. Hence it induces a homotopy from ∂−V toW 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 which isdisjoint from the frontier of ∂−M.

Then ∂V has defects, but ∂−V is still proper. Ahomotopy between N and N ′, as in the lemma, still induces a proper otopy between∂−V and W, so the Ind(V ) is still well-defined in this case also.

This case arises

10when M is embedded as a co-dimension zero manifold in such a way that it hascorners. Then the natural outward pointing normal in this situation is not definedon the corners.

But we still have the index defined if none of the corners is on thefrontier of ∂−M.Now our goal is to prove that non-zero vector fields have index equal to zero oncompact manifolds with boundary.Theorem 2.2. V is properly otopic to W if and only ifInd(∂−V ) = Ind(∂−W)for every connected component of ∂M.

So as a corollary in the case that ∂M isconnected, we have that V is properly otopic to W if and only if Ind(V ) = Ind(W).If V and W are both continuous, then “otopic” can be replaced by “homotopic” inthe above statements.Proof. The theorem is true for manifolds one dimension lower by lemma 1.1.

Aproper otopy of V to W induces a proper otopy from ∂−V to ∂−W in the arena ∂M.Hence Ind(∂−V ) = Ind(∂−W). Hence Ind(V ) = Ind(W) from (∗).

Conversely, wecan find a smooth collar ∂M × I of the boundary so that V restricted to this collarhas no defects. Then we otopy V to V ′ where V ′ is defined 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 find a properotopy 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 definea proper homotopy from V ′ to W ′. Here we assume W ′ has the same definitionrelative to W as V ′ has to V .

Thus W is properly otopic to V .Lemma 2.3. Suppose V is a proper vector field on a compact manifold M eachof whose components has a non-empty boundary.

Let ∂M × I be a collar of theboundary so small so that V has no defects on the collar. Then V restricted to Mminus 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 forevery t in I. Let W be the vector field on the collar defined by W(m, t) = ∂−Vt if

11(m, t) is a point in ∂−M×t. Then W is a proper otopy, proper since V has no defectson 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 fields are the same on their respective boundaries and χ(M) = χ(M ′).Lemma 2.4. Let V be a proper continuous vector field on M. Suppose that ∂−Vis properly otopic to some vector field W on ∂M.

Then there is a proper homotopyof V to a proper continuous vector field X so that ∂−X = W and the zeros of eachstage of the homotopy Vt are not changed.Proof. Use the otopy extension property to find a homotopy Ht from ∂V to avector field on ∂M which we shall call ∂X.

Let n(m, t) be a continuous real valuedfunction on ∂M × I which is positive on the open set T of the otopy between ∂−Vand W, zero on the frontier of T , and negative in the complement of the closure ofT , and so that n(m, 0) = n(m) where V (m) = n(m)N(m) + ∂V (m) defines n(m).Such a function exists by the Tietze extension theorem. Using n(m, t), we definea vector field X′ on ∂M × I by X′(m, t) = n(m, t)N(m) + Ht(m).

We adjoin thecollar to M as an external collar and extend the vector field V by X′ to get thecontinuous 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 chosen that X = V outside of a smallinternal collar.

Then the homotopy tX + (1 −t)V is the required homotopy whichdoes 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 define thehomotopy Wt(r) = V (tr). This homotopy introduces no zeros and shows that V ishomotopic to the constant vector field.

The constant vector field has index equalto zero, as can be seen by using (∗). If we have a ball diffeomorphic to the standardball, then the index of the vector field under the diffeomorphism is preserved, andhence it has the zero index.

If the ball is embedded with corners so that the cornersare not on the frontier of the set of inward pointing vectors of V , then the index isdefined and by lemma 2.3 it is equal to the index of V restricted to a smooth ball

12slightly inside the original ball. This index is zero.Theorem 2.6.

If V is a vector field with no defects on a compact manifold suchthat all the components have non-empty boundary, then Ind(V ) = 0.Proof. Now M can be triangulated and suppose we have proved the theorem formanifolds triangulated by k −1 n-simplicies.

