FINITE VOLUME FLOWS and MORSE THEORY

by

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

§0. Introduction. In this paper we present a new approach to Morse theory based on the de Rham - Federer theory of currents. The full classical theory is derived in a transparent way. The methods carry over uniformly to the equivariant and the holomorphic settings. Moreover, the methods are substantially stronger than the classical ones and have interesting applications to geometry. They lead, for example, to formulas relating characteristic forms and singularities of bundle maps.

The ideas came from addressing the following.

Question. Given a smooth flow φ_{t} : X → X on a manifold X, when does the limit

P(α) ≡ lim_{t→∞} φ_{t}^{∗}α

exist for a given smooth differential form α?

We do not demand that this limit be smooth. As an example consider the gradient flow of a linear function restricted to the unit sphere S^{n} ⊂ ^{n+1}. For any n-form α, P(α) = c[p] where c = ∫_{S^{n}} α and [p] is the point measure of the bottom critical point p.**

Our key to the problem is the concept of a finite volume flow – a flow for which the graph T of the relation x ≺ y, defined by the forward motion of the flow, has finite volume in X × X. Such flows are abundant. They include generic gradient flows, flows for which the graph is analytic (e.g., algebraic flows), and any flow with fixed points on S^{1}.

We show that for any flow of finite volume, the limit P exists for all α and defines a continuous linear operator

P : E^{∗}(X) ⟶ D′^{∗}(X)

from smooth forms to generalized forms , i.e., currents. Furthermore, this operator is chain homotopic to the inclusion I : E^{∗}(X) ↪ D′^{∗}(X). More precisely using the graph T we shall construct a continuous operator T : E^{∗}(X) ⟶ D′^{∗}(X) of degree -1 such that

(FME)  d ∘ T + T ∘ d = I − P.

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

Now let f : X → be a Morse function with critical set Cr(f), and suppose there is a riemannian metric on X for which the gradient flow φ_{t} of f has the following properties:

(i) φ_{t} is of finite volume.

(ii) The stable and unstable manifolds, S_{p}, U_{p} for p ∈ Cr(f), are of finite volume in X. (iii) p ≺ q ⇒ λ_{p} < λ_{q} for all p, q ∈ Cr(f), where λ_{p} denotes the index of p and where

p ≺ q means there is a piecewise flow line connecting p in forward time to q.

*The research of both authors was partially supported by the NSF **When α has compact support in ^{n} = S^{n} − {∞} this simple example is the useful “approximate identity” argument widespread in analysis.

Metrics yielding such gradient flows always exist. In fact Morse-Smale gradient systems have these properties [HL_{4}]. Under the hypotheses (1)–(3) we prove that the operator P has the following simple form

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

where ∫_{U_{p}} α = 0 if deg α ≠ λ_{p}. Thus P gives a retraction

P : E^{∗}(X) ⟶ S_{f} =^{def} span_{}{[S_{p}]}_{p∈Cr(f)}

onto the finite dimensional subspace of currents spanned by the stable manifolds of the flow. Using (FME) we see that S_{f} is d-invariant and show that H^{∗} (S_{f}) ≅ H_{de Rham}^{∗}(X). This immediately yields the strong Morse inequalities.

The restriction of d to S_{f} has the form

d[S_{p}] = ∑_{λ_{q}=λ_{p}−1} n_{pq}[S_{q}]

where, by a basic result of Federer, the constants n_{pq} are integers. One concludes that S_{f}^{Z} ≡ span_{Z} {[S_{p}]}_{p∈Cr(f)} is a finite-rank subcomplex of the integral currents I_{∗}(X) whose inclusion into I_{∗}(X) induces an isomorphism H(S_{f}^{Z}) ≅ H_{∗}(X; ). In the Morse-Smale case the constants n_{pq} can be computed explicitly by counting flow lines from p to q. This fact follows directly from Stokes’ Theorem [La].

