AFFINRE LINKING NUMBERS AND CAUSALITY RELATIONS FOR WAVE FRONTS VLADIMIR V. CHERNOV (TCHERNOV) AND YULI B. RUDYAK Abstract.Let M be an oriented n-dimensional manifold. We study the causal relations between the wave fronts W1 and W2 that originated at some points of M. We introduce a numerical topological invariant CR(W1, W2) (t* *he so-called causality relation invariant) that, in particular, gives the al* *gebraic number of times the wave front W1 passed through the point that was the source of W2 before the front W2 originated. This invariant can be easily calculated from the current picture of wave fronts on M without the knowl* *edge of the propagation law for the wave fronts. Moreover,_in fact we even do * *not need to know the topology_of M outside of a part M of M such that W1and W2are null-homotopic in M. Introduction Throughout this paper M is a smooth connected oriented manifold (not necessarily compact). Let W1 and W2 be two wave fronts which are propagating in M. (Generally, we assume that the fronts have different propagation laws.) We define a dangerous intersection between the fronts W1(t) and W2(t) at some moment of time t to be a point x where the fronts intersect and have the same direction of propagation (see Section 1 for the precise definition). A passage of the front W1(t) through the point that is the birth point of the front W2 at the moment of time t before the front W2 originated is called the baby- intersection. It turns out that we can associate to each dangerous intersection as well as to each baby-intersection a sign (i.e. a number 1 ). The sum of the signs up to a moment t is called a causality relation invariant and is denoted by CR (W1(t), W2(t)). In particular, suppose that W1 originated before W2 and that there are no dangerous intersections of W1(t) and W2(t) for all t, (see such examples in Section 1). Then the causality relation invariant tells us the algebraic number of times the first wave front reached the birth point of the second front before the second front originated. ___________ Date: July 16. 1 2 V. CHERNOV (TCHERNOV) AND YU. RUDYAK We are interested in reconstructing the value CR (W1(t), W2(t)) from the current shape of the wave fronts only, without the knowledge of the propagation laws, of the birth- points of the fronts, of the topology of M etc. It turns out that, having the current picture only, we can evaluate CR (W1(t), W2(t)) 2 Z modulo certain m 2 Z that depends on M. This m is zero if dim M is even. Furthermore, for M odd- dimensional this m is divisible by the order of the fundamental group of M. In particular, if ß1(M) is infinite then m = 0, i.e. we can completely evaluate CR from the current picture. The really bad case m = 1 (when we can not say anything about CR ) appears only when M is an odd-dimensional homotopy sphere. To evaluate CR modulo m, we introduce an affine linking number AL 2 Z=m which depends on the current picture only. Then we notice that CR and AL are congruent modulo m. Here the biggest technical difficulty appears since in order to define AL we must define the "linking number" for two spheres that are non-homologous to zero. ___ This theory has the following physical interpretation._Let M be the part of the manifold (the universe) M such that M contains the_current picture of wave fronts W1, W2 and W1, W2 are contractible in M . We transform the wave fronts via certain allowable moves to triv- ial fronts, i.e. small spherical fronts with the canonical orientation and coorientation, located far away from each other. The allowable moves should be thought of as generalized Reidemeister moves: they are the passages through generic singularities (in both directions) of wave fronts and dangerous intersection moves. We count the change of the invariant CR that occurs in the process of this formal deforma- tion, and it turns out that this change is congruent modulo m with the (unknown!) value CR (W1, W2) of the current picture. In particular, as we have already mentioned, if dim M is even or if ß1(M) is infinite, then we can completely compute CR from the current picture, without any knowledge of the propagations,_moments of birth of the fronts, and topology of M outside of M . The following observation seems to be interesting. Suppose that we have two pictures of two pairs of fronts (W1, W2) and (W10, W20) made at two unknown moments of time t0 and t1. (We assume that both pairs are free of dangerous intersection points.) Assume that we know that the propagation laws for the two fronts are such that the dan- gerous intersection points cannot appear during the propagation and that CR (W1(t0), W2(t0)) and CR (W10(t1), W20(t1)) are not comparable AFFINRE LINKING NUMBERS AND CAUSALITY RELATIONS FOR WAVE FRONTS 3 modulo m. Then we can conclude that the pairs (W1(t0), W2(t0)) and (W10(t1), W20(t1)) of wave fronts are not the pictures of the same pair of fronts taken at different moments of time. Notice that in these calculations we disregard the dangerous self- intersections of wave fronts. (In a sense this is similar to the theory of link homotopy where different components of links