MORSE THEORY and STOKES’ THEOREM

by

F. Reese Harvey and H. Blaine Lawson, Jr.*.

Abstract. We present a new, intrinsic approach to Morse Theory which has interesting applications in geometry. We show that a Morse function f on a manifold determines a submanifold T of the product X × X, and that (in the sense that Stokes theorem is valid) T has boundary consisting of the diagonal Δ ⊂ X × X and a sum

P = ∑_{p∈Cr(f)} U_{p} × S_{p}

where S_{p} and U_{p} are the stable and unstable manifolds at the critical point p. In the language of currents,

(Stokes Theorem)  ∂T = Δ − P.

This current (or kernel) equation on X × X is equivalent to an operator equation

(Chain Homotopy)  d ∘ T + T ∘ d = I − P,

where P is a chain map onto the finite complex of currents S_{f} spanned by (integration over) the stable manifolds of f. The operator P can be defined on an exterior form α by

P(α) = lim_{t→∞} φ_{t}^{∗}α

where φ_{t} is a gradient flow for f. The de Rham differential in the complex S_{f} is easily computed in terms of the flow lines. The chain homotopy equation also holds on certain integral chain complexes. Poincaré duality over follows from time-reversal in our operator equations. The method has many generalizations and applications. Residue theorems are established for functions with critical manifolds of higher dimension. The methods apply immediately to equivariant cohomology. Cup product formulas and a Lefschetztype theorem are proved for the Thom-Smale Complex. Other applications include a new proof of the Carrell-Lieberman Theorem and a proof of a local version of the MacPherson Formula for characteristic classes and bundle maps.

*The research of both authors was partially supported by the NSF

Table of Contents

0. Introduction

1. Finite Volume Flows

2. Morse-Stokes Gradients; Axioms

3. The Operator Equations

4. -complexes, /p -complexes

5. Poincaré Duality

6. Critical Submanifolds of Higher Dimension

7. The Relative Case

8. Holomorphic Flows and the Carrell-Liebermann-Sommese Theorem 9. Singularities and Characteristic Classes

10. Equivariant Morse Theory

11. Flat Bundles and Local Coefficients

12. Products

13. A Lefschetz Theorem for the Thom-Smale Complex

14. The Genericity Theorem Appendix: Currents and the Kernel Calculus

§0. Introduction. We present in this paper a new approach to Morse Theory which is stronger than the classical theory and has some interesting applications. It leads to formulas relating characteristic forms and singularities, and it unifies a body of results on holomorphic actions. It applies directly to the equivariant case. It also has the virtue of fitting neatly into the modern theory of invariants arising from topological quantum field theory.

This work resulted from addressing the following.

Question: Consider a flow φ_{t} : X → X generated by a smooth vector field on a compact manifold X. Under what cicumstances does the limit

α_{∞} ≡ lim_{t→∞} φ_{t}^{∗}α

exist for a given smooth differential form α on X?

We do not demand that α_{∞} be smooth. Even so, one would expect the answer to be “rarely, if ever”. However, we shall prove that for generic gradient dynamical systems, this limit does exist and has a beautiful, simple structure.

In fact we shall show that setting

(0.1)  P(α) =^{def} lim_{t→∞} φ^{∗}α

defines a continuous operator of degree 0

P : E^{∗}(X) ⟶ D′^{∗}(X)

from smooth forms to generalized forms, i.e., currents. Furthermore, we shall show that this operator is chain homotopic to the inclusion I : E^{∗}(X) ↪ D′^{∗}(X). That is, using the flow we shall construct a continuous operator

T : E^{∗}(X) ⟶ D′^{∗}(X)

of degree -1 such that

(0.2)  d ∘ T + T ∘ d = I − P.

By de Rham [deR], I induces an isomorphism in cohomology. Hence so does P.

The existence of P and T satisfying (0.1) and (0.2) is established for any flow of finite volume. This concept, which is central to our paper, is introduced in §2. A flow φ_{t} is said to have finite volume if the graph of the relation x ≺^{o} y, defined by the forward motion of the flow, has finite (n + 1)-dimensional volume in X × X, where n = dim(X). Any flow whose space-time graph

T_{φ} ≡ \{(t, φ_{\frac{1}{t}} (x), x) : 0 < t ≤ 1 and x ∈ X\} ⊂ R × X × X

is of finite volume has this property.

Consider now a Morse function f : X → R with (finite) critical set Cr(f). Suppose there is a riemannian metric on X for which the gradient flow φ_{t} of f has the following properties:

(1) φ_{t} is of finite volume.

(2) The stable and unstable manifolds, S_{p}, U_{p} for p ∈ Cr(f), are of finite volume in X.

(3) p ≺ q ⇒ λ_{p} < λ_{q} for all p, q ∈ Cr(f), where λ_{p} denotes the index of p and where

≺ is the closure of the relation ≺^{o}. Note that p ≺ q means there is a piecewise flow line connecting p in forward time to q.

We shall prove in §14 that such metrics always exist. They can be taken to be canonically flat in some neighborhood of Cr(f) (see §2) thereby making the gradient flow particularly tractible. In fact properties (1)–(3) are shown to be generic in the space of such metrics.

Under these assumptions the operator P is shown to have the following simple form:

(0.3)  P(α) = ∑_{p∈Cr(f)} r_{p}(α)[S_{p}]

for all α ∈ E^{∗}(X), where

r_{p}(α) = \{ \begin{matrix}∫_{U_{p}} α & if deg α = λ_{p} \\ 0 & otherwise \end{matrix}

and where [S_{p}] denotes the current defined by integration over S_{p}. Note that the image of P is the finite dimensional vector subspace

S_{f} = span_{R}{[S_{p}]}_{p∈Cr(f)}.

From (0.2) we see that S_{f} is d-invariant, i.e., that (S_{f}, d) is a complex, and we show that the inclusion S_{f} ⊂ D′^{∗}(X) induces an isomorphism

H^{∗} (S_{f}) ≅ H_{de Rham}^{∗}(X)

This immediately yields the classical strong Morse inequalities.

The exterior derivative restricted to S_{f} has the form

d[S_{p}] = ∑_{q∈Cr(f)} n_{pq}[S_{q}].

By a result of Federer the constants n_{pq} are integers, and so

S_{f}^{}= span_{}{[S_{p}]}_{p∈Cr(f)}

is a finite-rank subcomplex of the integral currents I_{∗}(X) on X. Using (0.2) we show that the inclusion S_{f}^{}⊂ I_{∗}(X) induces an isomorphism

H^{∗} (S_{f}^{}) ≅ H^{∗}(X; ).

In the special case of Morse-Smale flows the integers n_{pq} can be computed by counting flow lines from p to q. This follows from Stokes’ Theorem (cf. [La] and §4).

Poincaré duality (over ) is now directly deduced from time-reversal in the flow (§5). The analogous relative theorems for a Morse exhaustion function on a non-compact

manifold, including Alexander-Lefschetz duality, are proved in §7.

Our method of proof involves converting the operator equation (0.2) to a kernel equation

(0.4)  ∂T = [Δ] − P

on X × X, where Δ denotes the diagonal. There is a general correspondence between operators K : E^{∗}(X) → D′^{∗}(X) of degree ℓ and currents K of dimension n − ℓ on X × X ([HP], See Appendix A.) Under this transformation: I corresponds to [Δ], the pull-back of forms by φ_{t} corresponds to the graph of φ_{t}, and the chain homotopy d ∘ K + K ∘ d corresponds to the current boundary ∂K. Thus equation (0.4) carries back directly to equation (0.2).

The current T in (0.4) is simply defined by integration over the graph of the relation ≺^{o}. If the space-time graph T_{φ} has finite volume, then T = pr_{∗} T_{φ} where pr : R×X ×X → X ×X is the projection. Our finite-volume assumption on T implies that T is a rectifiable current and therefore that its current boundary is flat in the sense of Federer [F]. Applying the Federer Support Lemma, we conclude that

(0.5)  P = ∑_{p∈Cr(f)}[U_{p}] × [S_{p}].

Thus, T provides a homology between the diagonal and the sum of Künneth currents in X × X given by products of the unstable and stable manifolds of the flow. In other words, our Morse function gives a canonical chain approximation to the diagonal together with an explicit homology. This “transgression current” T = T_{f} plays a role in defining more subtle invariants of manifolds and knots.

The entire procedure outlined above can be applied to functions with non-degenerate critical manifolds, i.e., functions of Bott-type. Suppose f is such a function and F_{j}, j = 1, ..., ν are the connected components of the critical set. In this case the kernel P in (0.4) is written as a sum of fibre-products

(0.6)  P = ∑_{j=1}^{ν}[U_{j} ×_{F_{j}} S_{j}].

of the stable and unstable manifolds of the flow. Here the subcomplex image(P) is not finite-dimensional. However, for each smooth form α we have

P(α) = ∑_{j=1}^{ν} Res_{j}(α)[S_{j}]

where Res_{j}(α) is a smooth, integrable residue form on the manifold S_{j} computed directly in terms of α.

This approach works nicely in the case of holomorphic ^{∗}-actions with fixed-points on Kähler manifolds. Here there is an underlying function of Bott-type. One finds a complex analogue of the current T and replaces equation (0.4) with a ∂\overline{∂}-equation. General results of Sommese imply that T and all of the stable and unstable manifolds of the flow are subvarieties of finite-volume. One retrieves, in particular, classical results of Bialynicki-Birula [BB] and of Carrell-Lieberman-Sommese [CL],[CS]. The approach also fits directly into MacPherson’s Grassmann graph construction and Gillet-Soulé’s construction of transgression classes appearing in the refined Riemann-Roch Theorem [GS].

The arguments apply literally without change to the case of equivariant cohomology. It yields rapid calculations in certain cases and has been used by J. Latschev to derive a spectral sequence associated to functions in the equivariant case.

The method can be applied to derive an equation of forms and currents which relates the singularities of a smooth bundle map A : E → F to characteristic forms of E and F. For normal bundle maps, i.e., those which are transversal to the universal singularity sets in Hom(E, F), this yields local proof of a classical topological formula of MacPherson [Mac_{1,2}]. However, the method also yields formulas in many other cases, such as generic direct sum and tensor product mappings [HL_{3,4}]. This work, which is discussed briefly in §9, began in [HL_{1}] and led to the Morse Theory presented here.

Much has been written about assigning topological invariants of manifolds and knots to “Feynman graphs”. Primary invariants of this type, such as those discussed in [B] and [BC], can be constructed using operators P. The invariants of Kontsevich and Vasseliev (cf. [K], [BT]) involve the currents T.

It should be remarked that while the Morse Theory due to Ed Witten [W] involves the de Rham complex, it is distinctly different from the approach presented here. Witten considers the conjugates d_{t} of exterior differentiation d by the function e^{−tf} for t ≥ 0 and examines the asymptotics of the associated Hodge laplacians. Elliptic operators do not enter the story in our approach. Moreover, a crucial simplifying component of our approach (namely, the calculus of [HP] for the operator d) is not available for other operators such as the Laplacian or d_{t}. It would be interesting to find a more direct connection between the two theories.

The idea of using the stable manifolds of a generic gradient flow to give a cell structure to a manifold goes back to R. Thom [T] (See also [S] and [R].) F. Laudenbach was the first to consider the stable manifolds as de Rham currents [La]. He studied Morse-Smale flows and computed the boundary operator in the Thom-Smale complex by using Stokes’ Theorem as we do here. The concept of finite volume flows and its use in Morse Theory appeared in [HL_{5}]. This article expands certain themes in that paper.

The authors would like to thank Janko Latschev for many useful comments during the preparation of this work.

The second author would like to express gratitude to the Institut des Hautes Études Scientifiques for partial support during the preparation of this work.

§1. Finite Volume Flows. Let X be a compact smooth manifold of dimension n, and let φ_{t} : X → X be the flow generated by a smooth vector field V on X. Consider the operator P_{t} : E^{∗}(X) → E^{∗}(X) on the space of smooth differential forms which is given by pull-back

P_{t}(α) ≡ φ_{t}^{∗}(α).

We will exhibit a chain homotopy operator T_{t} : E^{k}(X) → E^{k−1}(X) satisfying

(A)  d ∘ T_{t} + T_{t} ∘ d = I − P_{t},

and show that under a “finite volume” condition it is possible to take the limit as t → ∞. We thereby obtain operators

(B)  T = lim_{t→∞} T_{t} and P = lim_{t→∞} P_{t}

satisfying:

(C)  d ∘ T + T ∘ d = I − P.

Now operator equations such as (A) and (C) are difficult to solve directly, but as we shall see in §3, they can be converted into current equations on the product manifold X ×X which are much more tractable. For example, equation (A) becomes the current equation

(A)  ∂T_{t} = [Δ] − P_{t}  on X × X

where Δ ⊂ X × X is the diagonal and P_{t} = [graph(φ_{t})] = {(φ_{t}(x), x) : x ∈ X} is the (reverse) graph of the diffeomorphism φ_{t}. The remaining conditions are that

(B)  T = lim_{t→∞} T_{t} and P ≡ lim_{t→∞} P_{t}

exist and satisfy

(C)  ∂T = [Δ] − P  on X × X.

Here T_{t}, T, and P are currents on X × X yet to be determined. Details of the above correspondence are discussed in §3. The remainder of this section is devoted to solving the second set of equations (A), (B), (C).

Equation (A) is particularly easy to solve. Consider the family of compact manifolds with boundary:

(1.1)  T_{t} ≡ {(s, φ_{s}(x), x) : 0 ≤ s ≤ t and x ∈ X}

contained in R × X × X. Obviously, T_{t} has the two boundary components {0} × Δ and {t} × P_{t}. Assume X is oriented (this condition will be dropped later) and orient T_{t} so that

(1.2)  ∂T_{t} = {0} × Δ − {t} × P_{t}.

Let pr : R × X × X → X × X denote projection and set

(1.3)  T_{t} ≡ (pr)_{∗}(T_{t}).

