This paper is the first in a series that investigates the Morse Theory and gradient flows on smooth compact manifolds with boundary, a special case of the well-developed Morse theory on stratified spaces (see [2] -[4] ). For us, however, the starting point and the source of inspiration is the 1929 paper of Morse [1] .

We intend to present to the reader a version of the Morse Theory in which the critical points remain behind the scene, while shaping the geometry of the boundary! Some of the concepts that animate our approach can be found in [5] , where they are adopted to the special environment a 3 D-gradient flows. These notions include stratified convexity or concavity of traversing flows in connection to the boundary of the manifold. That concavity serves as a measure of intrinsic complexity of a given manifold with respect to any traversing flow. Both convexity and concavity have strong topological implications.

Another central theme that will make its first brief appearance in this paper is the holographic properties of traversing flows on manifolds with boundary. The ultimate aim here is to reconstruct (perhaps, only partially) the bulk of the manifold and the dynamics of the flow on it from some residual structures on the boundary, thus the name “holography”.

In Section 2, for so-called boundary generic fields on (see Definition 2.1), we explore the Morse stratification of the boundary (see Formula (1) and [1] ), induced by the vector field on.

In Section 3, we investigate the degrees of freedom to change this stratification by deforming a given vector field within the space of gradient-like fields (Theorem 3.2, Corollary 3.2, and Corollary 3.3).

In Section 4, for vector fields on compact manifolds, we introduce the pivotal notion of boundary -convexity/-concavity, (see Definition 4.1). Then we explore some topological implications of the existence of a boundary -convex/-concave traversing field on (see Lemma 4.2, Corollary 4.2, Corollary 4.3, and Corollary 4.4).

Let denote the orthogonal projection of the field on the bundle tangent to the boundary. Occasionally, we can determine whether a given field is convex/concave just by observing the behavior of the -trajectories on the boundary (Theorem 4.1, Theorem 4.2). We view the possibility of such determination as an instance of a more general phenomenon, which we call “holography”. This phenomenon will occupy us fully in a different paper.

The Eliashberg surgery theory of folding maps [6] [7] helps us to describe the patterns of Morse stratifications for traversing 3-concave and 3-convex fields (Theorem 5.1, Conjecture 5.1, and Corollary 5.1).

Inspired by [1] , we start by introducing some basic notions and constructions that describe the way in which generic vector fields on a compact smooth manifold interact with its boundary.

Let be a compact smooth -dimensional manifold with a boundary. Let v be a smooth vec- tor field on which does not vanish on the boundary. As a rule, we assume that X is properly contained in a -dimensional manifold and that the field v extends to a field on so that. In fact, we always treat the pair as a germ of a space and a field in the vicinity of the given pair.

Often we will consider vector fields only with the isolated Morse-type singularities (zeros) located away from the boundary. This means that v, viewed as a section of the tangent bundle, is transversal its zero section. In other words, in the vicinity of each singular point, there is a local system of coordinates such that the field v can be represented as, where all.

To achieve some uniformity in our notations, let and.

The vector field v gives rise to a partition of the boundary into two sets: the locus, where the field is directed inward of, and, where it is directed outwards. We assume that, viewed as a section of the quotient line bundle over, is transversal to its zero section. This assumption implies that both sets are compact manifolds which share a common boundary . Evidently, is the locus where is tangent to the boundary.

Morse has noticed that, for a generic vector field ν, the tangent locus inherits a similar structure in con- nection to, as has in connection to (see [1] ). That is, gives rise to a partition of into two sets: the locus, where the field is directed inward of, and, where it is directed outward of. Again, let us assume that v, viewed as a section of the quotient line bundle over, is transversal to its zero section.

For generic fields, this structure replicates itself: the cuspidal locus is defined as the locus where v is tangent to; is divided into two manifolds, and. In, the field is directed inward of, in, outward of. We can repeat this construction until we reach the zero-dimensional stratum (see Figure 1 and Figure 2 which depict these strata for fields in).