The operator T above can be thought of as the fundamental solution for the de Rham complex provided by the Morse flow. It is the Morse-theoretic analogue of the Hodge operator d^{∗}∘G where G is the fundamental solution for the Laplacian (the Greens operator). The equation (FME) will be referred to as the fundamental Morse equation. As noted above it naturally implies the two homology isomorphisms (over and ) which together encapsule Morse Theory. The proof of this equation employs the kernel calculus of [HP] to convert current equations on X × X to operator equations.

The method introduced here has many applications. It was used in [HL_{2}] to derive a local version of a formula of MacPherson [Mac_{1,2}] which relates the singularities of a generic bundle map A : E → F to characteristic forms of E and F. However, the techniques apply as well to a significantly larger class of bundle maps called geometrically atomic. These include every real analytic bundle map as well as generic direct sums and tensor products of bundle maps. New explicit formulas relating curvature and singularities are derived in each case [HL_{3}]. This extends and simplifies previous work on singular connections and characteristic currents [HL_{1}].

The approach also works for holomorphic ^{∗}-actions with fixed-points on Kähler manifolds. One finds a complex analogue of the current T. General results of Sommese imply that all the stable and unstable manifolds of the flow are analytic subvarieties. One retrieves classical results of Bialynicki-Birula [BB] and Carrell-Lieberman-Sommese [CL],[CS]. The approach also fits into MacPherson’s Grassmann graph construction and the construction of transgression classes in the refined Riemann-Roch Theorem [GS].

The method has many other extensions. It applies to the multiplication and comultiplication operators in cohomology whose kernel is the triple diagonal in X × X × X (§11). This idea can be further elaborated as in [BC]. Similarly, the method fits into constructions of invariants of knots and 3-manifolds from certain “Feynman graphs” (cf. [K], [BT]).

The cell decomposition of a manifold by the stable manifolds of a gradient flow is due to Thom [T], [S]. That these stable manifolds embed into the de Rham complex of currents was first observed by Laudenbach [La].

The authors are indebted to Janko Latschev for many useful comments. Notation. For an n-manifold X, E^{k}(X) will denote the space of smooth k-forms on X, and D′^{k}(X) the space of k-forms with distribution coefficients (called the currents of dimension n − k on X). Under the obvious inclusion E^{k}(X) ⊂ D′^{k}(X), exterior differentiation d has an extension to all of D′^{k}(X), and d = (−1)^{k+1}∂ where ∂ denotes the current boundary. (See [deR] for full definitions.) We assume here that X is oriented, but this assumption is easily dropped and all results go through.

§1. Finite Volume Flows. Let X be a compact oriented manifold of dimension n, and let φ_{t} : X → X be the flow generated by a smooth vector field V on X. For each t > 0 consider the compact submanifold with boundary

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

oriented so that

∂T_{t} = {0} × Δ − {t} × P_{t}

where Δ ⊂ X × X is the diagonal and P_{t} = [graph(φ_{t})] = {(φ_{t}(x), x) : x ∈ X} is the graph of the diffeomorphism φ_{t}. Let pr : × X × X → X × X denote projection and consider the push-forward

T_{t} ≡ (pr)_{∗}(T_{t}).

of T_{t} as a de Rham current. Since ∂ commutes with (pr)_{∗}, we have that

∂T_{t} = [Δ] − P_{t}

Note that the current T_{t} can be equivalently defined by

T_{t} = Φ_{∗}([0, t] × X)

where Φ : × X → X × X is the smooth mapping defined by Φ(s, x) = (φ_{s}(x), x). This mapping is an immersion exactly on the subset × (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 non-negative symmetric tensor which is > 0 exactly on the subset × (X − Z(V)).

Definition 1.1. A flow φ_{t} on X is called a finite volume flow if ^{+} × (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, and let T_{t} be the family of currents defined above with

(A)  ∂T_{t} = [Δ] − P_{t}  on X × X.

Then both the limits

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

exist as currents, and taking the boundary of T gives the equation of currents

(C)  ∂T = [Δ] − P  on 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).

We shall see in the next section that Theorem 1.2 has an important reinterpretation in terms of operators on the differential forms on X.

