Connection matrix for Morse decompositions

Maciej J. Capiński

IMUJ Preprint 2000/24

Abstract

A new proof of the existence of a connection matrix for Morse decompositions is presented. The matrix is constructed by extending the matrix onto larger decompositions, by using the connecting homomorphism for the attractor repeller decomposition. As a result an entire matrix is obtained, together with a family of connecting isomorphisms.

Introduction. For some isolated invariant sets there exists a decomposition into smaller invariant sets, with a defined partial order between them called the Morse decomposition. For any Morse decomposition there exists an index filtration, containing index pairs of all the Morse sets from the decomposition ([?]). Using the index filtration we are able to define a family of Conley indexes of the Morse sets. A connection matrix for a Morse decomposition is a coboundary operator defined on the product of the Conley indexes of the Morse sets. From the connection matrix we are able to reconstruct information concerning the indexes of the isolated invariant subsets of our isolated invariant set and determine the existence of connecting orbits between them. The ideas listed above were introduced by Franzosa who in [?], [?], [?], [?], presented the proof of the existence of index filtrations and a connection matrix for Morse decompositions. The connection matrix theory was also extended by Richeson [?] from the theory for flows to the theory for discrete dynamical systems.

In this paper we present a new proof of the existence of a connection matrix for flows. The proof presented by Franzosa showed the existence, but did not explicitly write what this matrix should look like. In this proof we show a clear inductive method of constructing such a matrix.

1 Preliminaries

Throughout this paper we assume that the reader is familiar with the basic concepts of the Conley index theory as in [?]; in particular the concept of isolated invariant sets, isolating neighbourhoods, index pairs, attractor repeller pairs and the Conley index. For the definition of the Conley index we shall be using the Alexander-Spanier cohomology [?]. We will carry over the notations used in [?]. Let us recall some of them.

For a topological space X and a field Ξ we write

C^{n}(X) := {φ | φ : X^{n+1} → Ξ}.

We define a coboundary operator ∂^{n} : C^{n}(X) → C^{n+1}(X)

∂^{n}φ(x_{0}, …, x_{n+1}) := ∑_{r=0}^{n+1}(−1)^{r}φ(x_{0}, …, x̂_{r}, …, x_{n+1}),

where the hat means that the corresponding x is omitted. For a covering U of X we define U^{n+1} := ⋃_{U∈U} U^{n+1}. We say that a function φ is locally equal zero if there exists a covering U of X, such that φ is zero on U^{n+1}. We define

C_{0}^{n}(X) := {φ | φ ∈ C^{n}(X) and φ is locally zero},
C̄^{n}(X) := C^{n}(X)/C_{0}^{n}(X).

On C̄^{n}(X) there exists a coboundary operator ¯∂^{n} induced by ∂^{n}, ¯∂^{n}[φ] := [∂^{n}φ]. For f : X → Y there exists an induced function f^{#} : C^{n}(Y) → C^{n}(X),

f^{#}(φ)(x_{0}, …, x_{n}) := φ(f(x_{0}), …, f(x_{n})).

If f is a continuous function then the induced function f̄^{#} : C̄^{n}(Y) → C̄^{n}(X),

f̄^{#}[φ]_{C_{0}^{n}(Y)} := [f^{#}φ]_{C_{0}^{n}(X)}

is well defined. For a topological pair (X, A) we define the group of cochains on (X, A) as

C̄^{n}(X, A) := kerī^{#},

where ī^{#} is induced by an inclusion i : A → X. We can restrict the coboundary ¯∂^{n} to kerī^{#}, by which we get a coboundary on C̄^{n}(X, A). We define the group of cycles Z^{n}(X, A) := ker¯∂^{n} and the group of boundaries B^{n}(X, A) := im¯∂^{n−1}. Finally we define the Alexander Spanier cohomology H^{n}(X, A) := Z^{n}(X, A)/B^{n}(X, A). The details of these definitions are discussed in [?]. In our case we shall be dealing with the cohomology over a field Ξ, so all the cochain and cohomology groups are vector spaces.

Let S be an isolated invariant set for a locally compact metric space X and a local flow Γ. For an isolated invariant set S and its isolating neighborhood N there exists an index pair (P_{1}, P_{0}) contained in N. We define the Conley index of S as CH^{n}(S) := H^{n}(P_{1}, P_{0}). The definition is independent from the choice of N and (P_{1}, P_{0}) ([?]).