These considerations motivate

Definition 2.1 We say that a smooth field v on is boundary generic if:

· ,

· , viewed as a section of the tangent bundle, is transversal to its zero section,

· for each, the -generated stratum is a smooth submanifold of,

· the field, viewed as section of the quotient 1-bundle

,

is transversal to the zero section of for all.

We denote the space of smooth boundarygeneric vector fields on by the symbol.

Thus a boundary generic vector field on gives rise to two stratifications:

, the first one by closed submanifolds, the second one—by compact ones. Here .

For simplicity, the notations “” do not reflect the dependence of these strata on the vector field. When the field varies, we use a more accurate notation “”.

Remark 2.1. Replacing with affects the Morse stratification according to the formula:

where when, and otherwise.

We will postpone the proof of the theorem below until the second paper in this series of articles (see [8] , Theorem 3.4, an extension of Theorem 2.1 below). There we will develop the needed analytical tools.

Theorem 2.1 Boundary generic vector fields form an open and dense subset in the space of all smooth fields on.

Definition 2.2 We say that a smooth vector field on is of the gradient type (or gradient-like) for a smooth function if:

· the differential and the field vanish on the same locus,

· the function in,

· in the vicinity of, there exist a Rimannian metric on so that, the gradient field of in the metric.

Definition 2.3 A smooth function is called Morse function if its differential, viewed as a section of the cotangent bundle, is transversal to the zero section.

Recall that, for a Morse function on a compact -manifold X, the critical set

is finite, and each point has special local coordinates such that, where

for all (for example, see [9] ).

Definition 2.4 Let be a smooth function and its gradient-like vector field. We say that the pair is boundary generic if the field is boundary generic in the sense of Definition 2.1 and the res- trictions of to each stratum are Morse functions for all.

Lemma 2.1 Let be a compact smooth manifold, and a smooth manifold which is stratified by sub- manifolds. Let be the space of smooth maps which are transversal to each stratum. Put. Next consider the space of pairs such that

and has the property: are Morse functions for all. Then is open and dense in the

space.

Proof. Consider the space, where denotes the cotangent bundle of. The property is equivalent to the property of the section of the bundle

to be transversal to each (transversal) intersection of the -graph with each stratum. The latter property defines a open set in.

In order to validate density of in, we first perturb a given map to make it transversal to each stratum, and then perturb a given function to make the section of transversal to each manifold.

Theorem 2.2 The boundary generic1 Morse pairs on a compact manifold form an open and dense subset in the space of all smooth functions and their gradient-like fields.

Proof. By Theorem 2.1, the boundary generic fields form an open and dense set in the space of all fields.

Let be a complete flag in, formed by subspaces of codimension j. In the proof of Theorem 3.4 [8] , for every field, we will construct a smooth map such that. Moreover, is transversal to each, if and only if, is a boundary generic field. The construction of the map utilizes high order Lie derivatives of an auxiliary function as in Lemma 3.1 [8] .

Now the property of boundary generic Morse pairs to be open and dense in the space of all pairs follows from Lemma 2.1: just let, , , and in that lemma.

For the reader convenience, let us sketch now an alternative argument that establishes just the density of boundary generic Morse pairs in the space of all pairs. It does not rely on the construction of the map from [8] .

We start with a pair where and at the points of the set where. By a small perturbation of, we can assume the is a Morse function on and its gradient-like field.

Let be a compact regular neighborhood of in so small that. By Theorem 2.1, we can perturb to a new field so that is boundary generic in the sense of Definition 2.1 and still.

For a given, the condition defines an open cone in the space of all fields, subject to the constraint. Therefore can be chosen both boundary generic and gradient-like for. When is fixed, so are the stratifications.