Remark 1.3. 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 1.4. The immersion Φ : × (X − Z(V)) → X × X is an embedding outside the subset × 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) × X̃ and X̃ = 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. However, a gradient flow never has periodic points, and many such flows are of finite volume.

Note that any flow with fixed points on S^{1} has finite volume.

Remark 1.5. 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

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

Lemma 1.6. If X is analytic and T̂ ⊂ ×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.

Example 1.7. Consider S^{n} with two coordinate charts ^{n} and the coordinate change x = y/|y|^{2}. In the y-coordinates consider the translational flow φ_{t}(y) = y + tu where u ∈ ^{n} is a fixed unit vector. In the x-coordinates this becomes

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

To see that φ_{t} is finite volume flow let r = 1/t and note that T̂ is defined by algebraic equations so that Lemma 1.6 is applicable. The vector field V has only one zero and is therefore not a gradient vector field. It is however a limit of gradient vector fields.

§2. The Operator Equations. The current equations (A), (B), (C) of Proposition 1.2 can be translated into operator equations by a kernel calculus which was introduced in [HP] and which we shall now briefly review. For clarity of exposition we assume our manifolds to be orientable. However, the discussion here extends straightforwardly to the non-orientable case and to forms and currents with values in a flat bundle.

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

K(α) = (π_{X})_{∗}(K ∧ π_{Y}^{∗} α),

i.e., for a smooth form β of appropriate degree on X

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

Example 2.1. Suppose φ : X → Y is a smooth map, and let

P_{φ} = {(φ(x), x) : x ∈ X} ⊂ Y × X.

be the graph of φ with orientation induced from X. Then P_{φ} is just the pull-back operator P_{φ}(α) ≡ φ^{∗}(α) on differential forms α. In particular if X = Y and K = [Δ] ⊂ X × X is the diagonal (the graph of the identity), then K = I : E^{∗}(X) → D′^{∗}(X) is the standard inlcusion of the smooth forms into the currents on X.

Lemma 2.2. Suppose an operator T : E^{∗}(Y) → D′^{∗}(X) has kernel T ∈ D′^{∗}(Y × X). Suppose that T lowers degree by one, or equivalently, that deg(T) = dimY − 1. Then

(2.1)  The operator d ∘ T + T ∘ d has kernel ∂T.

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

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

since T(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}]

T T

d ∘ T + T ∘ d  ∂T

From this table Theorem 1.2 can be reformulated as follows.

Theorem 2.3. Let φ_{t} be a finite volume flow on a compact manifold X, and let P_{t} : E^{∗}(X) → E^{∗}(X) be the operator given by pull-back

P_{t}(α) ≡ φ_{t}^{∗}(α).

Let T_{t} : E^{k}(X) → E^{k−1}(X) be the family of operators associated to the currents T_{t} defined in §1. Then for all t one has that

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

Furthermore, the limits

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

exist as operators from forms to currents, and they satisfy the equation

(C)  d ∘ T + T ∘ d = I − P.

§3. 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

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

Fix a riemannian metric g on X and let φ_{t} denote the flow associated to ∇f. Suppose that at p ∈ Cr(f) there exists a canonical coordinate system (u, v) in which g = |du|^{2}+|dv|^{2}. Then in these coordinates the gradient flow is given by

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

Metrics g with this property at every p ∈ Cr(f) will be called f-tame.

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

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

If g is f-tame, then 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

(3.3)  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 contractible submanifolds (but not closed subsets) of X with

(3.4)  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 S_{p} via the splitting T_{p}X = T_{p}U_{p} ⊕ T_{p}S_{p}.

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 3.4 that this relation is simply the closure of the partial ordering of Remark 1.3.

Definition 3.1. The gradient flow of a Morse function f on a riemannian manifold X is called Morse-Stokes if the metric is f-tame and if:

(i) The flow 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 3.2. 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 metric are dense in the space of all f-tame metrics on X. See [HL_{4}].