are not allowed to intersect through possible deformations, but self-intersections are allowed.) The study of self-intersections of fronts on surfaces was initi- ated by the ground breaking work of Arnold [2], see also [4, 5, 9, 10, 14, 15, 17]. The methods developed in this paper allow us to calculate the algebraic number of dangerous self-intersection points that arise under the propagation of fronts on manifolds of arbitrary dimensions, we do it in a next paper. The following physical speculations related to the CR invariant seem to be possible. Assume that the space-time_is_topologically a product Mn x R, and that the observable universe M is so big that we are not able to see the current picture of wave fronts (due to the finiteness of the speed of light). The propagating fronts define the mapping of the cones C1, C2 (over the sphere Sn-1 parameterizing_the_fronts at every moment of time) into_M x_R._Let sec: M ! M x R_be_a section of the projection p__M: M xR ! M , and let Vi= Ci\sec(M ), i = 1, 2. Assume that the law of propagation is such that dangerous intersections do not occur. Then, similarly to the above, we can restore the number of baby- intersections from the picture of images of Viunder the projection p__M. The section sec can be thought of as the_picture_of the universe that we see as the light from the points of M reaches the observer, and thus Vi can be regarded as the picture of fronts that we actually see. Low [6, 7, 8] attacked a similar problem for M = R3, where he con- sidered some linking invariant for linked cones C1, C2 as above. In this case the linking numbers can be constructed directly via the approach of Tabachnikov [13], because R3 has the topological end. The paper is organized as follows. In Section 1 we discuss some preliminary information, in Section 2 we define the invariant CR , in Section 3 we prove homotopy theoretical results which we use in order to define the invariant AL , in Section 4 we define the invariant AL and fix the relation between CR and AL , in Section 5 we treat the case of propagation with respect to a certain Riemannian metric, in Section 6 we give some examples and applications. 4 V. CHERNOV (TCHERNOV) AND YU. RUDYAK 1.Preliminaries: propagation laws, propagations and dangerous intersections We denote by pT : T M ! M the tangent bundle over M. Let ____ s : M ! T M be the zero section of the tangent bundle. We set T M = T M \ s(M). The multiplicative group R+ of positive real numbers acts fiberwise on T M \ s(M) by multiplication, and we set ST M = (T M \ s(M))=R+ . ____ Let __p: T M ! ST M be the quotient map. Clearly, the projection pT : T M ! M yields the commutative diagram ____ _p T M ---! T M ---! ST M ? ? ? pT?y ?y ?ypr M _______M _______ M It is easy to see that pr : ST M ! M is a locally trivial bundle with the fiber Sn-1, we call the bundle the spherical tangent bundle. Given x 2 Mn, we denote by Sn-1x(or just by Sx) the fiber pr-1(x) over x of the spherical tangent bundle and by TxM the tangent space to M at x. Since M is orientable, the bundle pr : ST M ! M is also orientable and in order to orient ST M it suffices to orient the fiber Sn-1x. We do it as follows. Choose an orientation preserving chart for M centered at x and let S be a small (n - 1)-sphere centered at x. We equip S with the unique orientation o by requiring that the pair (o, outer normal vector to S) gives us the orientation of M. Given s 2 S, the radius-vector from x to s can be regarded as a nonzero tangent vector to M at x, i.e., as a point of Sn-1x. In this way we get a diffeomorphism _ : S ! Sx which gives us an orientation of Sn-1x. It is easy to see that this orientation of Sx does not depend of choice of the chart. Now, the pair (the orientation of M, the orientation of Sx) gives us an orientation of ST M which we fix forever. 1.1. Definition. We define a propagation law on M to be a smooth map ____ ____ L : T M x R x R ! T M ____ ____ (a time-dependent flow on T M ). Here L(u, s, t) 2 T M should be thought of as the point that corresponds to the position and the velocity vector at moment s+t of a perturbation whose position and the velocity vector at moment s was u. (We assume that a velocity of movement of a perturbation is either zero all the time or nonzero all the time.) Furthermore we assume that L(u, s, t) satisfies the following conditions: AFFINRE LINKING NUMBERS AND CAUSALITY RELATIONS FOR WAVE FRONTS 5 ____ a: L(u, s, 0) = u for all u_2_T_M ;___ b: 8s, t 2 R the map Ls,t: T M ! T M defined as Ls,t(u) = L(u, s, t) is a diffeomorphism; c: L(u, s,_t1_+ t2) = L L(u, s, t1), s + t1, t2 , 8s, t1, t2 2 R; d: 8u 2 T M and 8s0, t0 2 R _d_ fifi pT(L(u, s0, t))fi = L(u, s0, t0). dt t=t0 1.2. Definition. A propagation is a quadruple P = (L, x, T, V ) where L is a propagation law, x 2 M, T 2 R and V : Sn-1x! TxM \ s(x) is a smooth section of the R+ -bundle TxM \ s(x) ! Sn-1x. We fix an orientation preserving diffeomorphism Sn-1 ! Sn-1xand further in the text regard V as a mapping V : Sn-1 ! TxM \ s(x) . A propagation P = (L, x, T, V ) produces a wave front W (t) : Sn-1 ! M, t T as follows. Informally speaking, we