Since ∂ commutes with (pr)_{∗}, the push-forward of (1.2) by pr_{∗} gives equation (A). The current T_{t} can be equivalently defined by

(1.3’)  T_{t} = Φ_{∗}([0, t] × X)

where Φ : R × X → X × X is the smooth mapping given by Φ(s, x) = (φ_{s}(x), x). This mapping is an immersion exactly on the subset R × (X − Z(V)) where Z(V) = {x ∈ X : V (x) = 0}. Thus if we fix a riemannian metric g on X, then Φ^{∗}(g × g) is a symmetric positive semi-definite tensor whose associated volume is > 0 exactly on the subset R × (X − Z(V)).

This brings us to one of the central concepts of the paper.

Definition 1.1. A flow φ_{t} on X is called a finite volume flow if R^{+} × (X − Z(V)) has finite volume with respect to the metric induced by the immersion Φ. (This concept is independent of the choice of riemannian metric on X.)

Theorem 1.2. Let φ_{t} be a finite volume flow on a compact manifold X. Then both the limits

(B)  P ≡ lim_{t→∞}[graph φ_{t}]  and  T ≡ lim_{t→∞} T_{t}

exist as currents, and by taking the boundary of T we obtain the equation of currents

(C)  ∂T = [Δ] − P  on X × X.

relating P to the diagonal Δ in X × X.

Proof. Since φ_{t} is a finite-volume flow, the current T ≡ Φ_{∗}((0, ∞)×X) is the limit in the mass norm of the currents T_{t} = Φ_{∗}((0, t) × X) as t → ∞. The continuity of the boundary operator and equation (A) imply the existence of lim_{t→∞} P_{t} and also establish equation (C).

Remark 1.3. Since (φ_{t}(x), x) = (y, φ_{−t}(y)) if y = φ_{t}(x), it follows that

T^{∗} = Φ_{∗}((−∞, 0) × X)

is also a well-defined current for a finite-volume flow. It corresponds to the push-forward of T under the flip (y, x) ↦ (x, y) on X × X.

Remark 1.4. The immersion Φ : R × (X − Z(V)) → X × X is an embedding outside the subset R × P er(V) where

P er(V) = {x ∈ X : φ_{t}(x) = x for some t > 0 }

are the non-trivial periodic points of the flow. Thus, if P er(V) has measure zero, then T_{t} is given by integration over the embedded finite-volume submanifold Φ(R_{t}), where R_{t} = (0, t) × (X − Z(V) ∪ P er(V)). If furthermore the flow has finite volume, then T is given by integration over the embedded, finite-volume submanifold Φ(R_{∞}).

There is evidence that any flow with periodic points cannot have finite volume. Now a gradient flow never has periodic points, and many such flows are of finite volume (§14.). However, finite-volume flows are more general than gradient flows. For a first example, note that any flow with fixed points on S^{1} has finite volume.

Remark 1.5. If we define a relation on X × X by setting x ≺^{o} y if y = φ_{t}(x) for some 0 ≤ t < ∞, then T is just the (reversed) graph of this relation. This relation is always transitive and reflexive, and it is antisymmetric if and only if φ_{t} has no periodic orbits (i.e., ≺^{o} is a partial ordering precisely when φ_{t} has no periodic orbits).

Remark 1.6. A standard method for showing that a given flow is finite volume can be outlined as follows. Pick a coordinate change t ↦ ρ which sends +∞ to 0 and [t_{0}, ∞] to [0, ρ_{0}]. Then show that

(1.4)  T ≡ {(ρ, φ_{t(ρ)}(x), x) : 0 < ρ < ρ_{0}} has finite volume in R × X × X.

Pushing forward to X × X then yields the current T with finite mass.

Perhaps the most natural such coordinate change is r = 1/t. Another natural choice (if the flow is considered multiplicatively) is s = e^{−t}. Of course finite volume in the r coordinate insures finite volume in the s coordinate since r ↦ s = e^{−1/r} is a C^{∞}-map.

Many interesting flows can be seen to be finite volume as follows.

Proposition 1.7. If X is analytic and T ⊂ R × X × X is contained in a real analytic subvariety of dimension n + 1, then φ_{t} is a finite volume flow.

Proof. The manifold points of a real analytic subvariety have (locally) finite volume.

A flow need not be a gradient flow to be a finite volume flow.

Example 1.8. (The standard degenerate flow on S^{n}) Consider the translational flow φ_{t}(y) = y + tu on R^{n} where u ∈ R^{n} is a unit vector. We can identify R^{n} with S^{n} − {∞} so that φ_{t} extends to S^{n} as a finite volume flow. To do this choose coordinates x = y/|y|^{2} on R^{n} ≅ S^{n} − {0}. Then

(1.5)  φ_{t}(x) = \frac{|x|^{2}}{|x + t|x|^{2}u|^{2}} (x + t|x|^{2}u)

(The vector field V = φ_{t}^{·}|_{t=0} is given by V (y) = u on R^{n} = S^{n} − {∞}, and by V (x) = |x|^{2}u − 2⟨x, u⟩x on R^{n} = S^{n} − {0}.)

The flow φ_{t} is finite volume flow on S^{n}. To see this let r = 1/t and note that

T = {(r, z, x) : z = φ_{1/r}(x), 0 < r < ∞ and x ∈ R^{n} = S^{n} − {0}}

is defined by

z|rx + |x|^{2}u|^{2} = |x|^{2}r(rx + |x|^{2}u)

so that Proposition 1.7 is applicable. Note that ∞ (x = 0) is the only zero of V. Although V is not a gradient vector field, it is the limit of gradient vector fields.

Our next problem is to explicitly compute the current P and its associated operator P under additional assumptions on the flow. We shall show that when V is a “good” gradient vector field for a Morse function, the operator P is projection onto the finite complex of currents spanned by the stable manifolds of the flow. Furthermore, for any given Morse function the “good” gradients are generic (cf. §14).

§2. Morse-Stokes Gradients; Axioms. Let f ∈ C^{∞}(X) be a Morse function on a compact n-manifold X, and let Cr(f) denote the (finite) set of critical points of f. Recall that f is a Morse function if its Hessian at each critical point is non-degenerate. The standard Morse Lemma asserts that in a neighborhood of each p ∈ Cr(f) of index λ, there exist canonical local coordinates (u_{1}, ..., u_{λ}, v_{1}, ..., v_{n−λ}) for |u| < r, |v| < r with (u(p), v(p)) = (0, 0) such that

(2.1)  f(u, v) = f(p) − |u|^{2} + |v|^{2}.

Choose a riemannian metric on X and let φ_{t} denote the flow associated to ∇f. Suppose that this metric has the form |du|^{2} +|dv|^{2} in some canonical coordinate system (u, v) about each p ∈ Cr(f). Then in these coordinates the gradient flow is given by

φ_{t}(u, v) = (e^{−t}u, e^{t}v)

Metrics with this property will be called canonically flat near Cr(f) and the flow will be called f-tame.

Now to each p ∈ Cr(f) are associated the stable and unstable manifolds of the flow, defined respectively by

(2.2)  S_{p} = {x ∈ X : lim_{t→∞} φ_{t}(x) = p}  and  U_{p} = {x ∈ X : lim_{t→−∞} φ_{t}(x) = p}.

For coordinates (u, v) at p, chosen as above, we consider the disks

S_{p}(ϵ) = {(u, 0) : |u| < ϵ}  and  U_{p}(ϵ) = {(0, v) : |v| < ϵ}

and observe that

(2.3)  S_{p} = ⋃_{−∞<t<0} φ_{t} (S_{p}(ϵ))  and  U_{p} = ⋃_{0<t<+∞} φ_{t} (U_{p}(ϵ)) .

Hence, S_{p} and U_{p} are submanifolds (but not closed subsets) of X with

(2.4)  dimS_{p} = λ_{p}  and  dimU_{p} = n − λ_{p}

where λ_{p} is the index of the critical point p. For each p we choose an orientation on U_{p}. This gives an orientation on the normal bundle of S_{p} via the splitting T_{p}X = T_{p}U_{p} ⊕T_{p}S_{p}. We thereby obtain a kernel [U_{p}] × [S_{p}] on X × X (See A.9.).

The flow φ_{t} induces a partial ordering on X by setting x ≺ y if there is a continuous path consisting of a finite number of foward-time orbits, which begins with x and ends with y. We shall see in the proof of Lemma 2.4 that this is the closure of the partial ordering of Remark 1.5.

Definition 2.1. The gradient flow of a Morse function f on a riemannian manifold X is called Morse-Stokes if it is f-tame and:

(i) It is a finite-volume flow.

(ii) Each of the stable and unstable manifolds S_{p} and U_{p} for p ∈ Cr(f) has finite volume. (iii) If p ≺ q and p ≠ q then λ_{p} < λ_{q}, for all p, q ∈ Cr(f).

Remark 2.2. In Section 14 we shall prove that if the gradient flow of f is Morse-Smale, then it is Morse-Stokes. Furthermore, for any Morse function f on a compact manifold X there exist riemannnian metrics on X for which the gradient flow of f is Morse-Stokes. In fact these metrics are dense in the set of all metrics which are canonically flat near Cr(f).

Theorem 2.3. Let f ∈ C^{∞}(X) be a Morse function on a compact riemannian manifold whose gradient flow is Morse-Stokes. Then there is an equation of integral currents

(2.5)  ∂T = [Δ] − P

on X × X, where T is an embedded submanifold of finite volume, Δ ⊂ X × X is the diagonal, and where

(2.6)  P = ∑_{p∈Cr(f)}[U_{p}] × [S_{p}]

Proof. For each critical point p ∈ Cr(f) we define

Ũ_{p} =^{def} ⋃_{p≺q} U_{q} = {x ∈ X : p ≺ x}.

Lemma 2.4. Let f ∈ C^{∞}(X) be any Morse function whose gradient flow is of finite volume, and let spt P ⊂ X × X denote the support of the current P defined in Theorem 1.2. Then

spt P ⊂ ⋃_{p∈Cr(f)} Ũ_{p} × S_{p}.

Proof. Since P = lim_{t→∞} P_{t} and P_{t} = {(φ_{t}(x), x) : x ∈ X}, it is clear that (y, x) ∈ spt P only if there exist sequences x_{i} → x in X and s_{i} → ∞ in such that y_{i} ≡ φ_{s_{i}}(x_{i}) → y. Let L(x_{i}, y_{i}) denote the oriented flow line from x_{i} to y_{i}. Since the lengths of these lines are bounded, an elementary compactness argument shows that a subsequence converges to a piecewise flow line L(x, y) from x to y. By the continuity of the boundary operator on currents, ∂L(x, y) = [y] − [x]. Finally, since s_{i} → ∞ there must be at least one critical point on L(x, y), and we define p = lim_{s→∞} φ_{s}(x).

Consider now the subset

Σ ≡ ⋃_{\begin{matrix}p≺q \\ p≠q \end{matrix}} U_{q} × S_{p} ⊂ X × X.

and set Σ′ = Σ ∪ {(p, p) : p ∈ Cr(f)}. Note that from (2.4) and Axiom (iv) in Definition 2.1 that

(2.7)  Σ′ is a disjoint union of submanifolds of dimension ≤ n − 1.

Lemma 2.5. Let f ∈ C^{∞}(X) be a Morse function and φ_{t} an f-tame gradient flow. Consider the embedded submanifold

T = {(y, x) : x ∉ Cr(f), and y = φ_{t}(x) for some 0 < t < ∞}.

of X × X − Σ′. Then the closure \overline{T} of T is a proper C^{∞}-submanfold with boundary

∂\overline{T} = Δ − ∑_{p∈Cr(f)} U_{p} × S_{p}

in X × X − Σ′

Proof. We first show that it will suffice to prove the assertion in a neighborhood of (p, p) ∈ X × X for p ∈ Cr(f). Consider (ȳ, x̄) ∈ \overline{T} − T. If (ȳ, x̄) ∉ Σ′, then the proof of Lemma 2.4 shows that either (ȳ, x̄) ∈ Δ or (ȳ, x̄) ∈ U_{p} × S_{p} for some p ∈ Cr(f). Near points (x̄, x̄) ∈ Δ, x̄ ∉ Cr(f), one easily checks that \overline{T} is a submanifold with boundary Δ. If (ȳ, x̄) ∈ U_{p} × S_{p}, then for sufficiently large s > 0, the diffeomorphism ψ_{s}(y, x) ≡ (φ_{−s}(y), φ_{s}(x)) will map (ȳ, x̄) into any given neighborhood of (p, p). Note that ψ_{s} leaves the subset U_{p} × S_{p} invariant, and that ψ_{s}^{−1} maps T into T. Hence, if ∂\overline{T} = Δ − U_{p} × S_{p}, in a neighborhood of (p, p), then ∂\overline{T} = −U_{p} × S_{p} near (ȳ, x̄).

Now in a neighborhood O of (p, p) we may choose coordinates as in (2.1) so that T consists of points (y, x) = (ū, v̄, u, v) with ū = e^{−t}u and v̄ = e^{t}v for some 0 < t < ∞. Consequently, in O the set \overline{T} is given by the equations

ū = su and v = sv̄ for some 0 ≤ s ≤ 1.

This obviously defines a submanifold in O − {(p, p)} with boundary consisting of Δ and the set {ū = 0, v = 0} ≅ (U_{p} × S_{p}) ∩ O.

Lemma 2.5 has the following immediate consequence

(2.8)  spt {P − ∑_{p∈Cr(f)} U_{p} × S_{p}} ⊂ Σ′

We now apply the following important result of Federer.

Proposition 2.6. ([F, 4.1.20]) Let S be a flat current of dimension k in ^{n}. If the Hausdorff k-measure of the support of S is zero, then S = 0.

Recall that if S is a current defined by integration over an oriented submanifold of locally finite volume, then both S and dS are flat. (In fact for such currents having the further property that dS is supported in a submanifold N with dim(N) < dim(dS), Proposition 2.6 has an elementary proof [F, 4.1.15].) Combining Proposition 2.6 with (2.7) and (2.8) proves (2.6) and completes the proof of Theorem 2.3.