The previous lemma proves the casek = 1. We divide M by a manifold L of one lower dimension into manifolds M1and M2 each covered by fewer than k n-simplicies so that the theorem holds forthem.We arrange it so that L is orthogonal to ∂M.

We use lemma 2.4 to homotopy Vto a vector field with no defects so that the new V is pointing outside orthogonallyto ∂M at L ∩∂M. Then a simple counting argument shows that Ind(V ) = 0 sincethe restrictions of V to M1 and M2 have index zero.

This argument works if M hasno corners. If M has corners we find a collar of M which is a smooth embeddingof ∂M × t for all t but the last t = 1.

Then by lemma 2.3 above, we find that V ,restricted to the manifold bounded by ∂M × t for t close enough to 1, has the sameindex as V . That is zero.The counting argument goes as follows.

By induction, Ind(V |M1) = Ind(V |M2) =0. Thus Ind(∂−V1) = χ(M1) and Ind(∂−V2) = χ(M2).

Now Ind(∂−V ) = Ind(∂−V1)+Ind(∂−V2) −Ind(W) where W is the projection of V on the common part of theboundary of M1 and M2, that is L. This follows from repeated applications oflemma 1.6. Now Ind(W) = χ(L) since W points outwards at the boundary of L.HenceInd(∂−V ) = Ind(∂−V1) + Ind(∂−V2) −Ind(W) = χ(M1) + χ(M2) −χ(L) = χ(M).Hence Ind(V ) = 0 from (∗).3.

The index for open n-manifoldsLet 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 manifoldM which contains the defects of V .

We define(**)Ind(V ) = Ind(V |M).

13Lemma 3.1. Ind(V ) is well-defined.Proof.

If M and M ′ are two manifolds with boundary containing the defects, thereis a compact manifold with boundary M ′′ containing both M and M ′. The vectorfield V restricted to M ′′ −int(M) is a nowhere zero vector field.

Then Theorem 2.6implies that the index of V restricted to M ′′ −int(M) is zero. Now the index of Vrestricted 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 threemanifolds are compact manifolds with boundary so that the intersection of M1 andM2 consist of part of the boundary of M1 and is disjoint from the boundary of M.Suppose that V is a proper vector field defined on M which has no defects on theboundaries of M1 and M2. Then Ind(V ) = Ind(V1) + Ind(V2) where Vi = V |Mi.Proof.Ind(V ) = χ(M) −Ind(∂−V )= χ(M) −(Ind(∂−V1) + Ind(∂−V2) −Ind(∂−V1|L) −Ind(∂−V2|L))where L = M1 ∩M2.

NowInd(∂−V1|L) + Ind(∂−V2|L) = Ind(∂−V1|L) + Ind(∂+V1) = χ(L).ThusInd(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 field with domain U.

Suppose U is the unionof two open sets U1 and U2 such that the restriction of V to each of them and toU1 ∩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 thezeros of V which lie in U1 −U12 and U2 −U12 and U12 respectively. Then the indexof V is equal to the index of V restricted to the union of M1, M2, and M12.

Butthe index of V1 is the index of V restricted to M1 and M12, and the index of V2 isthe index of V restricted to M2 and M12, and the index of V12 is the index of Vrestricted to M12. Hence counting the index gives the equation (∗∗∗).

14Theorem 3.4. Given a connected arena N, two proper vector fields are propelyotopic if and only if they have the same index.

For every integer n there is a vectorfield whose index equals that integer.Proof. Suppose we have a proper otopy W with domain T on N × I.

Let Vt denoteW restricted to N × t. We show that there is some interval about t such that Vshas the same index for all s in the interval. Since the set of defects of the otopy iscompact we can find a compact manifold M so that M ×J, for some closed intervalJ, lies in T and contains the defects so that the defects avoid ∂M × J.

Thus theproper homotopy Vt on M × J preserves the index on M, and hence the properotopy on N × J preserves the index on N as t runs over J. Thus we have a finitesequence 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| vectorfields 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 isproperly homotopic to Wk. Now the defects of V form a compact set which arecontained in a compact manifold with boundary M so that V is defined and has nodefects on the boundary.