assume that at a moment of time T something happens at a point x 2 M and the perturbation caused by this event starts to radiate from the point x in all the di- rections according to a propagation law L with the initial velocities of propagation in TxM described by V . Formally, for t T we define the front W (t) to be the mapping W (t) := pT(L(V, T, t - T )) : Sn-1 ! M. ___ _____ We put W (t) = L(V, T, t-T ) and fW(t) = pOW (t). In this case we also say that the wave front has originated from the event (x, T ). Initially a front of an event is a smooth embedded sphere (because of 1.1(d)), but generically it soon acquires double points, folds, cusps, swallow tails, and other complicated singularities. A generic front is a singular hypersurface, whose set of singularities is a codimension two subset of M. We denote by "x : Sn-1 ! ST M any map of the form (1.1) Sn-1 - h--! Sn-1x ST M where h is a map of degree 1. Clearly, the homotopy class of "x is well-defined and does not depend on x. Let S be the space of smooth maps Sn-1 ! ST M that are homotopic to a map "x as in (1.1). Then S xS is the space of ordered pairs (f1, f2) with fi2 S. Put to be the discriminant in S xS, i.e. the subspace that consists of pairs (f1, f2) such that there exist y1, y2 2 Sn-1 with f1(y1) = f2(y2). 6 V. CHERNOV (TCHERNOV) AND YU. RUDYAK (We do not include into the maps that are singular in the com- mon sense but do not involve double points between the two different spheres.) 1.3. Definition. We define 0 to be a subset (stratum) of consisting of all the pairs (f1, f2) such that there exists precisely one pair of points y1, y2 2 Sn-1 such that: a: f1(y1) = f2(y2). And moreover this pair of points is such that: b: yi is a regular point of fi, i = 1, 2; c: (df1)(Ty1) \ (df2)(Ty2) = 0. Here dfi is the differential of fi and Tyiis the tangent space to Sn-1 at yi. 1.4. Construction. Let æ : (a, b) ! S x S be a path which intersects 0 in a point æ(t0). We also assume that æ(t0 - ffi, t0 + ffi) \ 0 = æ(t0) for ffi small enough. We construct a vector v = v(æ, t0, ffi) as follows. We regard æ(t0) as a pair (f1, f2) 2 S x S and consider the points y1, y2 as in 1.3. Set z = f1(y1) = f2(y2). Choose a small ffi > 0 and regard æ(t0 + ffi) as a pair (g1, g2) 2 S x S. Set zi = gi(yi), i = 1, 2. Take a chart for ST M that contains z and zi, i = 1, 2 and set v(æ, t0, ffi) := -!zz1- -!zz22 TzST M. 1.5. Definition. Let æ : (a, b) ! S x S be a path as in 1.4. We say that æ intersects 0 transversally for t = t0 if there exists ffi0 > 0 such that v(æ, t0, ffi) =2(df1)(Ty1Sn-1) (df2)(Ty2Sn-1) TzST M for all ffi 2 (0, ffi0). It is easy to see that the concept of transversal intersection does not depend on the choice of the chart. 1.6. Definition. A path æ : (a, b) ! S x S, -1 a < b 1 is said to be generic if a: æ(a, b) \ = æ(a, b) \ 0; b: the set J = {t|æ(t) \ 0 6= ;} (a, b) is an isolated subset of R; c: the path æ intersects 0 transversally for all t 2 J. As one can expect, every path can be turned into a generic one by a small deformation. We leave a proof to the reader. AFFINRE LINKING NUMBERS AND CAUSALITY RELATIONS FOR WAVE FRONTS 7 Let P1 = (L1, x1, T1, V1) and P2 = (L2, x2, T2, V2) be two propaga- tions. They define mappings ri: R ! S, i = 1, 2 as follows. ( __ pO Vi fort Ti, ri(t) = f Wi(t) fort > Ti. 1.7. Definition. A pair of propagations {P1, P2} is said to be generic if the path r = (r1, r2) : R ! S x S is generic and r(Ti) 62 , i = 1, 2. 1.8. Definition. Let {P1, P2} be a generic pair of propagations and let r : R ! S x S be as above. Then a moment t 2 R such that r(t) 2 corresponds either to the baby-intersection or to the case where t > max (T1, T2) and there exists y1, y2 2 Sn-1 such that W1(t)(y1) = W2(t)(y2) = z and fW1(t)(y1) = fW2(t)(y2) 2 Sn-1z, i.e. to the case where there is a double point of the two fronts W1(t) and W2(t) at which the directions of the propagations of the two fronts coincide. Such a double point z of two fronts is called a point of dangerous intersection. Notice that we do not exclude situations where the fronts are tangent at z, the so-called dangerous tangencies, cf. [2]. For many pairs of propagations the dangerous intersection points do not occur. Such pairs of propagations are called dangerous intersection free. Now we describe a source of examples of such pairs. ____ ____ 1.9. Source of Examples. Let L : T M x R x R ! T M be a propa- gation law. Suppose that there exists a section ____ es: ST M x R x R ! T M x R x R of the map __px 1 x 1 such that Im (es) consists_of the trajectories of L, i.e. if (u, s0, 0) 2 Im (es) for some u 2 T M and s0 2 R, then L(u, s0, t) 2 Im(es), for every t 2 R. Let P1 = (L, x1, T1, V1) andfP2i= (L, x2, T2, V2) be propagationsfsuchi that (Im (V1), T1, 0) Im(esfiSx ), (Im (V2), T2, 0) Im(esfi ) and 1,T1,0 Sx2,T2,0 r(Ti) 62 , i = 1, 2. Then it is easy to see that the pair (P1, P2) is dangerous intersections free. 1.10. Example. Propagations that are defined by a Riemannian met- ric. An interesting class of examples comes from the propagation de- fined by the geodesics of a complete Riemannian_metric g on M. In this case L(u, s, t) is just a point on T M that corresponds to a velocity vector at moment s + t of a geodesic curve that had a velocity vector u at the moment s. It is easy to see that in this example at every moment 8 V. CHERNOV (TCHERNOV) AND YU. RUDYAK of time the velocity vectors of the points on the wave front are perpen- dicular to the image of the front. Thus the dangerous intersections are precisely the dangerous tangencies. Furthermore, if both Im V1 and Im V2 are spheres of the same radius r then the dangerous intersections (= dangerous tangencies) do not occur, since spheres of radius r in all the tangent planes produce the section esdescribed above. 