Related to Proposition 2.6 is the following result of Federer ([F, 4.1.15], [F, 4.2.16(2)]).

Proposition 2.7. Let S be a flat current of dimension k with dS = 0 defined in a convex open subset U of ^{n}. If spt (S) ⊂ U ∩ ^{k}, then S = c[U ∩ ^{k}] for some c ∈ . Moreover, if S = dT where T is locally rectifiable, then c is an integer.

Propositions 2.6 and 2.7 are philosophically central to our paper.

Remark 2.8. Using the flow given in Example 1.8 and the methods above, one constructs an (n + 1)-current T on S^{n} × S^{n} with the property that

∂T = S^{n} × {∗} + {∗} × S^{n}.

This is a singular analogue of the form used by Bott and Taubes to study knot invariants [BT].

§3. The Operator Equations. We now explain how to pass from the current equations (A), (B), (C) to the operator equations (A), (B), (C) discussed in §1. This kernel calculus was introduced in [HP]. There is a brief appendix on currents with definitions and notation at the end of the paper. The discussion here includes the non-orientable case.

Let X and Y be compact manifolds, and let π_{Y} and π_{X} denote projection of Y ×X onto Y and X respectively. Then each partially twisted current (or kernel) K ∈ D^{′∗}(Y × X̃), determines an operator K : E^{∗}(Y) → D^{′∗}(X) by the formula

(3.1)  K(α) = (π_{X})_{∗}(K ∧ π_{Y}^{∗} α).

The formula (3.1) can be rewritten as

K(α)(β) = K(π_{Y}^{∗} α ∧ π_{X}^{∗}β)

where β ∈ Ẽ^{∗}(X) is a twisted form on X. This definition is motivated by the following example.

Example 3.1. Suppose φ : X → Y is a smooth map, and let P_{φ}(α) ≡ φ^{∗}(α) be the pull-back operator on on differential forms. Now a differential form φ^{∗}α defines a current by setting

(3.2)  (φ^{∗}α)(β) ≡ ∫_{X}(φ^{∗}α) ∧ β

for all twisted forms β ∈ Ẽ^{∗}(X). Consider the graph of φ given by graph φ = {(φ(x), x) : x ∈ X} ⊂ Y ×X. Integration over the graph of φ determines a kernel or (partially twisted) current on Y × X, by this same formula (3.2). Namely,

(3.3)  [graph φ](π_{Y}^{∗} α ∧ π_{X}^{∗}β) ≡ ∫_{X} φ^{∗}α ∧ β.

The left hand side equals

([graph φ] ∧ π_{Y}^{∗} α)(π_{X}^{∗}β) = ((π_{X})_{∗}([graph φ] ∧ π_{Y}^{∗} α))(β).

Therefore,

(3.4)  P_{φ}(α) = (π_{X})_{∗}([graph φ] ∧ π_{Y}^{∗} α).

To complete the transfer of the operator equations (A) and (C) to current equations on X × X, we need the following result.

Lemma 3.2. Suppose the operator K : E^{∗}(Y) → D^{′∗}(X) has kernel K ∈ D^{′∗}(Y × X̃). Suppose that K lowers degree by one, or equivalently, that deg(K) = dimY − 1. Then

(3.5)  The operator d ∘ K + K ∘ d has kernel ∂K.

Proof. The boundary operator ∂ is the dual of exterior differentiation. That is ∂K is defined by (∂K)(π_{Y}^{∗} α ∧ π_{X}^{∗}β) = K(d(π_{Y}^{∗} α ∧ π_{X}^{∗}β)). Also, by definition, (K(dα), β) ≡ K(π_{Y}^{∗} (dα) ∧ π_{X}^{∗}β), and

(d(K(α)), β) ≡ (−1)^{deg α}K(α)(dβ) = (−1)^{deg α}K(π_{Y}^{∗} (α) ∧ π_{X}^{∗}dβ),

since K(α) has degree equal to deg α − 1.

The results that we need are summarized in the following table.

Operators Kernels

I [Δ]

P_{t} ≡ φ_{t}^{∗}  P_{t} ≡ [graph φ_{t}]

K K

d ∘ K + K ∘ d  ∂K

Table 3.6

This completes the transfer of the operator equations (A), (B), (C) to the current equations (A), (B), (C). From Theorems 1.2 and 2.3 we immediately deduce the following.

Theorem 3.3. Let f ∈ C^{∞}(X) be a Morse function on a compact riemannian manifold X whose gradient flow φ_{t} is Morse-Stokes. Then for every differential form α ∈ E^{k}(X), 0 ≤ k ≤ n, one has

P(α) = lim_{t→∞} φ_{t}^{∗}α = ∑_{p∈Cr(f)} r_{p}(α)[S_{p}]

where the “residue” r_{p}(α) of α at p is defined by

r_{p}(α) = ∫_{U_{p}} α

if k = n − λ and 0 otherwise. Furthermore, there is an operator T of degree -1 on E^{∗}(X) with values in flat currents, such that

(3.7)  d ∘ T + T ∘ d = I − P

From Theorem 3.3, we see that P : E^{∗}(X) ⟶ D^{′∗}(X) maps onto the finite-dimensional subspace of currents

(3.8)  S_{f} =^{def} span_{}{[S_{p}]}_{p∈Cr(f)}

and that

(3.9)  P ∘ d = d ∘ P.

This together with (3.7) leads to the following.

Corollary 3.4. The subspace S_{f} is d-invariant and is therefore a subcomplex of D^{′∗}(X). The surjective linear map of cochain complexes

P̂ : E^{∗}(X) ⟶ S_{f}

defined by P induces an isomorphism

P̂ : H_{deR}^{∗}(X) ⟶^{≅} H^{∗}(S_{f}).

Proof. It follows immediately from Theorem 3.3 that P_{∗} = I_{∗} : H(E^{∗}(X)) → H(D^{′∗}(X)), and by [deR], I_{∗} is an isomorphism. Since i_{∗} ∘ P̂_{∗} = P_{∗}, where i : S_{f}^{∗} → D^{′∗}(X) denotes the inclusion, we see that P̂_{∗} is injective.

However, it does not follow formally that P̂_{∗} is surjective (as the example below shows). We must prove that every S ∈ S_{f}^{∗} with dS = 0 is of the form S = P(γ) where dγ = 0.

To do this we note the following consequence of 2.1(iii). For all p, q ∈ Cr(f) of the same index, one has that \overline{S}_{p} ∩ ∂U_{q} = ∅, \overline{S}_{p} ∩ \overline{U}_{q} = ∅ if p ≠ q, and \overline{S}_{p} ∩ \overline{U}_{q} = \{p\}. Now let S = ∑ n_{p}S_{p} be a k-cycle in S_{f}. Then by the above there is an open neighborhood N of spt (S) such that N ∩ ∂U_{q} = ∅ for all q of index k. Since by [deR] cohomology with compact supports can be computed by either smooth forms or all currents, there exists a current σ with compact support in N such that dσ = γ − S where γ is a smooth form. Now for each q of index k, U_{q} is a closed submanifold of N, and we can choose a family U_{ϵ} of smooth closed forms in N such that lim_{ϵ→0} U_{ϵ} = U_{q} in D′(N) and lim_{ϵ→0}(S_{p}, U_{ϵ}) = (S_{p}, U_{q}) = δ_{pq}[p] for all p ∈ Cr(f) of index k. This is accomplished via standard Thom form constructions for the normal bundle using canonical coordinates at q (e.g. [HL_{1}]). It follows that (γ − S, U_{q}) = lim_{ϵ→0}(γ − S, U_{ϵ}) = lim_{ϵ→0}(dσ, U_{ϵ}) = lim_{ϵ→0}(σ, dU_{ϵ}) = 0. We conclude that ∫_{U_{q}} γ = (S, U_{q}) = n_{q} for all q and so P(γ) = S as claimed.

Alternatively, one can choose an ϵ > 0 such that (spt S)_{ϵ} ∩ ∂U_{q} = ∅ for all q of index k, where (spt_{ϵ} S) denotes an ϵ-neighborhood of spt S. Applying the Federer smoothing homotopy [F, 4.1.18] gives a current H_{ϵ}(S) with spt H_{ϵ}(S) ⊂ (spt S)_{ϵ} and dH_{ϵ}(S) = γ − z where γ is a smooth closed form. Since (γ − S, U_{q}) = (dH_{ϵ}(S), U_{q}) = (H_{ϵ}(S), ∂U_{q}) = 0 we conclude as above that ∫_{U_{q}} γ = (S, U_{q}) = n_{q} for all q and therefore P(γ) = S.

Example. Consider the short exact sequence of complexes

0 ⟶ K = \{a\} ⟶ E = \{a, b\} \underrightarrow{_{P}} S = \{b′\} ⟶ 0

where a = db, db′ = 0 and P̂(b) = b′. Define D = {a, b, b′, c} with b′ = dc and let i : S → D denote the inclusion. Define T : E → D by T (a) = b and T (b) = −c, and I : E → D by I(a) = a, I(b) = b, and P : E → D by P = i ∘ P̂. Then dT + T d = I − P, but P̂_{∗} is not surjective on homology.

§4. -complexes, /p -complexes. We now observe that the complex (S_{f}, d) is actually defined over the integers. Consider the lattice

S_{f}^{}=^{def} span_{}{[S_{p}]}_{p∈Cr(f)}

and note that S_{f}^{}forms a subgroup of the integral currents I(X) on X.

Theorem 4.1. The lattice S_{f}^{}is preserved by exterior differentiation d, that is, (S_{f}^{}, d) is a subcomplex of (S_{f}, d). Furthermore, the inclusion of complexes (S_{f}^{}, d) ⊂ (I(X), d) induces an isomorphism

H(S_{f}^{}) ≅ H_{∗}(X; )

Proof. Corollary 3.4 implies that for any p ∈ Cr(f) we have

(4.1)  d[S_{p}] = ∑_{λ_{q}=λ_{p}−1} n_{p,q} [S_{q}]

for real numbers n_{p,q}. Furthermore, since [S_{p}] is rectifiable, we have n_{p,q} ∈ for all p, q by Remark 2.7, and the first assertion is proved.

Now the domain of the operator P extends to include any C^{1} chain c which is transversal to the submanifolds U_{p}, p ∈ Cr_{k}(f), while the domain of T extends to any C^{1} chain c for which c × X is transversal to T. Standard transversality arguments show that such chain groups (over ) compute H_{∗}(X; ). The result then follows from (3.7).

Corollary 4.2. Let G be a finitely generated abelian group. Then there are natural isomorphisms

H(S_{f}^{}⊗_{}G) ≅ H_{∗}(X; G).

Note that the proof of Theorem 4.1 gives an alternate proof of Corollary 3.4 Part one of Theorem 4.1 has an elementary proof when the flow is Morse-Smale, which

means by definition that S_{p} is transversal to U_{q} for all p, q ∈ Cr(f). Suppose the flow is Morse-Smale and that p, q ∈ Cr(f) are critical points with λ_{q} = λ_{p} − 1. Then U_{q} ∩ S_{p} is the union of a finite set of flow lines from q to p which we denote Γ_{p,q}. To each γ ∈ Γ_{p,q} we assign an index n_{γ} as follows. Let B_{ϵ} ⊂ S_{p} be a small ball centered at p in a canonical coordinate system (cf. (2.1) ), and let y be the point where γ meets ∂B_{ϵ}. The orientation of S_{p} induces an orientation on T_{y}(∂B_{ϵ}), which is identified by flowing backward along γ with T_{q}(S_{q}). If this identification preserves orientations we set n_{γ} = 1, and if not, n_{γ} = −1. As in [La] Stokes’ Theorem gives us the following (cf. [W]).

Proposition 4.3. When the gradient flow of f is Morse-Smale, the coefficients in (4.1) are given by

n_{p,q} = (−1)^{λ_{p}} ∑_{γ∈Γ_{p,q}} n_{γ}

Proof. Given a form α of degree λ_{p} − 1, we have

(−1)^{λ_{p}}d[S_{p}](α) = ∫_{S_{p}} dα = lim_{r→∞} ∫_{dS_{p}(r)} α

where S_{p}(r) = φ_{−r}(S_{p}(ϵ)) as in §3. It suffices to consider forms α with support near q where λ_{q} = λ_{p} − 1. Near such q, the set S_{p}(r), for large r, consists of a finite number of manifolds with boundary, transversal to U_{q}. There is one for each γ ∈ Γ_{p,q}. As r → ∞ along one such γ, dS_{p}(r) converges to ±S_{q} where the sign is determined by the agreement (or not) of the orientation of dS_{p}(r) with the chosen orientation of S_{q}.

Remark 4.4. The integers n_{p,q} have a simple definition in terms of currents. Set S_{p}(r) = φ_{−r}(S_{p}(ϵ)) and U_{q}(r) = φ_{r}(U_{q}(ϵ)) (cf. (2.3)). Then for all r sufficiently large

(4.2)  n_{p,q} = (−1)^{λ_{p}} ∫_{X}[U_{q}(r)] ∧ d [S_{p}(r)]

where the integral denotes evaluation on the fundamental class.

§5. Poincaré Duality. There is a simple proof of Poincaré duality in this context. Suppose that X is compact and oriented. Given two oriented submanifolds A and B of complementary dimensions in X which meet transversally in a finite number of points, we define A • B = ∫_{X}[A] ∧ [B] to be the algebraic number of intersections points (counting a point ±1 depending on orientations as usual). Let Cr_{k}(f) = {p ∈ Cr(f) : λ_{p} = k}. Then for any k we have

(5.1)  U_{q} • S_{p} = δ_{pq}  for all p, q ∈ Cr_{k}(f).

This gives a formal identification

(5.2)  U_{f}^{}=^{def} · {[U_{p}]}_{p∈Cr(f)} ≅ Hom (S_{f}^{}, ) .