Theorem 3.3. Let f ∈ C^{∞}(X) be a Morse function on a compact riemannian manifold whose gradient flow is Morse-Stokes. Then the current P in the fundamental equation

(3.5)  ∂T = [Δ] − P

from Theorem 1.2 is given by

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

where U_{p} and S_{p} are oriented so that U_{p} × S_{p} agrees in orientation with X at p.

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

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

Lemma 3.4. 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} exists 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 implies 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 define Σ′ = Σ ∪ {(p, p) : p ∈ Cr(f)}. Note that from (3.4) and Axiom (iii) in 3.1 that

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

Lemma 3.5. Consider the embedded submanifold

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

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

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

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 3.4 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 (3.2) 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 3.5 has the following immediate consequence

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

We now apply the following elementary but key result of Federer.

Proposition 3.6. ([F, 4.1.15]) Let [W] be a current in ^{n} defined by integration over a k-dimensional oriented submanifold W of locally finite volume. Suppose supp (d[W]) ⊂ ^{ℓ}, a linear subspace of dimension ℓ < k − 1. Then d[W] = 0.

Combining this with (3.7) and (3.8) proves (3.6) and completes the proof of Theorem 3.3.

§4. Morse Theory. Combining Theorems 2.3 and 3.3 gives the following.

Theorem 4.1. 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

(4.1)  d ∘ T + T ∘ d = I − P

Note that P : E^{∗}(X) ⟶ D′^{∗}(X) maps onto the finite-dimensional subspace

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

and that from (4.1) we have

P ∘ d = d ∘ P.

Theorem 4.2. 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 byP induces an isomorphism

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

Proof. It follows immediately from Theorem 4.1 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 examples show). 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 3.1(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}_{p} = \{p\}. Now let S = ∑ r_{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}) = r_{q} for all q and so P(γ) = S as claimed.

The vector space S_{f}^{∗} has a distinguished lattice

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

generated by the stable manifolds. Note that S_{f}^{}is a subgroup of the integral currents I(X) on X.

Theorem 4.3. The lattice S_{f}^{}is preserved by d, i.e., (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. Theorem 4.2 implies that for any p ∈ Cr(f) we have

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

for real numbers n_{p,q}. In particular, d[S_{p}] has finite mass, and so by [F; 4.2.16(2)] it is a rectifiable current. This implies that n_{p,q} ∈ for all p, q, and the first assertion is proved.

Now the domain of the operator P extends to include any C^{1} k-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; ). Furthermore they include S_{f}^{}. The result then follows from (4.1).

Corollary 4.4. Let G be a finitely generated abelian group. Then there are natural isomorphisms

H(S_{f}^{}⊗_{}G) ≅ H_{∗}(X; G).

The first assertion of Theorem 4.3 has a completely elementary proof whenever the flow is Morse-Smale, i.e., when 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. (3.1) ), 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. We then define

(4.4)  N_{p,q} =^{def} ∑_{γ∈Γ_{p,q}} n_{γ}.

As in [La] Stokes’ Theorem now directly gives us the following (cf. [W]).

Proposition 4.5. When the gradient flow of f is Morse-Smale, the coefficients in (4.3) are given by

n_{p,q} = (−1)^{λ_{p}} N_{p,q}

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 4.6. 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. (3.3)). Then for all r sufficiently large

(4.5)  N_{p,q} = ∫_{X}[U_{q}(r)] ∧ d [S_{p}(r)]

where the integral denotes evaluation on the fundamental class.

Next we observe that our Morse theory enables one to compute the full integral homology of X (including torsion) using smooth differential forms.

Definition 4.7. Let E_{f}^{}= P^{−1}(S_{f}^{}) denote the group of smooth forms with integral residues and E_{f}^{0} ≡ ker(P) the subgroup of forms with zero residues. Note that α ∈ E_{f}^{} iff ∫_{U_{p}} α ∈ for all p ∈ Cr(f). Since P commutes with d, both E_{f}^{}and E_{f}^{0} are d-invariant, and we have a short exact sequence of chain complexes

(4.6)  0 ⟶ E_{f}^{0} ⟶ E_{f}^{}⟶^{P} S_{f}^{}⟶ 0

showing that E_{f}^{}is an extension of the lattice S_{f}^{}by the vector space E_{f}^{0}.