Next, with being fixed, we perturb again to a new function so that and

are Morse functions for all j. The perturbation will be supported in the compact. We start

constructing inductively first from adjusting it on the 1-manifold and then moving sequentially to the strata with lower indices j. We pick each perturbation so small that the open condition is not violated. The existence of the desired -th perturbation is based on the fact that Morse functions on a compact manifold (in this case, on) form an open and dense subset in, the space of all smooth functions on, being equipped with the Whitney topology. Note that since is tangent

to along and, the restriction has no critical points in the vi-

cinity of. Thus we need to perturb only on a compact subset which has an

empty intersection with. This perturbation extends smoothly from to. Eventually, we reach the upper stratum, thus constructing a boundary generic approximation of the given pair.

All the changes of, but the first one, we have introduced so far are supported in, where

and. This proves that the boundary generic pairs form a dense set in the space of all pairs, where being a -gradient-like field, subject to the constraints:, and being a Morse function.

For a given Morse pair, we denote by the set of critical points of the function. For a boundary generic Morse pair, the finite critical set is divided into two complementary sets: the set of positive critical points and the set of negative ones (see Figure 3).

Remark 2.2. Note that when, it may happen that. However, if a component of is a closed manifold, then must have local extrema, in which case.

Consider a generic field and a Riemannian metric on. We denote by the orthogonal projection of the field on the tangent space. Note that if is a gradient field for a function in metric, then is automatically a gradient field for the restrictions and.

Take a smooth vector field on a compact -manifold with isolated singularities

. We denote by the localized index of at its typical singular point. In a

local chart, is defined as the degree of a map from a small -centered -sphere

to the unit -sphere. The map takes each point to the point.

We define the “global” index as the sum.

For a generic field and a Riemannian metric on, we form the fields on and define the global index of by the formula:

Let us revisit the beautiful Morse formulas [1] :

Theorem 2.3 (The Morse Law of Vector Fields) For a boundary generic vector field and a Riemannian metric on a -manifold, such that the singularities of the fields are isolated for all, the following two equivalent sets of formulas hold:

where stands for the Euler number of the appropriate space2.

For vector fields with symmetry, the Morse Law of Vector Fields has an equivariant generalization [10] . Here is its brief description: for a compact Lie group acting on a compact manifold, equipped with a

-equivariant field, we prove that the invariants can be interpreted as taking values in the

Burnside ring of the group (see [11] for the definitions). With this interpretation in place, the appearance of Formula (2) does not change.

Morse Formula (2) has an instant, but significant implication:

Corollary 2.1 Let be a smooth neighborhood of the zero set of a vector field on a compact -manifold. Assume that is boundary generic with respect to both boundaries, and. Then

Remark 2.3. Therefore, the numbers

can serve as “more and less localized” definitions of the index invariant.

An interesting discussion, connected to Theorem 2.3, its topological and geometrical implications, can be found in the paper of Gotlieb [12] . The “Topological Gauss-Bonnet Theorem” below is a sample of these re- sults.

Theorem 2.4 (Gotlieb) Let be a compact smooth -dimensional manifold and a smooth map which is a immersion in the vicinity of the boundary. Let be a Riemannian metric on which, in the vicinity of, is the pull-back of the Euclidean metric on. Consider a generic linear function such that the composite function has only isolated singularities in the interior of. Let be the gradient field of 3. Assume that is boundary generic.

Then the degree of the Gauss map

can be calculated either by integrating over the normal curvature (in the metric) of the hyper- surface, or in terms of the -induced stratification

by the formula

(3)4

Example 2.1. Let X be an orientable surface of genus g with a single boundary component. Let be an immersion, and let, and be as in Theorem 2.4.

Since Ф is an immersion everywhere (and not only in the vicinity of ∂X as Theorem 2.4 presumes), we get that. Thus. Then Theorem 2.4 claims that the degree of the Gauss map is equal to

Thus, the topological Gauss-Bonnet theorem, for immersions, reduces to the equation

.

So the number of -trajectories in that are tangent to, but are not singletons (they correspond to points of), as a function of genus, grows at least as fast as.