Therefore, taking the adjoint of d gives a differential δ on U_{f}^{}with the property that H_{n−∗}(U_{f}^{}, δ) ≅ H^{∗}(X; ). On the other hand the arguments of §§1–4 (with f replaced by −f) show that U_{f}^{}is d-invariant with H_{∗}(U_{f}^{}, d) ≅ H_{∗}(X; ). However, these two differentials on U_{f}^{}agree up to sign as we see in the next lemma.

Lemma 5.1. One has

(5.3)  (dU_{q}) • S_{p} = (−1)^{n−k} U_{q} • (dS_{p})

for all p ∈ Cr_{k}(f) and q ∈ Cr_{k−1}(f), and for any k.

Proof. One can see directly from the definition that the integers n_{p,q} are invariant (up to a global sign) under time-reversal in the flow. However, for a simple current-theoretic proof consider the 1-dimensional current [U_{q}(r)] ∧ [S_{p}(r)] consisting of a finite sum of oriented line-segments in the flow lines of Γ_{p,q} (cf. Remark 4.4). Note that

d ([U_{q}(r)] ∧ [S_{p}(r)]) = (d [U_{q}(r)]) ∧ [S_{p}(r)] + (−1)^{n−k+1}[U_{q}(r)] ∧ (d [S_{p}(r)])

and apply (4.2).

Corollary 5.2. (Poincaré Duality)

H^{n−k}(X; ) ≅ H_{k}(X; ) for all k.

Note 5.3. In our operator picture the Poincaré duality isomorphism can be realized in a nice way. Let

T_{tot} = T^{∗} + T = {(y, x) : y = φ_{t}(x) for some t ∈ }.

Then we obtain the operator equation

d ∘ T_{tot} + T_{tot} ∘ d = P − P̌

where

P = ∑_{p∈Cr(f)}[U_{p}] × [S_{p}]  and  P̌ = ∑_{p∈Cr(f)}[S_{p}] × [U_{p}].

This chain homotopy induces an isomorphism H_{∗}(U_{f}^{}) ≅ H_{∗}(C_{f}^{}), which after identifying U_{f}^{}with the cochain complex via (5.1) and (5.3), gives the duality isomorphism 5.2. When X is not oriented, a parallel analysis yields Poincaré duality with mod 2 coefficients.

§6. Critical Submanifolds of Higher Dimension. The methods introduced above apply in much greater generality. As seen in §1, one only needs the flow φ_{t} to be of finite volume to guarantee the existence of an operator P(α) = lim_{t→∞} φ_{t}^{∗}(α) which is chain homotopic to the identity. In this and the following sections we shall examine some important examples.

Let f : X → be a smooth function whose critical set (i.e., the set where df = 0) is a finite disjoint union

Cr(f) = ∐_{j=1}^{ν} F_{j}

of compact submanifolds F_{j} in X. We assume that Hess(f) is non-degenerate on the normal spaces to Cr(f). Then for any f-tame gradient flow (cf. [L]) there are stable and unstable manifolds

S_{j} = {x ∈ X : lim_{t→∞} φ_{t}(x) ∈ F_{j}}  and  U_{j} = {x ∈ X : lim_{t→−∞} φ_{t}(x) ∈ F_{j}}

with projections

(6.1)  S_{j} ⟶^{τ_{j}} F_{j} ⟵^{σ_{j}} U_{j}.

where

τ_{j}(x) = lim_{t→∞} φ_{t}(x)  and  σ_{j}(x) = lim_{t→−∞} φ_{t}(x).

For each j, let n_{j} = dim(F_{j}) and set λ_{j} = dim(S_{j}) − n_{j}. Then dim(U_{j}) = n − λ_{j}. For p ∈ F_{j} we define λ_{p} ≡ λ_{j} and n_{p} ≡ n_{j}.

Definition 6.1. The gradient flow φ_{t} of a smooth function f ∈ C^{∞}(X) on a riemannian manifold X is called a generalized Morse-Stokes flow if it is f-tame and:

(i) The critical set of f consists of a finite number of submanifolds F_{1}, ..., F_{ν} on the normals of which Hess(f) is non-degenerate.

(ii) The manifolds T, and T^{∗}, and the stable and unstable manifolds S_{j}, U_{j} for 1 ≤ j ≤ ν are submanifolds of finite volume. Furthermore, for each j, the fibres of the projections τ_{j} and σ_{j} are of uniformly bounded volume.

(iii) p ≺ q ⇒ λ_{p} + n_{p} < λ_{q} ∀p, q ∈ Cr(f).

These axioms are easily verified in a number of important cases in the algebraic and analytic category. The first main result concerning such flows is the following.

Theorem 6.2. Suppose φ_{t} is a gradient flow satisfying the generalized Morse-Stokes conditions 6.1 on a compact oriented manifold X. Then there is an equation of currents

∂T = [Δ] − P

on X × X, where T, Δ, and P are as in Theorem 1.2, and

(6.2)  P = ∑_{j=1}^{ν} [U_{j} ×_{F_{j}} S_{j}]

where U_{j} ×_{F_{j}} S_{j} ≡ {(y, x) ∈ U_{j} × S_{j} ⊂ X × X : σ_{j}(y) = τ_{j}(x)} denotes the fibre product of the projections (6.1)

Proof. The argument follows closely the proof of Theorem 2.3. Details are omitted.

Janko Latschev has found a Smale-type condition which yields this result in many cases where the hypothesis of 6.1 (iii) does not hold. In particular, Latschev’s condition implies only that: p ≺ q ⇒ λ_{p} < λ_{q}. Details appear in [L].

As in §3, this result can be translated into operator form.

Theorem 6.3. Let φ_{t} be a gradient flow satisfying the generalized Morse-Stokes Conditions on a manifold X as above. Then for all smooth forms α on X, the limit

P(α) = lim_{t→∞} φ_{t}^{∗}(α)

exists and defines a continuous linear operator P : E^{∗}(X) ⟶ D^{′∗}(X) with values in flat currents on X. This operator fits into a chain homotopy

(6.3)  d ∘ T + T ∘ d = I − P.

Furthermore, P is given by the formula

(6.4)  P(α) = ∑_{j=1}^{ν} Res_{j}(α) [S_{j}]

where

(6.5)  Res_{j}(α) ≡ τ_{j}^{∗} {(σ_{j})_{∗} (α|_{U_{j}})}

Proof. This is a direct consequence of Theorem 6.2 except for the formulae (6.4)-(6.5). To see this consider the pull-back square

(6.6)  \begin{matrix}U_{j} ×_{F_{j}} S_{j} & ⟶^{t_{j}} & U_{j} \\ ^{s_{j}} ↓ & & ↓^{σ_{j}} \\ S_{j} & ⟶^{τ_{j}} & F_{j} \end{matrix}

where t_{j} and s_{j} are the obvious projections. One sees from the definitions (cf §3) that

P(α) = ∑_{j=1}^{ν}(s_{j})_{∗} {(t_{j})^{∗} (α|_{U_{j}})} .

The commutativity of the diagram (6.6) allows us to rewrite these terms as in (6.5).

Corollary 6.4. Suppose that λ_{p} + n_{p} + 1 < λ_{q} for all critical points p ≺ q. Then the homology of X is spanned by the images of the groups H_{λ_{j}+ℓ}(\overline{S_{j}}) for j = 1, ..., ν and ℓ ≥ 0.

Proof. Under this hypothesis ∂(U_{j}×_{F_{j}} S_{j}) = 0 for all j, and so (6.2) yields a decomposition of P into operators that commute with d.

We can make this corollary more precise. Note that τ_{j} : S_{j} → F_{j} can be given the structure of a vector bundle of rank λ_{j}. The closure \overline{S_{j}} ⊂ X is a compactification of this bundle with a complicated structure at infinity. ( See [CJS] for example.) There is nevertheless a homomorphism Θ_{j} : H_{∗}(F_{j}) ⟶ H_{λ_{j}+∗}(\overline{S_{j}}) which after pushing forward to the one-point compactification of S_{j}, is the Thom isomorphism. This leads to the following (cf. [AB]).

Theorem 6.5. Suppose that λ_{p} + n_{p} + 1 < λ_{q} for all critical points p ≺ q and that X and all F_{j} and S_{j} are oriented. Then there is an isomorphism

H_{∗}(X) ≅ ⨁_{j} H_{∗−λ_{j}}(F_{j})

This result holds without the orientation assumptions if one takes homology with appropriately twisted coefficients. Much stronger versions of Theorems 6.3 and 6.5, are found in [L]. They include an extension to integral homology groups. Latschev also derives a spectral sequence associated to any Bott-Smale function satisfying a natural Smale-type transversality hypothesis. One virtue of this sequence is that the differentials are explicitly computable. Assuming for simplicity that everything is oriented, the E^{1}-term is given by

E_{p,q}^{1} = ⨁_{λ_{j}=p} H_{q}(F_{j}; )

and E_{p,q}^{k} ⇒ H_{∗}(X; )

§7. The Relative Case. In standard Morse Theory one often studies the change in the topology as one passes from {x : f(x) ≤ a} to {x : f(x) ≤ b}. Our approach is easily adapted to this case.

Let f : X → be a proper Morse function, where X is not necessarily compact, and suppose that X carries a metric as in 14.3. Let a < b be regular values of f and consider the compact manifold with boundary

Z ≡ f^{−1}([a, b]).

On Z we define a vector field V = (ψ ∘ f)∇f where ψ : [a, b] → [0, 1] is a smooth function satisfying: (i) ψ^{−1}(0) = {a, b}, (ii) ψ is linear on [a, a + ϵ] and [b − ϵ, b], (iii) ψ ≡ 1 on [a + 2ϵ, b − 2ϵ], (iv) f(Cr(f)) ∩ Z ⊂ (a + 2ϵ, b − 2ϵ) for some small ϵ > 0.

Let φ_{t} : Z → Z be the flow of V. Note that φ_{t} is complete and fixes the boundary ∂Z. By 14.3, the stable and unstable manifolds of each p ∈ Cr(f) have finite volume in Z, and so also does T ≡ {(y, x) : y = φ_{t}(x) for some t, 0 < t < ∞} ⊂ Z × Z.

We decompose the boundary

∂Z = f^{−1}(b) − f^{−1}(a) =^{def} ∂_{b} − ∂_{a}.

In analogy with the fibre products appearing in §6 we have the following submanifolds of Z × Z:

S(∂_{b}) = {(y, x) ∈ Z × Z : y ∈ ∂_{b} and lim_{t→∞} φ_{t}(x) = y}  and
U(∂_{a}) = {(y, x) ∈ Z × Z : x ∈ ∂_{a} and lim_{t→−∞} φ_{t}(y) = x}.

Theorem 7.1. On Z × Z there is an equation of integral currents

(7.1)  ∂T = [Δ] − ∑_{p∈Cr(f)∩Z}[U_{p}] × [S_{p}] − [U(∂_{a})] − [S(∂_{b})].

Proof. Consider T_{t} ⊂ × X × X defined as in (1.1) and note that

∂T_{t} = {0} × [Δ_{Z}] − {t} × [graph φ_{t}] − [0, t] × [Δ_{∂Z}].

Set T_{t} = pr_{∗} T_{t} and observe that since pr_{∗}([0, t] × [Δ_{∂Z}]) = 0,

∂T_{t} = [Δ] − [graphφ_{t}].

By hypothesis T = lim_{t→∞} T_{t} has finite volume, and a direct application of the arguments of §2 establishes (7.1).

Equation (7.1) can be translated to an operator equation. However, here we want the operator to act on the relative forms:

(7.2)  E^{∗}(Z, ∂_{b}) ≡ {α ∈ E^{∗}(Z) : α|_{∂_{b}}= 0} .

We begin with the case where f > a and so ∂_{a} = ∅.

Theorem 7.2. Suppose f : X ⟶ ^{+} is a proper Morse function. Let Z = f^{−1}(−∞, b] where b is a regular value of f, and consider the operator

P : E^{∗}(Z, ∂Z) ⟶ D^{′∗}(Z)

defined by

(7.3)  P(α) = lim_{t→∞} φ_{t}^{∗}(α)

where φ_{t} is the truncated gradient flow defined above. This operator is well defined and continuous. In fact, there exists a continuous operator T : E^{∗}(Z, ∂Z) ⟶ D^{′∗}(Z) of degree -1 with values in flat currents, such that

(7.4)  d ∘ T + T ∘ d = I − P

Furthermore, P is given by the formula

(7.5)  P(α) = ∑_{p∈Cr(f)∩Z} r_{p}(α)[S_{p}]

where r_{p}(α) = ∫_{U_{p}} α. In particular, P is a continuous chain mapping onto the finite dimensional complex

S_{f,Z} =^{def} span_{}{[S_{p}]}_{p∈Cr(f)∩Z},

with differential given as in 3.5, and P induces an isomorphism

H_{deR}^{∗}(Z, ∂Z) ⟶^{≅} H(S_{f,Z}).

Proof. This is deduced exactly as are Theorems 3.3, 3.5. We need only note that the operator given by [S(∂_{b})] is zero on E^{∗}(Z, ∂Z). This follows directly from the definition 3.1 and the fact that π_{1}(S(∂_{b})) = ∂_{b} = ∂Z.

In the more general case where ∂_{a} ≠ ∅ we compose our operators with the projection map π : E^{∗}(Z)′ ⟶ E^{∗}(Z, ∂Z)′, (which is adjoint to the inclusion E^{∗}(Z, ∂Z) ⊂ E^{∗}(Z)). In this case the operator corresponding to [U(∂_{a})] is zero. Specifically, letting π_{k} : X×X → X denote projection onto the k^{th} factor, we have that

(U(∂_{a})(α), β) = ((π_{1})_{∗} {π_{2}^{∗}α ∧ [U(∂_{a})]} , β) = 0

since π_{1}(U(∂_{a})) = ∂_{a} and β|_{∂_{a}}= 0. We conclude the following.