Proposition 4.8. The operator P induces an isomorphism P_{∗} : H(E_{f}^{}) ⟶^{≅} H(S_{f}^{}).

Proof. Note that (4.6) embeds into the short exact sequence 0 ⟶ E_{f}^{0} ⟶ E ⟶^{P} S_{f} ⟶ 0, and by 4.2 we know that P_{∗} : H(E) → H(S_{f}) is an isomorphism. Hence, H(E_{f}^{0}) = 0 and the result follows.

§5. Poincaré Duality. In this context there is a simple proof of Poincaré duality. Given two oriented submanifolds A and B of complementary dimensions in X which meet transversally in a finite number of points, let A • B = ∫_{X}[A] ∧ [B] denote the algebraic number of intersections points. Then for any k we have

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

where Cr_{k}(f) ≡ {p ∈ Cr(f) : λ_{p} = k}. 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 4.6). 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 (4.5).

Corollary 5.2. (Poincaré Duality)

H^{n−k}(X; ) ≅ H_{k}(X; ) for all k.

Note 5.3. The Poincaré duality isomorphism can be realized in our operator picture as follows. Consider the “total graph” of the flow: in X × X

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

with corresponding operator T_{tot}. Then one has 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_{∗}(S_{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. Generalizations. As seen in §2, for any flow φ_{t} of finite volume the operator P(α) = lim_{t→∞} φ_{t}^{∗}(α) exists and is chain homotopic to the identity. This situation occurs often. Suppose for example that f : X → is a smooth function whose critical set is a finite disjoint union

Cr(f) = ∐_{j=1}^{ν} F_{j}

of compact submanifolds F_{j} in X and that Hess(f) is non-degenerate on the normal spaces to Cr(f). Then for any f-tame gradient flow φ_{t} (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}}

for each 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 if 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).

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 3.3. Details are omitted.

In [L] Janko Latschev has found a Smale-type condition which yields this result in cases where when 6.1 (iii) fails. In particular, his condition implies only that: p ≺ q ⇒ λ_{p} < λ_{q}.

The result above can be translated into the following 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 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.

Each map 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 (cf. [CJS]). 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]).

Corollary 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})

One can drop the orientation assumptions by taking homology with appropriately twisted coefficients. Extensions of 6.3 and 6.5 to integral homology groups are found in [L]. Latschev also derives a spectral sequence, with geometrically computable differentials, associated to any Bott-Smale function satisfying a certain transversality hypothesis. Assuming 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; )

Remark 6.6. In standard Morse Theory one considers a proper exhaustion function f : X → and studies the change in topology as one passes from {x : f(x) ≤ a} to {x : f(x) ≤ b}. Our operator approach is easily adapted to this case (See [HL_{4}]).

§7. The local MacPherson formula and generalizations. The ideas above can be applied 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). 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). On the “affine chart” Hom(E, F), this flow has the form φ_{t}(A) = \frac{1}{t}A. (Note that here it is more natural to consider the flow multiplicatively (φ_{ts} = φ_{t} ∘ φ_{s}) than additively.)

Definition 7.1. The section α is said to be geometrically atomic if its radial span

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

has 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_{0}(Ω_{U}) where Φ_{0} is an Ad-invariant polynomial on 𝔤𝔩_{k}( ) and Ω_{U} is the curvature of the tautological k-plane bundle over G, one obtains formulas on X which relate the singularities of α to characteristic forms of E and F. For normal maps, maps which are transversal to the universal singularity sets, this process yields a local version of a classical formula of MacPherson [Mac_{1,2}]. In other cases, such as generic direct sum or tensor products mappings, the process yields new formulas. The method also applies whenever α is real analytic. Details appear in [HL_{2,3}].