1.11. Example. Propagation in a non-homogeneous and non- isotropic medium whose structure does not depend_on time. Assume that M is a Riemannian manifold and ~ : ST M ! T M isfaismooth section of the corresponding R+ -bundle such that Im(~fiSx) bounds a strictly convexfi domain in TxM for all x 2 M. The radius vector from s(x) to Im(mfiSx) in the given direction is the velocity vector of the distortion traveling in the direction. This information allows us to calculate for every smooth curve fl : [t1, t2] ! M the total time ø(fl) needed for the distortion to travel along this curve. Assume Im (V1) Im ~ and Im (V2) Im ~ and that propagation occurs according to the Huygens principle, i.e. distortion travels along the extremal curves of the functional ø on the space of smooth curves on M. It is clear that here we have a special case of the situation described in 1.9, and so the dangerous intersection points do not occur for such a pair of propagation. On the other hand, if the propagation happens according to the Huygens principle then at every point W (t)(x)f=iz the normal vector to the wave front is conjugate with respect to ~fiSzto the direction of the extremal curve along which the information travelled to this point, Arnold [3]. In particular, in this case the dangerous tangencies do not occur under the wave fronts propagation, since they are the dangerous intersections. 2. The causality relation invariant Recall that the standard sphere Sn-1 is assumed to be oriented. We say that a tangent frame r to Sn-1 is positive if it gives us the standard orientation of Sn-1. 2.1. Definition. Let æ be a path in SxS that intersects transversally in one point æ(t0) 2 0. We associate a sign eoe(æ, t0) to such a crossing as follows. We regard æ(t0) as a pair (f1, f2) 2 S x S and consider the points y1, y2 2 Sn-1 such that f1(y1) = f2(y2). Set z = f1(y1) = f2(y2). Let r1 and r2 be frames which are tangent to Sn-1 at y1 and y2, respectively, AFFINRE LINKING NUMBERS AND CAUSALITY RELATIONS FOR WAVE FRONTS 9 and both are assumed to be positive. Consider the frame {df1(r1), v, df2(r2)} at z 2 ST M where v is a vector described in 1.4. We put eoe(æ, t0) = 1 if this frame gives us the orientation of ST M, otherwise we put eoe(æ, t0) = -1. Because of the transversality and condition (c) from 1.3, the family {df1(r1), v, df2(r2)} is really a frame. Notice also that the vector v is not well- defined, but the above defined sign eoeis. Clearly if we traverse the path æ in the opposite direction then the sign of the crossing changes. 2.2. Definition. Suppose that a front W passes through a point x 2 M at the moment of time t0 in such a way that the velocity vector vx of the front W (t0) at x is transverse to W (t0), W (t0) restricted to a small neighborhood U of W -1(t0)(x) is an embedding, and x has only one preimage under W (t0). Recall that the manifold M is oriented. Let ox be the local orien- tation of W (t0) at x (i.e. the orientation of the tangent plane Tx to W (t0)). We say that the local orientation ox is positive, and write oe(W (t0), x) = 1 if the pair (ox, vx) gives us the orientation of M; oth- erwise we say that the local orientation of W (t) at x is negative and write oe(W (t0), x) = -1. Notice that the same wave front W (t) can contain two points x and y such ox is positive orientation while oy is the negative one, see Figure 1. Figure 1 Consider a generic pair (P1, P2) of propagations P1 = (L1, x1, V1, T1) and P2 = (L2, x2, V2, T2). In the text below we assume that T1 T2. The case where T1 > T2 is treated in a similar way. Let t > T2 be a generic moment of time, i.e. the one at which dangerous intersections do not occur. 10 V. CHERNOV (TCHERNOV) AND YU. RUDYAK Let ci, i 2 I N where T2 < ci < t be moments of time when dangerous intersections did occur. 2.3. Definition. We define oe(W1(ci), W2(ci)) as the sign eoeof the cor- responding passage of 0. Notice that oe(W1(ci), W2(ci)) is symmetric if n is even and skew- symmetric if n is odd. Let pj, j 2 J N be the moments of time when the front W1 passed through the point x2 before the front W2 originated. (Notice that pj < T2 and that oe(W1(pj), x2)) is well-defined since the pair of propagations is generic.) A straightforward verification shows that oe(W1(pj), x2) = eoe(æ, t0). where æ(t) = (fW1(t), "x2), t 2 (pj - ffi, pj + ffi). 