Theorem 7.3. The operators

T, P : E^{∗}(Z, ∂_{b}Z) ⟶ D^{′∗}(Z, ∂_{a}Z)

corresponding to the currents T, P from Theorem 7.2 satisfy the equation

d ∘ T + T ∘ d = I − P

where

P = ∑_{p∈Cr(f)∩Z} r_{p}[S_{p}]

and r_{p}(α) = ∫_{U_{p}} α. Thus, P gives a continuous chain mapping of the relative deRham complex E^{∗}(Z, ∂_{b}) onto the finite dimensional complex

S_{f,Z} =^{def} span_{}{[S_{p}]}_{p∈Cr(f)∩Z}

with differential given as in 3.5. This induces an isomorphism

H_{deR}^{∗}(Z, ∂_{b}Z) ⟶^{≅} H(S_{f,Z}).

Remark 7.4. Taking ∂_{b}Z = ∅ and ∂_{a}Z ≠ ∅ in Theorem 7.3 (and interchanging the roles of a and b) gives the version of Theorem 7.2 corresponding to going backwards in time. That is, one considers

P(α) =^{def} lim_{t→−∞} φ_{t}^{∗}α

and obtains a projection operator

P : E^{∗}(Z) ⟶ S_{f,Z} ⊂ D^{′∗}(Z, ∂_{b})

where S_{f,Z} is a finite complex defined over .

Remark 7.5. (Duality) Arguing exactly as in §5 one can retrieve the duality theorem

(7.6)  H^{k}(Z, ∂_{b}Z; ) ≅ H_{n−k}(Z, ∂_{a}Z : )

which gives, in the special case where either ∂_{a}Z = 0 or ∂_{b}Z = 0, the Lefschetz Duality Theorem.

§8. Holomorphic Flows and the Carrell-Lieberman-Sommese Theorem. The ideas in this paper have interesting consequences in the holomorphic case. Given a ^{∗}- action φ_{t} on a compact Kähler manifold X, there is a complex graph

Tl =^{def} {(t, φ_{t}(x), x) ∈ ^{∗} × X × X : t ∈ ^{∗} and x ∈ X} ⊂ ^{1}( ) × X × X

similar to the graphs considered above. It is a basic result of Sommese [So] that if φ_{t} has fixed-points, then Tl has finite volume and its closure \overline{Tl} in ^{1}( ) × X × X is an analytic subvariety.

∗-actions with fixed-points on Kähler manifolds are intimately related to Morse-Theory. The complex action φ_{t} can be decomposed into an “angular” S^{1}-action and a radial flow. Choosing an S^{1}-invariant Kähler metric and applying an argument of Frankel [F] gives a function f : X → of Bott-Morse type whose gradient generates the radial action. The methods of this paper can now be applied. In particular we find that \overline{Tl} defines a rational equivalence in X × X between the diagonal Δ and an analytic cycle P whose components are fibre products of stable and unstable manifolds over components of the fixed-point set.

When the fixed-points are all isolated, P becomes a sum of analytic Künneth components P = ∑ \overline{S}_{p} ×\overline{Ũ}_{p}. This immediately implies that the cohomology of X is freely generated by the stable subvarieties \{\overline{S}_{p}\}_{p∈Zero(φ)}. Furthermore, it follows that X is algebraic and that all cohomology theories on X (eg. algebraic cycles modulo rational equivalence, algebraic cycles modulo algebraic equivalence, singular cohomology) are naturally isomorphic. (cf. [BB], [ES], [Fr]).

When the fixed-point set has positive dimension, this method yields results of Carrell-Lieberman-Sommese for ^{∗}-actions ([CL], [CS]), which assert among other things that if dim(X^{^{∗}} ) = k, then H^{p,q}(X) = 0 for |p − q| > k.

§9. Singularities and Characteristic Classes. The ideas in this paper also have an interesting application to the study of curvature and singularities. Suppose

α : E ⟶ F

is a map between smooth vector bundles with connection over a manifold X. Let G = G_{k}(E ⊕ F) → X denote the Grassmann bundle of k-planes in E ⊕ F where k = rank(E) ≤ rank(F). There is a flow φ_{t} on G induced by the flow ψ_{t} : E ⊕ F → E ⊕ F where ψ_{t}(e, f) = (te, f). This is a very simple generalized Morse-Stokes flow on G (written multiplicatively rather than additively). On the “affine chart”Hom(E, F) ⊂ G, one has that φ_{t}(A) = \frac{1}{t}A. One considers the graph T of the flow in the fibre product of G with itself, and one obtains the equation ∂T = Δ − P where Δ is the fibre diagonal and P is a sum of fibre products of stable and unstable manifolds. In the affine chart Hom(E, F), the stable manifolds are exactly the universal singularity sets Σ_{ℓ} consisting of linear maps A : E → F of rank k − ℓ.

Passing to the associated operators as in §3 and applying the resulting chain homotopy to Chern-Weil characteristic forms of the tautological bundle U over G leads to a “universal McPherson Formula” on Hom(E, F). To pull this formula back to X we introduce the following concept.

Definition 9.1. The section α is said to be geometrically atomic if the submanifold

T (α) =^{def} \{\frac{1}{t}α(x) ∈ G : 0 < t ≤ 1 and x ∈ X\}

has locally finite volume in G.

This hypothesis is sufficient to guarantee the existence of lim_{t→0} α_{t}^{∗}ϕ where α_{t} ≡ \frac{1}{t}α and where ϕ is any differential form on G. Choosing ϕ = Φ(Ω_{U}) where Φ is an Ad-invariant polynomial on 𝔤𝔩_{k}( ) and Ω_{U} is the curvature of the tautological k-plane bundle over G, one obtains the following formula for the Φ characteristic form of E:

(9.1)  Φ(Ω_{E}) = ∑_{ℓ} Res_{ℓ,Φ} [Σ_{ℓ}(α)] + dT

where Σ_{ℓ}(α) is a current associated to the locus where α drops rank by ℓ, Res_{ℓ,Φ} is an explicit residue form defined along Σ_{ℓ}(α), and T is an L_{loc}^{1}-form on X.

For bundle maps which are normal, i.e., transversal to the universal singularity sets in Hom(E, F), this gives a local version of a basic formula of MacPherson [Mac_{1,2}]. However, many important classes of bundle mappings, such as direct sum and tensor product mappings, are not normal but are generically geometrically atomic. In these cases the explicit formulas given by (9.1) are new even at the global (cohomological) level.

The critical idea is to study the operator from forms on G to currents on X whose“kernel” is given by T (α). The method is analogous to that of §6. Details appear in [HL_{3,4}].

When the residues in (9.1) are constants, the form T represents a Cheeger-Simons differential character canonically associated to α and the connections on E and F (cf. [HL_{6}]).

§10. Equivariant Morse Theory. The ideas developed here carry over virtually intact to the setting of equivariant cohomology. The reason the method works directly is the simple but important fact that a closed, invariant submanifold is equivariantly closed (See Corollary 10.2 below). In this section we derive some consequences of the method which usefully apply to Morse functions arising in algebra and geometry (e.g., from moment map constructions). Deeper results have been obtained by J. Latschev [L]. We shall adopt the exposition of Cartan’s equivariant de Rham theory found in [BGV].

Let G be a compact Lie group with Lie algebra 𝔤 acting on a compact n-manifold X. An equivariant differential form on X is a G-equivariant polynomial map α : 𝔤 ⟶ E^{∗}(X). The set of such forms is denoted by

E_{G}^{∗}(X) = {S^{∗}(𝔤^{∗}) ⊗ E^{∗}(X)}^{G}

and is graded by defining elements of S^{p}(𝔤^{∗}) ⊗ E^{q}(X) to have total degree 2p + q. The equivariant differential d_{G} : E_{G}^{∗}(X) ⟶ E_{G}^{∗+1}(X) is defined by setting

(d_{G}α) (V) = dα(V) − i_{V}α(V)

for V ∈ 𝔤 where the vector field Ṽ is the image of V under the natural linear map 𝔤 ⟶ Γ(T X), and where i_{V} denotes contraction with Ṽ.

The complex D_{G}^{′∗}(X) of equivariant currents is defined analogously by passing from smooth forms to generalized forms, i.e., forms with distribution coefficients.

Suppose now that f ∈ C^{∞}(X) is a G-invariant function and X is provided with a Ginvariant riemannian metric. Then g_{∗}∇f = ∇f for all g ∈ G and so the gradient flow of f commutes with the action of G. Suppose this flow has finite volume and let

(10.1)  ∂T = Δ − P

denote the equation discussed in §1. Let G act on X ×X by the diagonal action g ·(x, y) = (gx, gy).

Lemma 10.1. The currents T and P are G-invariant and have the property that i_{V}T =

i_{V}P = 0 for all V ∈ 𝔤.

Proof. The current T corresponds to integration over the finite volume submanifold {(x, y) ∈ X × X − Δ : ∃t ∈ (0, ∞) s.t. y = φ_{t}(x)}. Since gφ_{t}(x) = φ_{t}(gx), the invariance of T is clear. The invariance of P then follows from (10.1). For the second assertion note that since V is tangent to T, (i_{V}T ) (ω) = ∫_{T} i_{V}ω = 0 for any n-form ω on X × X . From the equation d∘i_{V} +i_{V} ∘d = L_{V} (=Lie derivative) and the fact that L_{V}T = i_{V}Δ = 0, we conclude that i_{V}P = 0.

Corollary 10.2. Consider T ≡ 1 ⊗ T ∈ 1 ⊗ D^{′n−1}(X × X)^{G} as an equivariant current of total degree (n − 1) on X × X. Consider Δ and P similarly as equivariant currents of degree n. Then

(10.2)  ∂_{G}T = Δ − P.

in the complex of equivariant currents on X × X.

The dictionary given between operators and kernels in §1 carries over directly to this context. Currents in D_{G}^{′n−ℓ}(X × X) correspond to G-equivariant operators E^{∗}(X) ⟶ D^{′∗+ℓ}(X), and therefore give operators E_{G}^{∗}(X) ⟶ D_{G}^{′∗+ℓ}(X). Equation (10.2) corresponds to the operator equation

(10.3)  d_{G} ∘ T + T ∘ d_{G} = I − P,

and the discussion of §3 proves the following.

Proposition 10.3. Let X be a compact G-manifold and φ_{t} an invariant flow of finite volume on X. Then for all α ∈ E_{G}^{∗}(X) the limit

(10.4)  P(α) = lim_{t→∞} φ_{t}^{∗}α

exists and defines a continuous linear operator P : E_{G}^{∗}(X) ⟶ D_{G}^{′∗}(X) of degree 0, which is equivariantly chain homotopic to the identity on E_{G}^{∗}(X).

Applying the methods of §§2-3 and the fact that

S^{∗}(𝔤^{∗})^{G} ≅ H^{∗}(BG)

gives the following result.

Theorem 10.4. Let X be a compact riemannian G-manifold and f ∈ C^{∞}(X) an invariant Morse function whose gradient flow φ_{t} is Morse-Stokes. Then the continuous linear operator (10.4) defines a map of equivariant complexes

P : E_{G}^{∗}(X) ⟶ S^{∗}(𝔤^{∗})^{G} ⊗ S_{f}

where S_{f} = span{[S_{p}]}_{p∈Cr(f)} as in (3.8) and the differential on S^{∗}(𝔤^{∗})^{G} ⊗ S_{f} is 1 ⊗ ∂. The map P induces an isomorphism

H_{G}^{∗}(X) ⟶^{≅} H^{∗}(BG) ⊗ H^{∗}(X)

For an example of this phenomenon consider the torus acting on _{}^{n} via the standard action on homogeneous coordinates [z_{0}, ..., z_{n}], and set f([z]) = ∑ k|z_{k}|^{2}/∥z∥^{2}. One sees immediately the well-known fact that H_{G}^{∗}( _{}^{n}) is a free H^{∗}(BG)-module with one generator in each dimension 2k for k = 0, ..., n. This extends to generalized flag manifolds and to products.

It has been pointed out by Janko Latschev that there exists an invariant Morse function for which no choice of invariant metric gives a Morse-Stokes flow.

On the other hand the method applies to much more general functions and yields results as in §§5-9. Suppose for example that f is an invariant function whose critical set consists of a finite number of non-degenerate critical orbits O_{i} = G/H_{i}, i = 1, ..., N. Latschev (cf. [L], [ABr]) has established a spectral sequence E_{∗,∗}^{k} ⇒ H_{G}^{∗}(X) with computable differentials and (assuming for simplicity that everything is oriented)

E_{p,∗}^{1} = ⨁_{λ_{i}=p} H_{G}^{∗}(O_{i}) = ⨁_{λ_{i}=p} H^{∗}(BH_{i}).

§11. Flat bundles and local coefficients. Our method applies immediately to forms with coefficients in a flat bundle E → X. In this case the kernels of §2 are currents on X × X with coefficients in Hom(π_{1}^{∗}E, π_{2}^{∗}E). Given a Morse-Stokes flow φ_{t} on X we consider the kernel T_{E} = h ⊗ T where T is defined as in §2 and h : E_{φ_{t}(x)} → E_{x} is parallel translation along the flow line. One obtains the equation

∂T_{E} = Δ_{E} − P_{E}

where Δ_{E} = Id ⊗ Δ and P_{E} = ∑_{p} h_{p} ⊗ ([U_{p}] × [S_{p}]) with h_{p} : E_{y} → E_{x} given by parallel translation along the broken flow line. Thus, h_{p} corresponds to Id : E_{p} → E_{p} under the canonical trivializations E|_{U_{p}} ≅ U_{p} × E_{p} and E|_{S_{p}} ≅ S_{p} × E_{p}. We obtain the operator equation

(11.1)  d ∘ T_{E} + T_{E} ∘ d = I − P_{E}

where P_{E} maps onto the finite complex

S_{E} =^{def} ⨁_{p∈Cr(f)} E_{p} ⊗ [S_{p}]

by integration of forms over the unstable manifolds. The restriction of d to S_{E} is given as in 4.3 by d(e ⊗ [S_{p}]) = ∑ h_{p,q}(e)[S_{q}] where h_{p,q} = (−1)^{λ_{p}} ∑_{γ} h_{γ} and h_{γ} : E_{p} → E_{q} is parallel translation along γ ∈ Γ_{p,q}. By (11.1) the complex (S_{E}, d) computes H^{∗}(X; E).