§8. Equivariant Morse Theory. Our approach carries over to equivariant cohomology by using Cartan’s equivariant de Rham theory (cf. [BGV]). Suppose G is a compact Lie group with Lie algebra 𝔤 acting on a compact n-manifold X. By an equivariant differential form on X we mean a G-equivariant polynomial map α : 𝔤 ⟶ E^{∗}(X). The set of equivariant forms is denoted by

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

and is graded by declaring 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 i_{V} denotes contraction with the vector field Ṽ on X corresponding to V. The complex D_{G}^{′∗}(X) of equivariant currents is similarly defined by replacing smooth forms E^{∗}(X) by forms D^{′∗}(X) with distribution coefficients.

Consider a G-invariant function f ∈ C^{∞}(X) and suppose X is given a G-invariant riemannian metric. Suppose the (G-invariant) gradient flow has finite volume and let

(8.1)  ∂T = Δ − P

be the current equation derived in §3. Let G act on X × X by g · (x, y) = (gx, gy).

Lemma 8.1. The current T satisfies:

(i) g_{∗}T = T for all g ∈ G, and

(ii) i_{V}T = 0 for all V ∈ 𝔤.

So also does the current P.

Proof. The current T corresponds to integration over the finite volume submanifold {(x, y) ∈ X × X − Δ : y = φ_{t}(x) for some 0 < t < ∞}. Since gφ_{t}(x) = φ_{t}(gx), assertion (i) is clear. The invariance of T implies the invariance of P by (8.1). For assertion (ii) note that for any n-form β on X × X

(i_{V}T ) (β) = ∫_{T} i_{V}β = 0

since V is tangent to T. Similarly, i_{V}Δ = 0. Since d ∘ i_{V} + i_{V} ∘ d = L_{V} (Lie derivative) and L_{V}T = 0, we conclude that i_{V}P = 0.

Corollary 8.2. 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

(8.2)  ∂_{G}T = Δ − P  on X × X.

The correspondence between operators and kernels discussed in §2 carries over to the equivariant context. Currents in D_{G}^{′n−ℓ}(X × X) yield G-equivariant operators E^{∗}(X) ⟶ D^{′∗+ℓ}(X), and hence operators E_{G}^{∗}(X) ⟶ D_{G}^{′∗+ℓ}(X). Equations of type (8.2) translate into operator equations

(8.3)  d_{G} ∘ T + T ∘ d_{G} = I − P.

Applying the arguments of §4 proves the following.

Proposition 8.3. Let φ_{t} be an invariant flow on a compact G-manifold X. If φ_{t} has finite volume, then the limit

(8.4)  P(α) = lim_{t→∞} φ_{t}^{∗}α

exists for all α ∈ E_{G}^{∗}(X) 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).

The fact that S^{∗}(𝔤^{∗})^{G} ≅ H^{∗}(BG) leads as in §4 to the following result.

Theorem 8.4. Let f ∈ C^{∞}(X) be an invariant Morse function on a compact riemnannian G-manifold whose gradient flow φ_{t} is Morse-Stokes. Then the continuous linear operator (8.4) 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 (4.2) and where the differential on S^{∗}(𝔤^{∗})^{G} ⊗ S_{f} is 1 ⊗ ∂. This map induces an isomorphism

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

Examples of this phenomenon arise in moment map constructions. For a simple example consider G = (S^{1})^{n+1}/Δ 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 all generalized flag manifolds and to products.

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

However, the method applies to quite general functions and yields results as in §§6-7. 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 (the generic case). There is a spectral sequence with (assuming for simplicity that everything is oriented)

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

and computable differentials such that E_{∗,∗}^{k} ⇒ H_{G}^{∗}(X) (cf. [L]).

§9. Holomorphic Flows and the Carrell-Lieberman-Sommese Theorem. This method also applies to the holomorphic case. For example, 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

analogous to the graphs considered above. The following is a result of Sommese [So].

Theorem 9.1. If φ_{t} has fixed-points, then Tl has finite volume and its closure \overline{Tl} in 1( ) × X × X is an analytic subvariety.