2.4. Definition. We set X X CR (W1(t), W2(t)) = oe(W1(ci), W2(ci)) + oe(W1(pj), x2) 2 Z i2I j2J and call it the causality relation invariant for the fronts W1(t) and W2(t) at a given momentPof time t. (If in fact T1 > T2, then the second sum should be k2K(-1)dimM oe(W2(qk), x1), where qk, k 2 K N are the moments of time when the front W2 passed through the point x1 before the front W1 originated. One can easily verify that (-1)dimM oe(W2(qk), x1) coincides with the sign of the corresponding crossing of 0 by the path r = (r1, r2).) If the above pair (P1, P2) of propagations is dangerous intersections free, then X CR (W1(t), W2(t)) = oe(W1(pj), x2). j2J It is easy to see that in this case CR (W1(t), W2(t)) does not depend on t provided t > max (T1, T2), and thus it is invariant under the propaga- tion. In particular, if CR (W1(t), W2(t)) is non-zero then we know for a fact that the perturbation caused by the first signal has reached the source point of the second signal before the second signal originated. Moreover, if CR (W1(t), W2(t)) = k 6= 0, then we can say for sure that the first wave front has passed through the source point of the second front at least k times before the second signal originated. (Of course it could be that it did pass more times, because it could have passed k + l times with a positive sign and l times with a negative sign.) In case of generic propagations P1, P2 and CR (W1(t), W2(t)) = k 6= 0 we can conclude that the sum of the number of baby-intersections and of the number of dangerous intersections is at least k. This probably AFFINRE LINKING NUMBERS AND CAUSALITY RELATIONS FOR WAVE FRONTS 11 could be interpreted as the quantity that measures either how much faster the first front is than the second so that they could become dan- gerously intersected or how many times the first front did pass through the source of the second front before the second front originated. 3.Homotopy properties of maps to ST M 3.1. Definition.fGivenia map ff : S1 x Sn-1 ! ST M, we say that ff is special if fffi*xSn-1has the form "x for some x 2 M, see (1.1). Here * 2 S1 is the base point. 3.2. Definition. Given an n-dimensional manifold N and a map fi : N ! ST M, we define d(fi) to be the degree of the map prOfi : N ! M 3.3. Lemma. Let ff : S1 x Sn-1 ! ST M be a special map. Then there exists a map fi : Sn ! ST M such that d(fi) = d(ff). Proof. We regard Sn-1 as a pointed space. Consider a map eff: S1 x Sn-1 ! STfMisuch that:fi 1: efffi*xSn-1=ffffi*xSn-1,ifi 2: efffiS1x*= fffiS1x*,fi 3: eff|txSn-1= "fieff(tx*). We regard S1xSn-1 as the CW -complex with four cells e0, e1, en-1, en, dim ek = k. It is easy to see that the maps effand ff coincide on the (n - 1)-skeleton. Thus, the maps ff and eff(restricted to the n-cell) together yield a map fi : Sn ! ST M. Clearly d(eff) = 0, and therefore d(fi) = d(ff). 3.4. Lemma. Suppose that there exists a map fi : Sn ! ST M with d(fi) 6= 0. Then the Euler class Ø 2 Hn(M) of the tangent bundle T M ! M is zero. Proof. We setf = prOfi : Sn ! M. First, notice that Hn(M) = Z because d(fi) 6= 0. So, again since d(fi) 6= 0, we conclude f*Ø 6= 0 whenever Ø 6= 0. Now the result follows because f*Ø is the obstruction to the lifting of f to ST M, while fi is such a lifting of f. 3.5. Lemma. Let M be a 2k-dimensional oriented manifold and fi : S2k ! ST M be a map with d(fi) 6= 0. Then the Euler class of the tangent bundle T M ! M is non-zero. Proof. We set f = prOfi : S2k ! M and d = d(fi). Notice that M is closed because d(fi) 6= 0. Let f! : H*(M) ! H*(S2k) be the transfer map, see e.g. [11, V.2.12]. Since f*(f*y\x) = y\f*x for all x 2 H*(S2k) and y 2 H*(M), we conclude that f*f!(z) = dz, for all z 2 H*(M). 12 V. CHERNOV (TCHERNOV) AND YU. RUDYAK In particular, dHi(M) = 0 for 0 < i < 2k, i.e. Hi(M; Q) = 0 for 0 < i < 2k. So, the Euler characteristic of M is 2, and thus the Euler class of the tangent bundle is non-zero (in fact, 2). 3.6. Corollary. If M is an even-dimensional oriented manifold, then d(fi) = 0 for every fi : Sn ! ST M. Proof. This is a direct consequence of Lemma 3.4 and 3.5. Let deg : ßn(M) ! Z be the degree homomorphism, i.e., the ho- momorphism which assigns the degree degf to the homotopy class of a map f : Sn ! M. (In fact, it coincides with the Hurewicz ho- momorphism h : ßn(M) ! Hn(M) for M closed and is zero for M non-closed.) 3.7. Definition. Given a connected oriented manifold Mn, we define an Abelian group AAA(M) and a homomorphism q = qM : Z ! AAA(M) as follows. If n is even then AAA(M) = Z and q = 1Z. If n is odd then AAA(M) is the cokernel of the degree homomorphism deg : ßn(M) ! Z and q : Z ! AAA(M) is the canonical epimorphism. Notice that AAA(M) = Z for odd- dimensional non-closed manifolds. 3.8. Proposition. Let Mn be a closed odd-dimensional manifold as in Definition 3.7. Then the following holds: (i) If AAA(M) = 0 then M is a homotopy sphere. (ii) If ß1(M) is infinite then AAA(M) = Z. Proof. (i) If AAA(M) = 0 then there exists a map Sn ! M of degree 1. Since every map of degree 1 induces epimorphism of fundamen- tal groups and homology groups, we conclude that M is a homotopy sphere. (ii) This follows because every map Sn ! M passes through the universal covering, and thus the Hurewicz homomorphism is trivial. 