We may 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 thetangent bundle of M. Using the transversality theorem, we can smoothly homotopythe cross-section so that it is transversal to the zero section of the tangent bundlekeeping the cross-section fixed over the boundary.

The dimensions are such thatthe intersection consists of a finite number of points. Thus we proper otopy V toa vector field with only a finite number of zeros.

Now we put small open ballsaround each of these zeros. The index of the vector field on the ball around each ofthese zeros is either 1 or −1.

This follows from transversality, but we do not needthat fact. We may find a diffeomorphic n-ball which contains exactly |k| zeros sothat around these zeros the vector field restricts to Wk.

The two vector fields havethe same index on the n-ball and thus are properly homotopic, since from (∗) theindex on the boundary of the inward pointing ∂−vector fields is the same, and soby 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

15two vector fields originally on the n-ball. Then using the sequence of homotopiesand otopies, we can piece together a proper otopy of V to Wk.Corollary 3.5.

The proper homotopy classes of continuous proper vector fields ona compact manifold with connected boundary is in one-to-one correspondence withthe integers via the index.Lemma 3.8. The index of a vector field on an open manifold is invariant underdiffeomorphism.Lemma 3.9.

The index of a vector field V on a closed manifold M whose domainis the whole of M is equal to χ(M).Proof. First otopy V to the zero vector field.

Then homotopy the zero vector fieldto a vector field V ′ so that it is a non-zero vector field on a small n-ball B about apoint. Now let V1 be V ′ on the n-ball and let V2 be V ′ on the complement.

ThenInd(V1) = 0, so Ind(∂−V1) = 1. Now Ind(∂−V2) = (−1)n−1.

SoInd(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).4. The Index of particlesLet V be a vector field on an arena N. Let D be the set of defects of V .

ThenD breaks up into a set of connected components Di. We define an index for eachcomponent 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 isolatedparticle.

For isolated particles we can find a compact manifold M containing Diand no other defects. Then we define(****)Ind(Di) = Ind(V |M).Now if we have a finite number of particles Di in the domain of V , then Ind(V ) =Pi Ind(Di).

However it is possible that V is a proper vector field and there are aninfinite number of Di. Then at least one of the Di is not isolated in D. But theindex of V is still defined.

This event is very rare in practical situations. A one

16dimensional example occurs when M is the interval [−1, 1] and the vector field Vis defined by V (x) = x sin(1/x) for x ̸= 0 and V (0) = 0. Then 0 is a connectedcomponent of the defects which is not open in the set of zeros of V .If we have an otopy Vt, we imagine the components of the defects Dt as changingunder time.

We can say that Dti at time t transforms without radiation into Dsjat time s if there is a compact connected component T of the defects of the otopyfrom time t to time s so that T intersects N ×t in exactly Dti and T intersects N ×sexactly at Dsi. The index of Dti is 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 afinite number of particles Cj at time s by a compact T , the sum of the indices areconserved.

That is(1)XInd(Ci) =XInd(Dj).Thus the idea of otopy allows us to make precise the concept of defects movingwith time and changing with time and undergoing collisions. The index is conservedunder 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 = χ(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 ifInd V = Ind W. 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 Vand W are continuous proper vector fields on M. Then V is properly homotopicto W if and only if Ind V = Ind W. For any integer n there is a continuous propervector field W so that n = Ind W.(5)If M is a closed compact manifold and V is a vector field whose domain is allof M, then Ind V = χ(M).

17Proof. Property (3) and (4) are Theorem 3.4 and Corollary 3.5 respectively for thehomotopy 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 sothat 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. (8)Suppose V is a proper vector field and the set of defects consists of a finitenumber of connected components Di.

Then Ind V = PiInd(Di).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 × W bea vector field on A×B defined 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 arenas.

Then V isotopic to Vn where Vn is restricted to a finite set of open sets in A homeomorphicto the interior of Ik when k = dim A and 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|Ik is ±1 respectively (by induction on (9)).So Ind (V ×W) = (Ind Vn × Wn) = Pi,jInd(Vn|Iki ) × (Wn|Iℓj).Now it is easy to see thatInd(Vn|Iki ) × (Wn|Iℓj)) = Ind(Vn|Iki ) · Ind(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)-manifolds.