On the other hand, by the Whitney index formula [13] , the degree of can be also calculated as, where denotes the number of positive/negative self-intersections of the curve, and.

By a theorem of L. Guth [14] , the total number of self-intersections. Moreover, this lower bound is realized by an immersion! Therefore, for any immersion, the total number of self-intersections of the curve can be estimated in terms of the boundary-tangent -trajectories:

and for some special immersion, we get

Corollary 2.2 Let be a compact -manifold with boundary, which is properly contained in an

open -manifold. Let be a smooth map which is a immersion in the vicinity of the

boundary. Let be a Riemannian metric on which, in the vicinity of, is the pull-back of the Euclidean metric on.

Let be a linear function, and its composition with the map. Form the gradient field in. Assume that the pair is boundary generic in the sense of Definition 2.4.

For each, consider a -small tubular neighborhood of the manifold in. Then is an immersion. This setting gives rise to the Gauss map, defined by the

formula, where and is the unit vector inward normal to at.

Then the degree of the Gauss map can be calculated either by integrating (with respect to the

-measure) over the normal curvature of the hypersurface, or in terms of the

-induced stratum:

Proof. We will apply Theorem 2.4 to the field in to conclude that

Since in, , and the last term of this equation reduces to.

Remark 2.4. Of course, for an odd-dimensional, the Euler number, and so is. When is even-dimensional (i.e.,), the integral in Equation (4) can be expressed in terms of intrinsic Riemannian geometry of the manifold, namely, in terms of the Pfaffian. The Pfafian is a -differential form, whose construction utilizes the curvature tensor on the manifold (see [15] ). So, when,

Given a boundary generic field on, we introduce a sequence of basic degree-type invariants

which are intimately linked, via the Morse Formula (2), to the invariants.

We use a Riemannian metric on to produce the orthogonal projection of the field on the tangent subspace.

Let be the bundle of unit -spheres associated with the tangent bundle of the manifold. We denote by the restriction of the bundle to the subspace.

For each k, consider two fields, the inward normal field to in and, as sections of the sphere bundle (remember, is tangent to along so that along!). Assume that the sections ν and are transversal in the space. This transversality can be achieved by a perturbation of (equivalently, by a perturbation of the metric), supported in the vicinity of the singularity locus. Indeed, the intersections occur where the field is positively proportional to, that is, where. The later locus is exactly the locus. The perturbation does not affect the stratification. Assuming the transversality of the intersection, the locus is zero- dimensional.

We define the integer as the algebraic intersection number of two -cycles, and, in the ambient manifold of dimension.

Lemma 2.2 For a boundary generic field on a Riemannian manifold, the following formula holds:

Proof. We already have noticed that the intersection set projects bijectively under the map onto the locus, where the component of vanishes and points in- ward of. It takes more work to see that the sign attached to the transversal intersection point is, where is the index (the localized degree) of the field in the vicinity of its singularity. Thus. By the Morse Formula (2), the claim of the lemma follows.

Corollary 2.3 The integer depends only on the singular locus of and on the local indices of its points.

Question 2.1. How to compute in the terms of Riemannian geometry and in the spirit of Theorem 2.4 and Corollary 2.2?

For a boundary generic field and a fixed metric on, each manifold comes equipped with a preferred normal framing of the normal bundle: just consider the unitary inward normal field of in, then the unitary inward normal field of in, being restricted to, then the unitary inward normal field of in, being restricted to, and so on.

Via the Pontryagin construction [16] , this framing generates a continuous map. Its homotopy class is an element of the cohomotopy set. If, then we define to be the trivial map that takes to the base point in.

Unfortunately, as we will see soon,! However, when, each of the two loci is a closed manifold. Then we can apply the Pontryagin construction only to, say, to get a map. This application leads directly to the following proposition.

Corollary 2.4 Consider a boundary generic vector field such that and a metric, defined in the vicinity of in. Then these data give rise to continuous map.