4. The affine linking invariant AL as a reduction of CR 4.1. Definition. We define 1 to be the subset (stratum) of con- sisting of all the pairs (f1, f2) such that there exists precisely two pairs of points y1, y2 2 Sn-1 as in 1.3. Here we assume that the two double points of the image are distinct. Notice that, in a sense, i is a stratum of codimension i in . In particular, a generic path in S x S intersects 0 in a finite number of points, and a generic disk in S x S intersects 1 in a finite number of points. A generic path fl : [0, 1] ! S xS that connects two points in S xS \ intersects 0 in finitely many points fl(tj), j 2 J N and all the AFFINRE LINKING NUMBERS AND CAUSALITY RELATIONS FOR WAVE FRONTS 13 intersection points are of the types described in 2.1. Put X (4.1) AL(fl) = oe(fl, tj) 2 Z. j2J fi fi We let A = {(x,fy)i2 R2 fix2 + y2 1}, B1f=i{(x, y) 2 A fixy = 0}, B2 = {(x, y) 2 A fix = 0}, B3 = {(x, y) 2 A fiy = 0}, B4 = {(0, 0)}, B5 = ;. We define a regular disk in S x S as a generically embedded disk D such that the triple (D, D \ 0, D \ 1) is homeomorphic to a triple (A, B, C) where B is one of Bi's and C B4. 4.2. Lemma. Let fi be a generic loop that bounds a regular disk in S x S. Then AL(fi) = 0. Proof. Straightforward. 4.3. Lemma. Let fi be a generic loop that bounds a disk in S x S. Then AL(fi) = 0. Proof. Without loss of generality we can (using a small deformation of the disk) assume the disk is the union of regular ones, cf. Arnold [2], [1]. Now the proof follows from Lemma 4.2. Notice that S x S is path connected. 4.4. Corollary. The invariant AL induces a well-defined homomor- phism AL : ß1(S x S, *) ! Z. Proof. Since every element of ß1(S x S, *) can be represented by a generic loop, the proof follows from Lemma 4.3. Let x1, x2 be two distinct points of M. Let ff : S1 x Sn-1 ! ST M be a special map (see Definition 3.1) such that the composition Sn-1 S1 x Sn-1 ff-!ST M has the form "x1, and let e : S1x Sn-1 ! ST M be the map of the form proj n-1 "x2 S1 x Sn-1 --! S --! ST M Then (ff, e) is a loop in (S x S, *). 4.5. Lemma. AL[(ff, e)] = d(ff). Proof. Notice that AL[(ff, e)] is the intersection index of the cycles ff(S1 x Sn-1) and Sn-1x2. This index coincides with the degree of the map prOff because the last one is equal to the algebraic number of the preimages of x2. 14 V. CHERNOV (TCHERNOV) AND YU. RUDYAK 4.6. Definition. Choose a point * = (f01, f02) 2 S x S \ and put AL (*) to be a constant k 2 Z. (It is easy to see from the following proof that this choice is the only ambiguity in the definition of the AL -invariant.) Take an arbitrary point f = (f11, f12) 2 S x S \ and choose a generic path fl going from * to f. We set AL (f) = q(k + AL(fl)) 2 AAA(M) and call AL the affine linking invariant. Here q is the epimorphism from Definition 3.7. 4.7. Theorem. The function AL : ß0(S x S \ ) ! AAA(M) is well- defined and increases by 1 2 AAA(M) under the positive transverse pas- sage through the stratum 0. Furthermore, the above property determines the function AL uniquely up to an additive constant. Proof. To show that AL is well-defined we must verify that the defi- nition is independent on the choice of the path fl that goes from * to f. This is the same as to show that q( AL(')) = 0 for every closed generic loop ' at *. Since AL : ß1(S xS, *) ! Z is a homomorphism, it suffices to prove that q( AL(')) = 0 for all generators ' of ß1(S x S, *). We can and shall assume that the maps f01, f02: Sn-1 ! ST M are the maps "x1, "x2, respectively, where x1, x2 are two distinct points in M. Then the classes [(ff, e)] and similar classes [(e, ff)] generate the group ß1(S x S, *). By Lemma 4.5 we have AL[(ff, e)] = d(ff). Clearly, d(ff) = 0 if M is an non-closed manifold. So, we assume M to be closed. Now, for n even d(ff) = 0 by Corollary 3.6, while for n odd q(d(ff)) = 0 by Lemma 3.3. Let (P1, P2) be a generic pair of propagations, and let t be a moment of time when dangerous intersection do not occur. Let q : Z ! AAA(M) be the epimorphism described in Definition 3.7. 4.8. Theorem. The invariants CR and AL are related as follows: q CR (W1(t), W2(t) = AL fW1(t), fW2(t) - AL (V1, V2). Proof. The Theorem follows because the signs defined for dangerous intersections (as well as for baby-intersections) are exactly the sign of the corresponding crossings of 0. If the pair of propagations is dangerous intersections free then the invariant AL (fW1(t), fW2(t)) 2 AAAgives us the number of times the wave AFFINRE LINKING NUMBERS AND CAUSALITY RELATIONS FOR WAVE FRONTS 15 fronts W1 has passed through the point x2 before the wave front W2 has originated. It is also easy to see that if the wave front W2 did not originated yet then AL (fW1(t), fW2(t)) is the number of times the wave front W1 has passed through x2 at the moment of time t. The invariant AL works especially nice for even dimensional mani- folds, since in these cases AAA= Z. 5. Causality relation invariant in the case of the propagation according to Riemannian metrics. As it was noticed in 1.10, if a propagation happens according to a complete Riemannian metric and Im V is a sphere then the velocities of the