Now

18using (2) we haveInd(−V ) = χ(M) −Ind(∂−(−V ))by (2)= χ(M) −Ind(−∂+V )by definition of ∂−V and ∂+V= χ(M) −(−1)n−1Ind(∂+(V ))by induction= χ(M) + (−1)n(χ(∂M) −Ind(∂−V ))sinceχ(∂M) = Ind(∂−V ) + Ind(∂+V ).If n is even thenInd(−V ) = χ(M) + (0 −Ind(∂−V )) = Ind Vby (2).If n is odd thenInd(−V ) = χ(M) −(2χ(M) −Ind(∂−V ))= −(χ(M) −Ind(∂−V )) = −Ind Vby (2)(11)Suppose M is a compact sub-manifold of Rn of 0-codimension. Let f : M →Rn be a map so that f(∂M) does not contain the origin.

Define a proper vectorfield V f on M by V f(m) = f(m). Then Ind V f = deg f ′ where f ′ : ∂M →Sn−1by f ′(m) =f(m)∥f(m)∥.Proof.

We homotopy f if necessary so that ⃗0 is a regular value. Then f −1(⃗0) isa finite set of points.

There is a neighborhood of f −1(0) of small balls so thatf : ∂(ball) →Rn −0 ∼= Sn−1. Now, in each of these small balls, f has either degree1 or −1.

If degree 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 casesInd(V f|ball) = ±1 = deg f|∂(ball). NowInd(V f) =XInd V f|(ball)by proper otopy=Xdeg f|∂(balls) = deg f ′.

(12)Suppose f : M →Rn where M ⊂Rn is a codimension zero compact mani-fold. Define Vf(m) = m −f(m).

Then Ind Vf = fixed point index of f (assumingno fixed points on ∂M)

19Proof. The fixed point index is defined to be the degree of the map m →m−f(m)∥m−f(m)∥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 smoothmap. Let V be a vector field on M. Define the pullback vector field f ∗(V ) by⟨f ∗V (m),⃗vm⟩= ⟨V (f(m)), f∗(⃗vm)⟩.Then if f : M m →Rn so that f∗|∂M has maximal rank and f(∂M) contains nozeros of V , thenInd f ∗V =Xviwi + (χ(M) −deg ˆN)where vi = Ind(xi) where xi is the ith zero of V , wi is the winding number of f|∂Mabout xi, and ˆN : ∂M →Sn−1 is the normal (or Gauss) map.Proof.

In paper [G5].

20ReferencesG1. Daniel H. Gottlieb, A certain subgroup of the fundamental group, Amer.

J. Math.

87 (1966),1233–1237.G2., A de Moivre formula for fixed point theory, ATAS do 5◦Encontro Brasiliero deTopologia, Universidade de S˜ao Paulo, S˜ao Carlos, S.P. Brasil 31 (1988), 59–67.G3., A de Moivre like formula for fixed point theory, Proceedings of the Fixed PointTheory Seminar at the 1986 International Congress of Mathematicians, R. F. Brown (editor),Contemporary Mathematics, AMS Providence, Rhode Island 72, 99–106.G4., On the index of pullback vector fields, Proc.

of the 2nd Siegen Topology Symposium,August 1987, Ulrich Koschorke (editor), Lecture Notes of Mathematics, Springer Verlag, NewYork.G5., Zeroes of pullback vector fields and fixed point theory for bodies, Algebraic topology,Proc. of Intl.

Conference March 21–24, 1988, Contemporary Mathematics 96, 168–180.G6., Vector fields and classical theorems of topology, Renconti del Seminario Matematicoe Fisico, Milano.M. Marston Morse, Singular points of vector fields under general boundary conditions, Amer.

J.Math 51 (1929), 165–178.P. Charles C. Pugh, A generalized Poincare index formula, Topology 7 (1968), 217–226.Purdue University

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