Reversing time in the flow shows that the complex U_{E} = ⨁_{p} E_{p}^{∗} ⊗ [U_{p}] with differential defined as above computes H^{∗}(X; E^{∗}). As in §5 the obvious dual pairing of these complexes establishes the generalized Poincaré duality. Furthermore, one can extend all this to integral currents twisted by representations of π_{1}(X) in GL_{n}( ) or GL_{n}( /p ) and obtain duality with local coefficient systems.

§12. Products. Our method has a number of interesting extensions. For example, consider the triple diagonal Δ_{3} ⊂ X × X × X as the kernel of the wedge-product operator

∧ : E^{∗}(X) ⊗ E^{∗}(X) → E^{∗}(X).

Let f and f′ be functions with Morse-Stokes flows φ_{t} and φ′_{t} respectively. Assume that for all (p, p′) ∈ Cr(f)×Cr(f′) the stable manifolds S_{p} and S′_{p′} intersect transversely in a manifold of finite volume, and similarly for the unstable manifolds U_{p} and U′_{p′}. Degenerating Δ_{3} gives a kernel

T ≡ {(φ_{t}(x), φ′_{t}(x), x) ∈ X × X × X : x ∈ X and 0 ≤ t < ∞}

and a corresponding operator T : E^{∗}(X) ⊗ E^{∗}(X) → E^{∗}(X) of degree -1. One calculates that ∂T = Δ_{3} − M where

M = ∑_{(p,p′)∈Cr(f)×Cr(f′)}[U_{p}] × [U_{p′}] × [S_{p} ∩ S′_{p′}].

The corresponding operator M : E^{∗}(X) ⊗ E^{∗}(X) → E^{∗}(X) is given by

(12.1)  M(α, β) = ∑_{(p,p′)∈Cr(f)×Cr(f′)} (∫_{U_{p}} α) (∫_{U_{p′}} β) [S_{p} ∩ S′_{p′}].

The arguments of §§1-3 adapt to prove the following.

Theorem 12.1. There is an equation of operators ∧−M = d∘T+T∘d from E^{∗}(X ×X) to E^{∗}(X) (where ∧ denotes restriction to the diagonal). In particular for α, β ∈ E^{∗}(X) we have the chain homotopy

(12.2)  α ∧ β − M(α, β) = dT(α, β) + T(dα, β) + (−1)^{degα}T(α, dβ)

between the wedge product and the operator (12.1).

Note that the operator M has range in the finite dimensional vector space

M =^{def} span_{}{ [S_{p} ∩ S′_{p′}]} _{(p,p′)∈Cr(f)×Cr(f′)}.

It converts a pair of smooth forms α, β into a linear combination of the pairwise intersections of the stable manifolds [S_{p}] and [S′_{p′}]. If dα = dβ = 0, then

(12.3)  M(α, β) = α ∧ β − dT(α, β),

and so M(α, β) is a cycle homologous to the wedge product α ∧ β. This operator also has the following properties.

Theorem 12.2. The operator M maps onto the subspace M. Furthermore, for forms α, β ∈ E^{∗}(X), it satisfies the equation

(12.4)  dM(α, β) = M(dα, β) + (−1)^{degα}M(α, dβ)

Proof. To see that M is onto (as a linear map from E^{∗}(X × X)) it suffices to see that for each non-empty intersection S_{p} ∩ S′_{p′} the current [S_{p} ∩ S′_{p′}] is in the range. However, by transversality we see that if S_{p} ∩S′_{p′} ≠ ∅, then λ_{p} +λ_{p′} ≥ n, and so n ≥ (n−λ_{p})+(n−λ_{p′}). Therefore, by transversality we have dim(U_{q} ∩ U′_{q′}) ≤ 0 for all q, q′. It follows that we can find differential forms α and β such that ∫_{U_{p}} α = ∫_{U′_{p′}} β = 1 and ∫_{U_{q}} α = ∫_{U′_{q′}} β = 0 for all

q ≠ p and q′ ≠ p′. Then by (12.1), M(α, β) = [S_{p} ∩ S′_{p′}] and the assertion is proved.

Equation (12.4) follows from the fact that dM = 0, which implies that d∘M+M∘d = 0, together with the standard formula for d(α ∧ β).

It follows immediately that d(M) ⊂ M. In fact one can see from the transversality assumptions that for (p, p′) ∈ Cr(f) × Cr(f′) one has

(12.5)  d [S_{p} ∩ S′_{p′}] = ∑_{q∈Cr(f)} n_{pq}[S_{q} ∩ U′_{p′}] + (−1)^{n−λ_{p}} ∑_{q′∈Cr(f′)} n′_{p′q′}[S_{p} ∩ U′_{q′}]

where the n_{pq} are defined as in §4. Thus we retrieve the cup product over the integers in the Morse complex.

Example 12.3. A fundamental example of a pair satisfying our hypotheses is given by f, −f where the gradient flow is Morse-Stokes. In this case U′_{p} = S_{p} and S′_{p} = U_{p} for all p ∈ Cr(f) = Cr(−f). Thus formula (12.1) becomes

M(α, β) = ∑_{p,p′∈Cr(f)} (∫_{U_{p}} α) (∫_{S_{p′}} β) [S_{p} ∩ U_{p′}].

In particular we have the following.

Proposition 12.4. Suppose the gradient flow of f is Morse-Stokes. Then for any cycle Y in X which is transversal to all the U_{p} and S_{p}, p ∈ Cr(f), we have the formula

(12.6)  ∫_{Y} α ∧ β = ∑_{p,p′∈Cr(f)} (∫_{U_{p}} α) (∫_{S_{p′}} β) [S_{p} ∩ U_{p′} ∩ Y].

whenever dα = dβ = 0.

Example 12.5. Suppose deg α = n − deg β = k. Then (cf. [La, Prop. 12])

∫_{X} α ∧ β = ∑_{p∈Cr_{k}(f)} (∫_{U_{p}} α) (∫_{S_{p′}} β)

§13. A Lefschetz Theorem for the Thom-Smale complex. Let X be a compact oriented riemannian manifold, and let U_{p}, S_{p}, p ∈ Cr(f) be the unstable and stable manifolds of a Morse-Stokes flow on X, oriented as in §2. Recall the Lefschetz number of a smooth mapping F : X → X defined by

Lef(F) = ∑_{i}(−1)^{i}trace {F_{∗} : H_{i}(X; ) → H_{i}(X; )}

Theorem 13.1. Suppose F : X → X is a smooth mapping such that F maps S_{p} transversally to U_{q} for all q ≻ p (i.e., F |_{S_{p}} is transversal to U_{p} and F (S_{p}) ∩ U_{q} = ∅ for all q ≻ p and q ≠ p). Let C_{p} = {x ∈ S_{p} : F (x) ∈ U_{p}}. Then

Lef(F) = ∑_{p∈Cr(f)} (−1)^{λ_{p}} ∑_{x∈C_{p}} σ_{x}(F)

where

σ_{x}(F) =^{def} \{ \begin{matrix}1  if F_{∗}T_{x}(S_{p}) agrees in orientation with the normal space to U_{p} at F (x) \\ −1  otherwise. \end{matrix}

Proof. Our transversality assumption implies that the graph Γ_{F} = {(F (x), x) : x ∈ X} in X × X meets the cycle P only in its regular points ⋃_{p} U_{p} × S_{p} and it is transversal there. We recall that Lef(F) = [Δ] • [Γ_{F}] in X × X. By (2.5) and the fact that Γ_{F} and P meet nicely, we conclude that Lef(F) = [P] • [Γ_{F}] which is easily computed as claimed.

Note that when F = Id, the hypotheses are satisfied and we get the standard computation of the Euler characteristic from the Morse complex.

§14. The Genericity Theorem. Let f ∈ C^{∞}(X) be a Morse function on a compact manifold X. In this section we shall prove that there exists a riemannian metric on X for which the gradient flow satisfies the Morse-Stokes conditions of §2. The authors have subsequently discovered that some of the material in this section could be reduced by appealing to a paper of Laudenbach [La]. For completeness we have kept the details. To begin we recall the following.

Definition 14.1. The gradient flow of f for a riemannian metric on X is called Morse-Smale if the stable and unstable manifolds S_{p} and U_{q} intersect transversely for all p, q ∈ Cr(f).

To simplify arguments we shall demand a little more. Recall that at each p ∈ Cr(f) there exist canonical local coordinate systems (u, v) : O_{p} ⟶^{≅} V_{p} where

(14.1)  V_{p} = {(u, v) ∈ ^{λ_{p}} × ^{n−λ_{p}} : |u|^{2} ≤ r_{p} and |v|^{2} ≤ r_{p}}

such that u(p) = v(p) = 0 and f(u, v) = f(p) − |u|^{2} + |v|^{2}.

Definition 14.2. A riemannian metric ds^{2} is said to be canonically flat near Cr(f) if ds^{2} = |du|^{2} + |dv|^{2} in some canonical linear coordinate system about each p ∈ Cr(f).

We shall prove the following.

Theorem 14.3. Let f ∈ C^{∞}(X) be a Morse function on a compact manifold X. Suppose X is given a riemannian metric which is canonically flat near Cr(f) and for which the gradient flow φ_{t} is Morse-Smale. Then φ_{t} satisfies the Morse-Stokes conditions 2.1

Theorem 14.4. If f ∈ C^{∞}(X) is a Morse function and ds^{2} is any riemannian metric on X, then ds^{2} can be modified outside some neighborhood of Cr(f) so that φ_{t} becomes Morse-Smale. In fact this modification can be made arbitrarily small in the C^{∞}-topology.

Taken together these theorems prove the following.

Theorem 14.5. Given any Morse function f on a compact manifold X, there exists a riemannian metric on X for which the gradient flow is Morse-Stokes.

Furthermore, this metric can be chosen to be canonically flat near Cr(f).

Proof of Theorem 14.3. We first observe that if the flow of f is Morse-Smale, then

(14.2)  p ≺ q  ⇒  λ_{p} < λ_{q}

for all p, q ∈ Cr(f). To see this suppose p and q are joined by an (unbroken) flow line ℓ. Then U_{p}∩S_{q} ⊃ ℓ and so by the transversality condition, dim(U_{p}∩S_{q}) = (n−λ_{p})+λ_{q}−n ≥ 1.

Now let a_{1} < ⋯ < a_{m} be the critical values of f. For each a_{k} let Cr(f, a_{k}) ⊂ Cr(f) be the set of critical points of f with critical value a_{k}. Each p ∈ Cr(f, a_{k}) has a canonical local coordinate system as in (14.1) where the metric is flat. We may assume that the radius r_{p} is the same for all p ∈ Cr(f, a_{k}). Call this radius r_{k}. By shrinking the neighborhoods O_{p} we may assume that these canonical coordinate systems are pairwise disjoint and that a_{k+1} − r_{k+1} > a_{k} + r_{k} for all k. Furthermore, by multiplying f by some scalar a >> 1 and further shrinking the O_{p} we can assume that r_{p} = 2 for all p ∈ Cr(f).

Now our manifold decomposes into “blocks”:

(14.3)  X = P_{0} ∪ Q_{0} ∪ P_{1} ∪ Q_{1} ∪ P_{2} ∪ Q_{2} ∪ … P_{m}

where

P_{k} = f^{−1}[a_{k} − 1, a_{k} + 1]  and  Q_{k} = f^{−1}[a_{k} + 1, a_{k+1} − 1].

Note that P_{k} and Q_{k} are compact manifolds with boundary. The manifolds P_{k} can be further decomposed. Let O′_{p} ⊂ O_{p} be the subset defined by the equations |u||v| ≤ 1 and −1 ≤ |v|^{2} − |u|^{2} ≤ 1. Then

P_{k} = R_{k} ∪ ⋃_{p∈Cr(f,a_{k})} O′_{p}

where R_{k} is the closure of P_{k} − ⋃_{p} O′_{p}.

Let ψ_{s} be the (incomplete) flow on X − Cr(f) generated by W ≡ grad f/∥grad f∥^{2}, so that

f(ψ_{s}(x)) = f(x) + s

whenever ψ_{s}(x) is defined. Using this vector field in the obvious way (cf. [M_{1}]) we obtain smooth product structures

(14.4)  Q_{k} ≅ (∂^{−}Q_{k}) × [0, 1]

(14.5)  R_{k} ≅ (∂^{−}R_{k}) × [0, 1]

where ∂^{−}Q_{k} = Q_{k} ∩ f^{−1}(a_{k} + 1) and ∂^{−}R_{k} = R_{k} ∩ f^{−1}(a_{k} − 1). Note that ∂^{−}R_{k} is a compact manifold with non-empty boundary.

Consider now the unstable manifold U_{p} for some p ∈ Cr(f, a_{k}) and some k. We shall show that vol(U_{p}) < ∞. To begin note that U_{p} ∩ O′_{p} ≅ {(0, v) ∈ V_{p} : |v| ≤ 1} is a

smoothly embedded closed disk of dimension ℓ =^{def} n − λ_{p} and clearly has finite ℓ-volume. Its boundary U_{p}∩(∂^{−}Q_{k}) is a smoothly embedded sphere, and via (14.4) we have U_{p}∩Q_{k} ≅ (U_{p} ∩ ∂^{−}Q_{k}) × [0, 1] which also has finite ℓ-volume.

We now show that U_{p} ∩ P_{k+1} has finite ℓ-volume. To begin note that via (14.5) we have a smooth product U_{p} ∩R_{k+1} ≅ (U_{p} ∩∂^{−}R_{k+1})×[0, 1] (which extends beyond the boundary of R_{k+1} so we needn’t worry about how U_{p} meets this boundary). Since U_{p} ∩ ∂^{−}R_{k+1} is a subset of a compact (ℓ − 1)-manifold, we see that U_{p} ∩ R_{k+1} has finite ℓ-volume.