points of the front are always orthogonal to the front. So, if each of two propagations happens according to a complete Riemannian metric then dangerous tangency points and the dangerous intersection points are the same thing. In this section we deal only with this case and we provide an especially nice way of calculation of the CR invariant. We need some preliminaries. Let W be a wave front, and let x 2 Im W (t) be a non-singular point of W (t). For sake of simplicity we denote Tx ImW (t) just by T . Let O be a small neighborhood of x in M, and let U = O \Im W (t). Without loss of generality we can and shall assume that the injectivity radius is big enough ( 3) for all points of O. The Riemannian metric g on M produces a unique symmetric con- nection on M. So, for every a 2 O, the parallel transport along the geodesic segment (connecting x and a) gives us an isomorphism (5.1) øa : TaM ! TxM Furthermore, we can regard every sphere Sa 2 ST M, a 2 M as the unit sphere in TaM, and so ST M can be regarded as the total space of the unit sphere subbundle of T M. Since the connection respects the Riemannian metric, we conclude that øa(Sa) = Sx. 5.1. Definition. (a) We define ß : pr-1(O) ! Sx as follows. A point z 2 pr-1(O) is a pair (a, ,) with a = pr(z) and , 2 Sa, and we set ß(z) = øa(,) with øa as in (5.1). (b) Given u 2 U, let `(u) 2 Im fW be the point with pr(`(u)) = u. In this way we get a map ` : U ! fW. We set z = `(x). Given e 2 T , we set eW := d`(e), eW 2 TzIm fW TzST M. It is clear that d` : T ! TzST M is an isomorphism. 16 V. CHERNOV (TCHERNOV) AND YU. RUDYAK (c) Let z 2 ST M be the point described in (b). We define the horizontal section H : O ! ST M of pr by setting H(a) = ø-1a(z) 2 Sa ST M. Furthermore, given w 2 TaM, a 2 O, we set wH = dH(w) 2 TH(a)ST M. Clearly, wH can be characterized by the properties (d pr)(wH ) = w, dß(wH ) = 0. (d) We define the Gauss map G = GW : U ! Sx by setting G(u) = øu(nu), u 2 U where nu is the unit normal vector to U at u. (e) Let z 2 ST M be the point described in (b). Given w 2 TxM, we define wS 2 TzTxM as follows. We regard z as the vector z 2 TxM. Furthermore, we regard TxM as the affine space T affover the vector space TxM and consider the parallel shift Pz : T aff! T aff, a 7! a + z. Let o 2 T affcorrespond to the origin of the vector space TxM. Using the obvious identification TxM = ToT aff, we regard w as the tangent vector wo 2 ToT aff, and we set wS = dPz(wo) 2 TzT aff= TzTxM. Notice that if e 2 T then eS 2 TzSx. (This is where the notation comes from: eS is the spherical lifting of e.) 5.2. Lemma. For every e 2 T we have eW - eH = dG(e). Proof. First, notice that G = ß O ` : U ! Sx, because `(u) can be regarded as the unit normal vector to U at u. So, dß(eW ) = dG(e). Now, (d pr)(eW - dG(e)) = e - 0 = e and dß(eW - dG(e)) = dG(e) - dG(e) = 0. Thus, eW - dG(e) = eH . 5.3. Proposition. Let e 2 T , and let n be the normal vector field to U in M. Then dG(e) = (ren)S. Here r denotes the covariant differentiation operation on M. Proof. Let fl : (-ffi, ffi) ! U be a curve with `fl(0) = e. We define the curve i : (-ffi, ffi) ! Sx by setting i(t) to be the end of the vector øfl(t)nfl(t). Since d fifi ren = __ øfl(t)nfl(t)- nxfi dt t=0 AFFINRE LINKING NUMBERS AND CAUSALITY RELATIONS FOR WAVE FRONTS 17 we conclude that `i(0) = (ren)S. On the other hand, G O fl = i, and thus dG(e) = dG(f`l(0)) = (G `O fl)(0) = `i(0) = (ren)S. 5.4. Corollary. eW - eH = (ren)S. Proof. This is the direct consequence of 5.2 and 5.3. Consider the Weingarten operator (5.2) A = AW : T ! T, A(e) = ren. The Corollary 5.4 can now be written as follows: (5.3) eW - eH = (Ae)S. Now let W1 and W2 be two wave fronts, and let x 2 M be a point of dangerous tangency of W1(t) and W2(t). We assume that the cor- responding pair of propagations is generic. Again, we denote by T the common tangent plane TxWi(t), i = 1, 2. Let Ai := AWi : T ! T, i = 1, 2 be the Weingarten operators considered in (5.2). We set B = A1 - A2. It is well known that each Ai is a self-adjoint operator, [12, Ch. 7], and therefore B is. Let k1, . .,.kn-1 be the eigenvalues (with multiplicities) of B. 5.5. Proposition. Ker B = 0, and so ki6= 0 for all i. Proof. Let e 6= 0 be a vector with Be = 0. Then eW1 - eW2 = (eW1 - eH ) - (eW2 - eH ) = (A1e - A2e)S = (Be)S = 0. i.e. Tx ImW1 \ Tx ImW2 6= 0. But this is impossible because the pair of propagations is assumed to be generic (look conditions 1.6(b) and 1.3(c)). In particular, detB = k1. .k.n6= 0. 5.6. Definition (Alternative definition of the sign oe(W1(t), W2(t))). We put "(W1(t), W2(t) = 1 if both fronts have the same local orienta- tions at x (as defined in 2.2) and "(W1(t), W2(t)) = -1 if the fronts have opposite local orientations. Now we set boe(W1(t), W2(t)) = "(W1(t), W2(t)) sign(det B) sign(|v1| - |v2|) where vi is the velocity vector of Wi(t) at x. 5.7. Theorem. boe(W1(t), W2(t)) = oe(W1(t), W2(t)). 18 V. CHERNOV (TCHERNOV) AND YU. RUDYAK Proof. Given a vector e 2 T , we set e0 = eW1 and e00= eW2 . Choose a basis {e1, . .,.en-1} of T containing of the eigenvectors of B, i.e., Bei= kiei. Because of equality (5.3)we have (5.4) e00i- e0i= (Bei)S = (kiei)S = kieSi. We can and shall assume that the frame {e1, . .,.en-1} gives the posi- tive (local) orientation of Wi(t), i = 1, 2 at x. Take the polyvector p := e01^ . .