The relation of ^{∗}-actions to Morse-Theory is classical. The action φ_{t} decomposes into an “angular” S^{1}-action and a radial flow. Averaging a Kähler metric over S^{1} and applying an argument of Frankel [Fr], we find a function f : X → of Bott-Morse type whose gradient generates the radial action. Theorem 9.1 implies that the associated current T on X × X, which is a real slice of Tl, is real analytic and hence of finite volume. One can then apply the methods of this paper, in particular those of §6.

Thus when φ_{t} has fixed-points, \overline{Tl} gives a rational equivalence between the diagonal Δ in X × X and an analytic cycle P whose components consist of fibre products of stable and unstable manifolds over components of the fixed-point set of the action.

When the fixed-points are all isolated, P becomes a sum of analytic Künneth components P = ∑ \overline{S}_{p} × \overline{Ũ}_{p}, and we recover the well-known fact that the cohomology of X is freely generated by the stable subvarieties \{\overline{S}_{p}\}_{p∈Zero(φ)}. 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. (See [BB], [ES], [Fr]).

When the fixed-point set has positive dimension, one can recover 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.

§10. Local coefficient systems. 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 §3 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

(10.1)  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 (10.1) 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 dual pairing of these complexes establishes Poincaré duality. Furthermore, this extends to integral currents twisted by representations of π_{1}(X) in GL_{n}(Z) or GL_{n}(Z/pZ) and gives duality with local coefficient systems.

§11. Cohomology operations. Our method has many 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) → D^{′∗}(X) of degree -1. One calculates that ∂T = Δ_{3}−M where M = ∑_{(p,p′)} [U_{p}]×[U′_{p′}]×[S_{p}∩S′_{p′}]. The corresponding operator M : E^{∗}(X) ⊗ E^{∗}(X) → D^{′∗}(X) is given by

(11.1)  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 11.1. There is an equation of operators ∧−M = d∘T+T∘d from E^{∗}(X ×X) to D^{′∗}(X) (where ∧ denotes restriction to the diagonal). In particular for α, β ∈ E^{∗}(X) we have the chain homotopy

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

Note that the operator M has range in the finite dimensional vector space M =^{def} span_{}{ [S_{p} ∩ S′_{p′}]}_{(p,p′)}. 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 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 maps onto the subspace M, and for forms α, β ∈ E^{∗}(X), it satisfies the equation

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

It follows that d(M) ⊂ M. Furthermore, for (p, p′) ∈ Cr(f) × Cr(f′) one has

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

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

A basic example of a pair satisfying these hypotheses is simply f, −f where the gradient flow is Morse-Smale. Here U′_{p} = S_{p} and S′_{p} = U_{p} for all p ∈ Cr(f), and we have:

Proposition 11.2. Suppose the gradient flow of f is Morse-Smale and Y is a cycle in X which is transversal to U_{p} ∩ S_{p′} for all p, p′ ∈ Cr(f). Then for any closed forms α, β

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

Analogous formulas are derived by the same method for the comultiplication operator. Following ideas in [BC] one can extend these constructions to other cohomology operations.

§12. Knot invariants. From the finite volume flow 1.7 our methods construct 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 [BT], and can be used to develop a combinatorial version of their study of knot invariants.

§13. Differential characters. The ring of differential characters on a smooth manifold, introduced by Cheeger and Simons [CS] in 1973, has played an important role in geometry. In deRham-Federer formulations of the theory (see [HL_{5}]), differential characters are represented by sparks. These are currents T with the property that dT = α−R where α is smooth and R is integrally flat. Our Morse Theory systematically produces such creatures.

Consider the group Z_{f}^{}= E_{f}^{}∩ker d of closed forms with integral residues (See §4). Then T : Z_{f}^{}→ D′ is a continuous map with the property that dT(α) = α − P(α) for all α. Thus T gives a continuous map of Z_{f}^{}into the space of sparks, and therefore also into the ring of differential characters on X.

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