The homotopy class is independent of the choice of and a homotopy of

within the open subspace of, defined by the constraint.

In particular, when, we get an element

and when, an element

If, we can interpret also as an element of the homotopy group.

The elements and have another classical interpretation as elements of oriented framed

cobordism set. In fact, the pair defines the trivial element in. In

contrast, if, then the bordism class may be nontrivial.

Let us recall the definition of framed cobordisms (for example, see [17] ). Let be oriented closed smooth -dimensional submanifolds of a compact -manifold, whose normal bundles and are equipped with framings and, respectively.

We say that two pairs and define the same element in, if there is a compact -dimensional oriented submanifold whose normal bundle admits a framing so that:

1),

2) the restriction of to coincides with, and the restriction of to coincides with.

Then the Pontryagin construction establishes a bijection, where. If both sets admit a structure of abelian groups and the bijection becomes a group isomorphism.

Now we are in position to explain why. Consider the obvious embedding

We can isotop in to a regular embedding

such that:

1), and

2) the inward normal field is parallel to the factor in the product.

Note that for, all the normal fields are preserved under the imbedding. So,

for any, the normal framing of in extends to a normal framing

of in. Therefore as an element of the framed bordisms of. As a

result, when, we get in (equivalently, in).

Let be a smooth compact -manifold with boundary. A boundary generic field (see Definition 2.1) gives rise to two stratifications (1).

We are going to investigate how the stratification changes as a result of deforming the vector field.

Lemma 3.1 Let be a closed submanifold of a manifold and a closed manifold. Consider a family of maps such that each is transversal to. All the manifolds, maps, and families of maps are assumed to be smooth.

Then all the submanifolds are isotopic in. In particular, the intersections and are diffeomorphic.

Proof. Let be the map defined by the family. Thanks to the transversality hypothesis, is transversal to and is a submanifold of whose boundary is

Let be a vector field on, normal to each codimension 1 submanifold in. In the construction of, we evidently rely on the property of each being transversal to. Since and, each w-trajectory that originates at a point of must reach in finite time. Therefore, employing the w-flow, is diffeomorphic to, and the -image of that product structure in defines a smooth isotopy between and in. This isotopy extends to an ambient isotopy of itself [18] .

Note that these arguments fail in general if ether M or N have boundaries. However, under additional assumptions (such asbeing t-independent and), the relative versions of the lemma are valid.

Theorem 3.1 The diffeomorphism type of each stratum is constant within each path-connected component of the space of boundary generic fields.

Proof. If two generic fields, and, are connected by a continuous path, then they can be connected by a path such that the dependence of the field on is smooth. The argument is based on the property of generic fields to form an open set in the space of all fields (Theorem 2.1), the smooth partition of unity technique (which utilizes the compactness of manifold), and the standard techniques of approximating continuos functions with the smooth ones.

Thus it suffices to consider a smooth 1-parameter family of vector fields, connecting to. Since any generic field, viewed as a section of the vector bundle, is transversal its zero section, we may apply Lemma 3.1 (with, being the zero section of, , and) to conclude that all the submanifolds are isotopic in.

Since each divides into a pair of complementary domains, and, and since their polarity ±is determined by the inward/outward direction of, which changes continuously with, the ambient isotopy of (which takes to) must take to. The isotopy extends to an isotopy.

A similar argument applies to lower strata. Indeed, with the isotopy that takes to in place, consider the two sections, and, of the bundle

, both sections being transversal to the zero section of. Applying again Lemma 3.1, we conclude that the loci and are isotopic in (recall that these

loci are exactly the transversal intersections of two sections and of with its zero section).

Again, an isotopy that takes to must take to. The isotopy extends to an isotopy which preserves the pair. So, the pairs and are diffeomorphic via the composite isotopy.

This reasoning can be recycled to prove that all the pairs and are diffeomorphic via a single isotopy of. This argument is carried explicitly in the proof of Theorem 3.4 from [8] .