It remains to show that U_{p} ∩ O′_{q} has finite volume for q ∈ Cr(f, a_{k}). This is equivalent to showing that U_{p} ∩ O′′_{q} has finite volume, where O′′_{q} is defined by |u| ≤ 1 and |v| ≤ 1. (To see this push inward along the flow.) For simplicity, from here on we shall denote O′′_{q} by O_{q} and V′′_{q} by V_{q}.

Since q ≻ p we know from (14.2) that

dimU_{p} > dimU_{q}.

In our local coordinate box V_{q} = {(u, v) ∈ ^{λ_{q}} × ^{n−λ_{q}} : |u| ≤ 1, |v| ≤ 1} the flow is generated by

(14.6)  φ_{t}(u, v) = (e^{−t}u, e^{t}v)

We decompose ∂V_{q} into two pieces:

A = {(u, v) : |u| = 1, |v| ≤ 1}  and  B = {(u, v) : |u| ≤ 1, |v| = 1}.

There are subsets

A_{0} = A∩S_{q} = {(u, 0) : |u| = 1} ≅ S^{λ_{q}−1}, B_{0} = B∩U_{q} = {(0, v) : |v| = 1} ≅ S^{n−λ_{q}−1}.

The flow determines a diffeomorphism

Φ : A − A_{0} ⟶^{≅} B − B_{0}

given by

Φ(u, v) = (|v|u, \frac{1}{|v|}v)

Note that Φ(u, v) is the unique point in B which lies on the flow line through (u, v). Let Ã ⟶^{π_{A}} A be the “oriented blow-up” of A along A_{0} where A_{0} is replaced by the

oriented normal lines to A_{0}. Let B̃ ⟶^{π_{B}} B be defined similarly. Ã has coordinates (û, v̂, t) where |û| = 1, |v̂| = 1 and 0 ≤ t ≤ 1 and π_{A}(û, v̂, t) = (û, tv̂). B̃ has the same coordinates with π_{B}(û, v̂, t) = (tû, v̂).

Φ lifts to a map Φ̃ : Ã ⟶ B̃ which in these coordinates is the identity map (and so, in particular, a diffeomorphism).

Now since U_{p} is transversal to S_{q} we know that:

(14.7)  (U_{p} ∩ S_{q}) ∩ V_{q} ≅ C(U_{p} ∩ A_{0})

where U_{p} ∩A_{0} is a compact submanifold of codimension λ_{p} in A_{0}, and where for any subset Y ⊂ A_{0}, C(Y) is the cone on Y defined by

C(Y) = {(tu, 0) : u ∈ Y and 0 ≤ t ≤ 1}

Now transversality also tells us that in a neighborhood of

Y ≡ U_{p} ∩ A_{0}

the set U_{p} ∩ A is of the form Y × ^{n−λ_{q}}. Furthermore, U_{p} ∩ A has a “smooth proper transform” to Ã, i.e., the lift of U_{p} ∩ (A − A_{0}) to Ã has closure which is a smooth manifold with boundary diffeomorphic to Y × S^{n−λ_{q}−1}. Denote this closure by Ũ_{p} ∩ A.

We can now describe the structure of \overline{U}_{p} in our coordinate box. To begin we observe that

\overline{U_{p} ∩ B} = π_{B}Φ̃(Ũ_{p} ∩ A)

and recall that Φ̃ is a diffeomorphism. It follows that \overline{U_{p} ∩ B} is a C^{∞} stratified set with two strata. The top stratum is U_{p} ∩ B. The singular stratum is exactly B_{0} ≅ S^{n−λ_{q}−1}. In a neighborhood of this singular stratum, \overline{U_{p} ∩ B} is diffeomorphic to C(Y) × S^{n−λ_{q}−1} where Y = U_{p} ∩ A_{0}.

Since \overline{U_{p} ∩ B} is the image of a manifold of finite (ℓ − 1)-volume (namely Ũ_{p} ∩ A) under the smooth proper map π_{B} ∘ Φ̃, it follows that \overline{U_{p} ∩ B} has finite (ℓ − 1)-volume.

Conclusion 14.6 The closure of U_{p} ∩ ∂^{−}Q_{k+1} is a compact C^{∞}-stratified set of finite (ℓ − 1)-volume, whose top stratum is U_{p} ∩ ∂^{−}Q_{k+1} and whose singular strata are exactly the spheres U_{q} ∩ ∂^{−}Q_{k+1} for q ∈ Cr(f, a_{k+1}).

From the above analysis we can also conclude that U_{p} has finite volume in V_{q}. Set

Ṽ_{q} = S^{λ_{q}−1} × S^{n−λ_{q}−1} × [0, 1] × [0, 1]

and consider the map Π : Ṽ_{q} → V_{q} given by

Π(û, v̂, s, t) = (sû, tv̂)

where |û| = |v̂| = 1. We identify Ã with the subset where s = 1 (and B̃ with t = 1). Then Π restricts to be the projections π_{A} and π_{B}.

Consider the mapping

Ψ : Ṽ_{q} ⟶ Ã

given by

Ψ(û, v̂, s, t) = (û, v̂, 1, st).

Then Ψ has the following two important properties. Let M ⊂ Ã be a closed submanifold with ∂M ⊂ ∂Ã which is transversal to ∂_{0}Ã = S^{λ_{q}−1} × S^{n−λ_{q}−1} × {1} × {0}. Then

Ψ is strongly transversal to M (See below),

Π (Ψ^{−1}(M)) = V_{q} ∩ ⋃_{t≥0} φ_{t} (π_{A}M)

By strong transversality we mean that if we extend M beyond ∂_{0}Ã by adding a collar, and if we extend Π to S^{λ_{q}−1} × S^{n−λ_{q}−1} × (−ϵ, 1] × (−ϵ, 1] using the same algebraic formula, then the extended Π is transversal to the extended M. Hence, (Π^{−1}(M), ∂Π^{−1}(M)) is a smooth submanifold with corners neatly embedded into (Ṽ_{q}, ∂Ṽ_{q}).

In particular, if dim(M) = ℓ−1, thenΨ^{−1}(M) has finite ℓ-volume and so does Π(Ψ^{−1}(M))

Remark 14.7. Note that the above comments also apply if M is a compact C^{∞} stratified set in Ã which in a collar neighborhood ∂_{0}Ã × [0, ϵ) of ∂_{0}Ã is of the form M_{0} × [0, ϵ).

We have now passed the first critical level beyond a_{k} and the closure of U_{p} has exited as a compact C^{∞} stratified set, whose singularities are the submanifolds U_{q} for q ≻ p and q ∈ Cr(f, a_{k+1}).

We now compound the process. The product structure (14.4) shows that U_{p} ∩ Q_{k+1} has finite ℓ-volume (since it is a product of a smooth stratified set of finite (ℓ − 1)-volume with [0, 1]). For similar reasons, U_{p} ∩ R_{k+2} has finite ℓ-volume, and it remains to examine what happens as we pass through the canonical coordinate boxes of critical points q at level k + 2.

At points of U_{p} ∩ S_{q} for q ∈ Cr(f, a_{k+2}), the above analysis can be applied locally. However, to prove finite volume one must also consider points x ∈ \overline{U}_{p} ∩ S_{q} where \overline{U}_{p} is singular. At such points x the singular set of \overline{U}_{p} ∩ S_{q} consists of all points of U_{q′} in a neighborhood of x, for some q′ ≺ q. Now U_{q′} is transversal to S_{q}, and \overline{U}_{p} is locally of the form C(Y) × U_{q′} for some manifold Y. Consequently \overline{U}_{p} ∩ S_{q} is locally of the form C(Y) × ^{i} and \overline{U}_{p} is locally of the form C(Y) × ^{i} × ^{n−λ_{q}} for some i > 0. In particular, \overline{U}_{p} ∩ A has a smooth proper transform to Ã. This proper transform has a neat collar structure at the boundary as discussed in 14.7.

Applying the analysis above one concludes that U_{p} ∩∂^{−}Q_{k+2} is a compact C^{∞} stratified set of finite ℓ-volume whose singular strata consist precisely of the sets U_{q} ∩ ∂^{−}Q_{k+2} for critical points q ≻ p of level ≤ k + 2. The same analysis also shows (using Remark 14.7) that U_{q} has finite ℓ-volume in a neighborhood of each critical point of level k + 2.

One can now proceed inductively through the critical levels to prove that U_{p} has finite ℓ-volume.

Since S_{p} is the unstable manifold for -grad(f) = grad(−f), we have also proved that each S_{p} has finite volume.

It remains to prove that the graph T has finite volume. For this we first observe that the arguments above apply directly to prove the following result. We say that the gradient flow of a smooth function F with non-degenerate critical manifolds is tame in a neighborhood of each critical point there are coordinates (x, u, v) such that ∇F = (0, −2u, 2v).

Proposition 14.8. Let Φ_{t} be a tame gradient flow of a proper function F : X → with non-degenerate critical manifolds. Let c be a non-critical value of F and suppose that Σ ⊂ {F ≤ c} is defined as the backward time image of a compact manifold Σ_{0} ⊂ {F = c}. Suppose that Σ is transversal to all the unstable manifolds of the flow. Then Σ has finite volume.

We now consider the gradient flow

Φ_{t}(s, x, y) = (e^{t}s, φ_{−t}(y), x)

of the function

F (s, x, y) = \frac{1}{2}s^{2} − f(y)

on × X × X (with metric ds^{2} on ). The critical set of F consists of the non-degenerate critical manifolds

P_{p} = {0} × {p} × X

for p ∈ Cr(f). The stable and unstable manifolds at P_{p} are given by

S_{p} = {0} × U_{p} × X  and  U_{p} = × S_{p} × X.

We consider the invariant manifold

T̃ ≡ {(e^{−s}, φ_{s}(x), x) : −∞ < s < ∞ and x ∈ X}

and the subdomain

T ≡ {(e^{−s}, φ_{s}(x), x) : 0 ≤ s < ∞ and x ∈ X} .

Note that T̃ is merely the union of orbits passing through Δ = {(1, x, x) : x ∈ X} and

T = ⋃_{s≤0} Φ_{s}(Δ)

is the union of the backward time orbits of Φ which begin at Δ. Recall (cf. Remark 1.6) that vol(T ) < ∞ ⇒ vol(T) < ∞. Thus to complete the proof w shall prove vol(T ) < ∞ by applying Proposition 14.8.

To begin, a straightforward check verifies that

(14.10)  T̃ is transversal to S_{p} and to U_{p} for all p.

Now the intersection

T̃(c) =^{def} T̃ ∩ {F = c},

is always a smooth submanifold since T̃ is Φ_{t}-invariant, i.e., grad(F |_{T}) = grad(F)|_{T} ≠ 0 on T̃. Furthermore, if c > max|f|, then T̃(c) is compact. To see this note that T̃(c) is not compact iff there exists a sequence (s_{j}, x_{j}) ∈ × X with

F (e^{s_{j}}, x_{j}, φ_{s_{j}}(x_{j})) = \frac{1}{2}(e^{2s_{j}} − f(φ_{−s_{j}}(x)) = c

for all j and for which there is no convergent subsequence. Clear for such a sequence we have s_{j} → −∞ and so e^{2s_{j}} → 0 implying c ≤ max|f| as claimed. It therefore follows from Proposition 14.8 that for any c > max|f|, the submanifold T̃_{≤c} ≡ T̃ ∩ {F ≤ c} has finite volume in × X × X.

Since max_{T} F = \frac{1}{2} + max(f), we have T ⊂ T̃_{≤c} for all c > max|f| + \frac{1}{2}. Thus T has finite volume, and the proof is complete.

Proof of Theorem 14.4. The following proof is inspired by arguments of Milnor given for a similar result [M_{2}, §4]. Consider the block decomposition (14.3) and the product structure 14.4 given by the flow. Proceeding in order from k = 1 we shall modify the metric on each subset

∂^{−}Q_{k} × (\frac{1}{3}, \frac{2}{3}) ⊂ ∂^{−}Q_{k} × [0, 1]

so that under the new gradient flow (which agrees with the old one outside ∂^{−}Q_{k} ×(\frac{1}{3}, \frac{2}{3})), the unstable manifolds entering ∂^{−}Q_{k} × {0} become transversal to the stable manifolds at ∂^{−}Q_{k} × {1}. By invariance under the flow this implies that each unstable manifold which meets Q_{k} is transversal everywhere on X to each stable manifold which meets Q_{k}. Modifying the metric at level k does not change the unstable manifolds below ∂^{−}Q_{k}. It also does not change the stable manifolds which originate below level k. From this one sees that after successively modifying the metric at level k for k = 1, ..., m − 1, we have established the result.

We now show how to modify the metric. Fix a level k and consider the submanifolds

M_{p} = U_{p} ∩ ∂^{−}Q_{k}  and  N_{q} = S_{q} ∩ ∂^{−}Q_{k}

for p, q ∈ Cr(f). We want to change the metric over ∂^{−}Q_{k} × (\frac{1}{3}, \frac{2}{3}) so that after pushing each M_{p} × {0} forward by the new gradient flow, it becomes transversal to N_{q} × {1} for all q. We shall do this as follows. We shall construct a family of deformations of the metric, smoothly parameterized by an open subset U ⊂ ^{N} for some N. This family will induce a smooth mapping

U × (∂^{−}Q_{k} × {0}) ⟶^{Θ} ∂^{−}Q_{k} × {1}

such that each Θ_{u}(•) ≡ Θ(u, •) is the diffeomorphism induced by the new gradient flow (in the product structure of the old gradient flow).

By Sard’s Theorem for Families it will suffice to prove that Θ|_{U×M_{p}} is transversal to N_{q} for all p, q (since then for almost all choices of u ∈ U, we have that Θ_{u}|_{M_{p}} is transversal to N_{q} for each p, q). This condition will follow automatically if we show that

(14.11)  \frac{∂Θ}{∂u} is surjective at all points of U × ∂^{−}Q_{k}

In fact it will suffice to show that there exists u_{0} ∈ U such that

(14.12)  \frac{∂Θ}{∂u} is surjective at all points of \{u_{0}\} × ∂^{−}Q_{k}

for then, by the compactness of ∂^{−}Q_{k}, condition (14.11) will hold with U replaced by a small neighborhood of u_{0} in U.