^.e0n-1^ v ^ e001^ . .^.e00n-1 where v is the vector defined in 1.4. Then p 6= 0 since the pair of prop- agations is assumed to be generic. Notice that p gives us an orientation of ST M, and we say that p is positive if this orientation coincides with the original one, otherwise we say that p is negative. According to Definition 2.3, the sign of p is equal to "(W1(t), W2(t))oe(W1(t), W2(t)). So, we must prove that sign of the polyvector p is equal to the sign of (det B) sign(|v1| - |v2|). To be definite, we assume that |v1| > |v2| and prove that the sign of p is equal to the sign of detB. Since e00i= e0i+ kieSiand detB = k1. .k.n-1, we conclude that e01^. .^.e0n-1^v^e001^. .^.e00n-1= (det B)e01^. .^.e0n-1^v^eS1^. .^.eSn-1. So, it remains to prove that the polyvector e01^ . .^.e0n-1^ v ^ eS1^ . .^.eSn-1 is positive. Since e01^ . .^.e0n-1^ v ^ eS1^ . .^.eSn-16= 0, we conclude that the family {eS1, . .,.eSn-1} generate TzSx. It is easy to see that the frame {eS1, . .,.eSn-1} gives the original orientation of Sx, since the frame {e1, . .,.en-1} gives the positive local orientation of each of the fronts at x. So, it remains to prove that the polyvector (d pr)(e01^ . .^.e0n-1^ v) = e1 ^ . .^.en-1 ^ (d pr)v is positive, i.e. that it gives the original orientation of M. Recall that the polyvector e1 ^ . .^.en-1 ^ (v1 - v2) is positive since |v1| > |v2|. Taking into account that the vector v1-v2 is orthogonal to T and points into the direction of both propagations at x, it sufices to prove that > 0. But this is clear because W1 is faster then W2 at (x, t), and so the point pr(z1) (in notation of 1.4) is further then the point pr(z2) from T . AFFINRE LINKING NUMBERS AND CAUSALITY RELATIONS FOR WAVE FRONTS 19 5.8. Example. Suppose that the fronts propagate as it is shown in Figure 2. Figure 2 Let W1 be the "right" front. Then, clearly, oe(W1(t), W2(t)) = -"(W1(t), W2(t)). The negative sign appears because W2 is faster the W1. 6. Examples To illustrate the usage of the affine linking invariant consider the following examples. 6.1. Example. Here we show how to apply AL to determining the causality relation. Let M be a smooth oriented n-dimensional mani- fold that is not an odd-dimensional homotopy sphere. Let W1, W2 be the wave fronts that originated on M long time ago and were propagat- ing according to the dangerous intersections free pair of propagations {P1, P2}. Assume that the current picture of wave fronts W1(t), W2(t) is the one shown in Figure 3 with the velocity vectors normal to the two spheres shown in Figure 3. Then a straightforward calculation shows that AL (fW1(t), fW2(t)) - AL (V1, V2) = 1 6= 0 (we used the notation as in Theorem 4.8), and thus the first wave front reached the birth point of the second front before the second front originated. (The sign of 1 in this example depends on which of the two fronts shown in Figure 3 is W1 in the case where n is odd and is always a plus sign when n is even.) This seems to demonstrate that AL is a very powerful invariant be- cause in this case we know neither the propagation laws nor when and where the fronts originated. In fact, in this example we can make this conclusion even without the knowledge of the topology of M outside of the depicted part of it. 20 V. CHERNOV (TCHERNOV) AND YU. RUDYAK Figure 3 6.2. Example. Here we show how to apply AL to estimating of the number of times the wave front passed though a given point between the two moments of time. Assume that we have a wave front W that propagates on M and that M is not an odd-dimensional homotopy sphere. Assume that at a certain moment of time the picture of the wave front was the one shown in Figure 4.a and later it developed into the shape shown in Figure 4.b. (The Figure 4.b depicts a sphere that can be obtained from the trivially embedded sphere by passing three times through a point.) Figure 4 Let fW(t1), fW(t2) : Sn-1 ! ST M be the liftings of the fronts shown in Figure 4.a and b respectively. A straightforward calculation shows AFFINRE LINKING NUMBERS AND CAUSALITY RELATIONS FOR WAVE FRONTS 21 that AL (fW (t2), "x) - AL (fW (t1), "x) = 3 2 AAA(M) for every map "x : Sn-1 ! ST M as in (1.1). Thus if the dimension of the ambient manifold is even, or ß1(M) is infinite, then W has passed at least three times through the point x between the time moments shown in Figure 4.a and 4b. Once again, this conclusion does not depend on the topology of M outside of the part of it depicted in Figure 4, on the time passed between the two pictures taken, and on the propagation law. Acknowledgments: The first author was supported by the free- term research money from the Dartmouth College. The second author was supported by the free-term research money from the University of Florida, Gainesville. The authors are very thankful to Robert Caldwell and Jacobo Pe- jsachowicz for useful discussions. References [1]V.I. 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