Corollary 3.1 Let be a -dimensional compact smooth manifold with boundary.

Within each path-connected component of the space of generic fields, the numbers, as

well as the numbers, are constant.

Proof. The claim follows instantly from Theorem 3.1 and Lemma 2.2.

For a manifold with nonempty boundary, by deforming any given function and its gradient-like field, we can expel the isolated -singularities from. This can be achieved by the appropriate “finger moves” which originate at points of the boundary and engulf the isolated singularities of. The result of these manipulations leads to

Lemma 3.2 Any connected -manifold with a non-empty boundary admits a Morse function with no critical points in the interior of and such that is a Morse function. Such functions form an open nonempty set in the space of all smooth functions on.

As a result, the gradient-like vector fields on form an open nonempty set in the space of all vector fields on.

Proof. Let us sketch the main idea of the argument. Start with a Morse function. Connect each critical point in the interior of by a smooth path to a point on the boundary in such a way that a system of non-intersecting paths is generated. Then delete from small regular neighborhoods of those paths (“dig a system of dead-end tunnels”) and restrict to the remaining portion of. Smoothen the entrances of the tunnels so that the boundary of will be a smooth manifold which is diffeomorphic to. We got a nonsingular function on. A slight perturbation of on will not introduce critical points in the interior of and will deliver a Morse function on its boundary. Indeed, recall that the sets of Morse functions on and are open and dense in the spaces and of all smooth functions, respectively (for example, see [9] ). Of course is an open condition imposed on a vector field on a compact manifold. On the other hand, if, then any field, sufficiently close to, will have the property. The previous arguments show that the set of gradient-like non-vanishing fields is nonempty. So it is an open nonempty subspace in the space of all all vector fields on.

Eliminating isolated critical points of a given function on a manifold with boundary is not “a free lunch”: the elimination introduces new critical points of the restricted function. This is a persistent theme throughout our program:

Expelling critical points of gradient flows from a manifold leaves crucial residual geometry on its boundary.

This boundary-confined geometry allows for a reconstruction of the topology of.

Ideas like these will be developed in the future papers from this series. Meanwhile, the following lemma gives a taste of things to come.

Lemma 3.3 Let be a Morse function with no local extrema in the interior of a -manifold. Then an elimination by a finger move5 of each -critical point of the Morse index results in the introduction of new critical points of positive type and new critical points of negative type for the modified function.

Proof. Let be a Morse singularity of in the interior of. Denote by a sphere which bounds a

small disk centered on and such that is a Morse function. Without loss of generality, we can

assume that, in the Morse coordinates, is given by, while with all the being distinct. Then has only Morse-type singularities at the points where the coordinate axes pierce the sphere. With respect to the pair, these points come in two flavors: positive and negative. The two types are separated by the hypersurface of the cone

In the vicinity of, the intersection is exactly the locus

so that the -gradient field (tangent to along) is transversal to, the product of two spheres. Therefore, in the vicinity of,!

The function has exactly critical points of the positive type and exactly

critical points of the negative type. We shall denote these sets by and the two domains in which divides—by.

Let be a local maximum of. Note that it is possible to connect to a non-singular (for) point by a smooth path along which f is increasing. Indeed, any non-extendable path such that either approaches a critical point or reaches the boundary. By a small perturbation, we can insure that avoids all the (hyperbolic) critical points in the interior of X (by the hypothesis, f has no local maxima/minima in the interior of X). Thus can be extended until it reaches the boundary at a point.

Drilling a narrow tunnel, diffeomorphic to the product, along does not change the topology of X; the function retains almost the same list of singularities at the boundary as the function has: more accurately, the local maximum at disappears in and a negative critical point of index 1 of appears near the -end of the tunnel. Thus we have modified f and have eliminated the critical point in the interior of X at the cost of introducing on the boundary critical points of positive type and critical points of negative type.