We first construct such a map locally on ∂^{−}Q_{k}. Let x = (x_{1}, ..., x_{n−1}) : O ⟶^{≈} ^{n−1} be a local coordinate chart on Q_{k} (which maps onto ^{n−1}). We then have coordinates (x, t) ∈ ^{n−1} × [0, 1] for ∂^{−}Q_{k} × [0, 1] where f(x, t) = t. In these coordinates the given riemannian metric has the form

⟨·, ·⟩_{(x,t)} = (·, ·)_{(x,t)} + dt^{2}

where (·, ·) is a family of inner products on ^{n−1}.

Let 𝔐 denote the space of symmetric n × n-matrices. Fix a non-trivial smooth function ψ : [0, 1] → [0, 1] with support in (\frac{1}{3}, \frac{2}{3}). For each A ∈ 𝔐 with eigenvalues of absolute value < 1, we define a new metric ⟨·, ·⟩_{A} by

(14.13)  ⟨·, ·⟩_{A} = ⟨·, ·⟩ + ψ⟨A(·), ·⟩.

Let V = V (A) be the gradient of f in this new metric. Write V = V′ + V_{0}∂/∂t where V′ is tangent to ^{n−1}. Then for any vector field W = W′ + W_{0}∂/∂t we have that

W · f = ⟨V, W ⟩_{A} == ⟨∂/∂t, W ⟩ = ⟨(I + ψ(t)A)V, W ⟩,

which implies that

\frac{∂}{∂t} = e_{n} = (I + ψ(t)A)V.

Our map Θ is given by taking the ^{n−1}-component of the integral of the vector field V = (I + ψ(t)A)^{−1}e_{n}. We write the integal of V as (Θ_{A}(x, s), τ_{A}(x, s)) with respect to the decomposition ^{n−1} ×[0, 1] ⊂ ^{n−1} × . Since V is translation invariant in the x-variables, we have

τ_{A}(x, s) = τ_{A}(s) and Θ_{A}(x, s) = Θ_{A}(s) + x

where Θ_{A}(s) = Θ_{A}(0, s). Note that

\frac{d}{ds}τ_{A}(s) = n-component of (I + ψ(τ_{A}(s))A)^{−1} · e_{n}
= (n, n)-component of ∑_{k=0}^{∞} ψ(τ_{A}(s))^{k}A^{k}.

Similarly

\frac{d}{ds}Θ_{A}(s) = pr \{(I + ψ(τ_{A}(s))A)^{−1} · e_{n}\}

where pr : ^{n} → ^{n−1} is the linear projection. Hence for all A sufficiently small,

\begin{aligned}Θ_{A}(1) & = pr ∫_{0}^{1}(I + ψ(τ_{A}(s))A)^{−1} · e_{n} ds \\ & = pr ∫_{0}^{1} ∑_{k=0}^{∞} ψ(τ_{A}(s))^{k}A^{k} · e_{n} ds. \end{aligned}

Therefore, for each i, j with 1 ≤ i, j ≤ n we have

\frac{∂Θ_{A}}{∂A_{i,j}} |_{A=0} = pr ∫_{0}^{1} ∑_{k=0}^{∞} \frac{∂}{∂A_{i,j}} ψ(τ_{A}(s))^{k}A^{k} · e_{n} ds|_{A=0}

= pr ∫_{0}^{1} {−ψ(τ_{0}(s))E_{i,j}} · e_{n} ds

= − ∫_{0}^{1} ψ(s) ds \{ \begin{matrix}E_{in} & if i > j = n \\ 0 & otherwise. \end{matrix}

where E_{ij} is the elementary (i, j)-matrix. This proves the surjectivity of ∂Θ/∂A in this coordinate system.

We now modify this family by replacing A with ξ(x)A where ξ ∈ C_{0}^{∞}( ^{n−1}) satisfies ξ(x) = 1 for |x| ≤ 2. This family of deformations now extends trivially to all of ∂^{−}Q_{k} and agrees with the one above in a neighborhood of B_{1} = {|x| ≤ 1}.

We now choose a finite family of such local coordinate systems x^{k} : O_{k} → ^{n−1}, k = 1, ..., ν such that the open sets (x^{k})^{−1}(B_{1}) cover ∂^{−}Q_{k}. Let α_{k} = ξ_{k}A_{k}, for A_{k} ∈ 𝔐, be the global section of End(T (∂^{−}Q_{k})) defined above, and consider the deformations of the metric given by

⟨·, ·⟩_{A_{1},...,A_{ν}} = ⟨·, ·⟩ + ψ(t) ∑_{j=1}^{ν}⟨ξ_{j}A_{j}(·), ·⟩

for A^{→} = (A_{1}, ..., A_{ν}) ∈ 𝔐^{ν}. Then our calculation above shows that

\frac{∂Θ_{A^{→}}}{∂A^{→}} |_{A^{→}=0} is surjective

as desired.

Appendix: Currents and the Kernel Calculus

Let Z be a compact smooth n-dimensional manifold which is not necessarily orientable. In addition to the space E^{k}(Z), of smooth differential forms of degree k, one may consider the space Ẽ^{k}(Z), of twisted smooth forms on Z of degree k. These are sections of the bundle Λ^{k}T^{∗}Z ⊗_{}O_{Z} where O_{Z} is the orientation line bundle for Z. Since the transition functions for O_{Z} are _{2} = {−1, +1} ⊂ valued, exterior differentiation d is naturally defined on any twisted form α̃ ∈ Ẽ^{k}(Z). The resulting cohomology groups are the de Rham groups H^{k}(Z; O_{Z}) of Z with coefficients in the flat line bundle O_{Z}.

A basic fact is that for any twisted n-form α̃ ∈ Ẽ^{n}(Z) the integral ∫_{Z} α̃ is well defined. (This generalizes the usual definition since, if Z is oriented, then Ẽ^{k}(Z) ≅ E^{k}(Z) are identified.) Following de Rham and Schwartz [Sc] we have the following.

Definition A.1. The space of currents of degree k on Z is the topological dual space

D^{′k}(Z) =^{def} Ẽ^{n−k}(Z)′

of the space of twisted (n − k)-forms on Z.

Currents of degree k are a generalization of differential forms of degree k. In fact there is an embedding

(A.1)  E^{k}(Z) ↪ D^{′k}(Z)

which associates to α ∈ E^{k}(Z), the current defined by

(A.2)  α(β̃) ≡ ∫_{Z} α ∧ β̃,  for all β̃ ∈ Ẽ^{n−k}(Z).

Since ∫_{Z} dα ∧ β̃ = ∫_{Z} d(α ∧ β̃) − (−1)^{k} ∫_{Z} α ∧ dβ̃ = (−1)^{k−1} ∫_{Z} α ∧ dβ̃, it is natural to define the exterior derivative of a current T ∈ D^{′∗}(Z) by:

(A.3)  (dT)(β̃) = (−1)^{k−1}T (dβ̃),  for all β̃ ∈ Ẽ^{n−k}(Z).

Thus the de Rham complex of smooth forms (E^{∗}(Z), d) is a subcomplex of the de Rham complex of currents (D^{′∗}(Z), d). (In fact D^{′∗}(Z) are exactly the distributional sections of Λ^{∗}T^{∗}Z, and d the natural extension of exterior differentiation to these sections.)

Example A.2. Let S be a codimension-k submanifold of Z which has finite volume and oriented normal bundle. The identification O_{Z} |_{S}= O_{S} enables us to pull back twisted forms via the immersion i : S → Z, and so S determines a current [S] ∈ D^{′∗}(Z) by integration:

(A.4)  [S](β̃) = ∫_{S} i^{∗}(β̃),  for all β̃ ∈ Ẽ^{∗}(Z).

It is sometiimes also useful to consider the boundary operator ∂ which is defined to be the dual of d on Ẽ^{∗}(Z), i.e.

∂ = (−1)^{k−1}d  on  D^{′k}(Z).

Remark A.3. Note that in general an oriented compact submanifold does not define a current, but does define a twisted current, i.e., a linear functional on (untwisted) differential forms.

If T ∈ D^{′p}(Z) and α ∈ E^{q}(Z) then the wedge product

(A.5)  T ∧ α ∈ D^{′p+q}(Z) is defined by (T ∧ α)(β̃) = T (α ∧ β̃).

More generally, for any flat bundle E → Z we have the spaces E^{∗}(Z; E) of differential forms on Z with coefficients in E, and their extensions

(A.6)  D^{′∗}(Z; E) =^{def} E^{n−∗}(Z; E^{∗} ⊗ O_{Z})′

to currents with coefficients in E.

Next we wish to represent operators on differential forms by currents on a product space. Let Y and X be manifolds with

dim(Y) = n  and  dim(X) = m

and note that O_{Y ×X} = π_{Y}^{∗} O_{Y} ⊗ π_{X}^{∗}O_{X} where π_{Y} and π_{X} denote the projections to Y and X. Consider the space of differential forms E^{∗}(Y × X̃) =^{def} E^{∗}(Y × X; π_{X}^{∗}O_{X}) on Y × X twisted by the orientation line bundle π_{X}^{∗}O_{X}. For example (π_{Y}^{∗} α) ∧ (π_{X}^{∗} β̃) ∈ E^{∗}(Y × X̃), if α ∈ E^{∗}(Y), β̃ ∈ Ẽ^{∗}(X). The topological dual spaces

(A.7)  D^{′∗}(Ỹ × X) ≡ D^{′∗}(Y × X; π_{Y}^{∗} O_{Y}) ≡ E^{n+m−∗}(Y × X̃)′,

is called the space of kernels for operators from E^{∗}(Y) to D^{′∗}(X). Each kernel K ∈ D^{′n−r}(Ỹ × X) determines an operator K : E^{∗}(Y) → D^{′∗−r}(X) by setting

(A.8)  K(α)(β̃) = K(π_{Y}^{∗} α ∧ π_{X}^{∗} β̃).

Since π_{X}^{∗} : Ẽ^{∗}(X) → E^{∗}(Y × X̃) the dual map (π_{X})_{∗} : D^{′∗}(Ỹ × X) → D^{′∗}(X) pushes forward kernels on Y × X to currents on X. Now the right hand side of (A.7) can be rewritten as ((π_{X})_{∗}(K ∧ π_{Y}^{∗} α))(β̃)), so that

(A.8)′  K(α) = (π_{X})_{∗}(K ∧ π_{Y}^{∗} α)

provides an alternate “pull-push” definition of K.

Proposition A.4. Suppose K ∈ D^{′n−r}(Ỹ × X) is a kernel whose operator K lowers degree by r. Then the kernel ∂K ∈ D^{′n−r+1}(Ỹ × X), determines the operator

(A.10)  K ∘ d + (−1)^{r−1}d ∘ K.

Proof. By definition (∂K)(π_{Y}^{∗} α ∧ π_{X}^{∗} β̃) = K(d(π_{Y}^{∗} α ∧ π_{X}^{∗} β̃) which equals K(π_{Y}^{∗} dα ∧ π_{X}^{∗} β̃)+(−1)^{deg α}K(π_{Y}^{∗} α∧π_{X}^{∗}dβ̃). But (K(dα))(β̃) = K(π_{Y}^{∗} dα∧π_{X}^{∗} β̃), and (d(K(α)))(β̃) ≡ (−1)^{deg α−r−1}(K(α))(dβ̃) = (−1)^{deg α−r−1}K(π_{Y}^{∗} α ∧ π_{X}^{∗}dβ̃).

Now we list some examples of kernels and their corresponding operators.

Example A.5. Let Σ ⊂ Y × X be a finite volume submanifold of codimension-q with a given isomorphism π_{X}^{∗}O_{X} |_{Σ}≅ O_{Σ}. Then integration of forms over Σ defines a current [Σ] ∈ D^{′q}(Ỹ × X).

Example A.6. (The Identity) The identity operator I : E^{∗}(X) ⟶ E^{∗}(X) on forms is represented by the kernel I = [Δ], corresponding to integration over the diagonal Δ ⊂ X × X, since α(β̃) ≡ ∫_{X} α ∧ β̃ = ∫_{Δ} π_{Y}^{∗} α ∧ π_{X}^{∗} β̃ = [Δ](π_{Y}^{∗} α ∧ π_{X}^{∗} β̃).

Example A.7. (The Pull Back) Suppose φ : X ⟶ Y. Then the operator P_{φ} α ≡ φ^{∗}α is represented by the kernel P_{φ} = [graph φ]; that is

P_{φ} = (π_{X})_{∗}([graph φ] ∧ π_{Y}^{∗} α).

Example A.8. (Projection onto ψ along φ̃). Suppose ψ ∈ E^{∗}(X) and φ̃ ∈ Ẽ^{∗}(Y). Then K = π_{Y}^{∗} φ̃ ∧ π_{X}^{∗}ψ ∈ D^{′∗}(Ỹ × X) is a kernel inducing the operator

K(α) = ± (∫_{Y} φ̃ ∧ α) ψ,

where ± equals (−1)^{(deg ψ)(deg α)}. Note K(π_{Y}^{∗} α ∧ π_{X}^{∗} β̃) = ∫_{Y ×X} π_{Y}^{∗} φ̃ ∧ π_{X}^{∗}ψ ∧ π_{Y}^{∗} α ∧ π_{X}^{∗} β̃.

Example A.9. (Projection onto [S] along [U]). Suppose S is a submanifold with oriented normal bundle in X and [U] is an oiented submanifold of Y. Let [S] denote the current in X and [U] the twisted current in Y determined by integration. Then K = [U] × [S] ∈ D^{′∗}(Ỹ × X) is a kernel and the corresponding operator is K(α) = ± (∫_{U} α) [S], where ± equals (−1)^{(deg S)(deg α)} = (−1)^{(dimY −dimS)(dimU)}.

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