Let P be a finite set and < a partial order on P.

Definition 1.1 M = M(P, <) = {M_{p} ⊂ S : p ∈ P } is called a Morse decomposition of S if:

1. ∀p ∈ P, M_{p} is an isolated invariant set,

2. ∀x ∈ S \ ⋃ M_{p} ∃p, q ∈ P : q < p and x ∈ C(M_{p}, M_{q}; S).

Where C(M_{p}, M_{q}; S) is the set of connecting orbits from M_{p} to M_{q} in S.

If we take (P, ≺) such that when q < p then q ≺ p, the definition of the Morse decomposition with (P, ≺) is also correct, therefore we can extend (P, <) to a linear order. From now on we will assume that P = {1, 2, …, m} and that < is the natural order. By A(<) we shall denote the set of attracting intervals, that is all the intervals I = [1, k], for some k in P. We say that the two intervals I, J are adjacent, if for every i ∈ I and j ∈ J, i < j and IJ := I ∪ J is an interval.

For any interval I in P we define

M_{I} := ( ⋃_{p∈I} M_{p}) ∪ ( ⋃_{p,q∈I} C(M_{p}, M_{q}; S)).

For any two adjacent intervals I, J the pair (M_{I}, M_{J}), is an attractor repeller decomposition of M_{IJ}. An immediate consequence of this fact is that for all I ∈ A(<) the pair (M_{I}, M_{P \I}) is an attractor repeller decomposition of M_{P} . This fact inspires the following definition, which is a natural extension of the idea of the index pair.

Definition 1.2 A collection of compact sets {N(I) | I ∈ A(<)} is called an index filtration for M if

1. for each I ∈ A(<) (N(I), N(∅)) is an index pair for M(I),

2. for each I_{1}, I_{2} in A(<), N(I_{1} ∩ I_{2}) = N(I_{1}) ∩ N(I_{2}) and N(I_{1} ∪ I_{2}) = N(I_{1}) ∪ N(I_{2}) .

Theorem 1.3 For every Morse decomposition there exists an index filtration.

The proof of this theorem is presented in [?].

For an index filtration {N(I) | I ∈ A(<)} of M we will use the following notation N_{k} := N([1, k]), N_{0} := N(∅).

Index filtrations have the following properties

1) (N_{k}, N_{i−1}) is an index pair for M_{[i,k]},

2) N_{0} ⊂ N_{1} ⊂ … ⊂ N_{n}.

We shall now introduce the definition of a connection matrix for a partially ordered Morse decomposition. First let us introduce some notations. Let

CΔ(I) := ⨁_{p∈I} CH^{∗}(M_{p}).

We can treat any linear function Δ(I) : CΔ(I) → CΔ(I) as a matrix

(Δ_{pq} : CH^{∗}(M_{q}) → CH^{∗}(M_{p}) | p, q ∈ I).

If Δ(I) is a coboundary then by HΔ^{∗}(I) we shall denote the cohomology group H^{∗}(CΔ(I), Δ(I)).

We say that a matrix (a_{k i})_{k,i∈P} is lower triangular if for every k < i, a_{k i} = 0.

Definition 1.4 A lower triangular coboundary operator Δ(P) : CΔ(P) → CΔ(P) is called a connection matrix for M(P, <) if there exists a family of isomorphisms {F^{∗}(I) : CH^{∗}(M_{I}) → H^{∗}Δ(I) | I is an interval in P } such that for any two adjacent intervals I,J in P, the diagram

… → CH^{∗}(M_{J}) → CH^{∗}(M_{IJ}) → CH^{∗}(M_{I}) → …
↓ F^{∗}(J) ↓ F^{∗}(IJ) ↓ F^{∗}(I)
… → HΔ^{∗}(J) → HΔ^{∗}(IJ) → HΔ^{∗}(I) → …

commutes.

2 Main results

Let us assume that C_{1} and C_{2} are cochain groups and that there exists a lower triangular coboundary operator ∂ = (∂_{ki})_{k,i=1,2} on C_{1} ⊕ C_{2}. The sequence

0 → C_{2} → C_{1} ⊕ C_{2} → C_{1} → 0

is exact, and there exists a connecting homomorphism δ^{∗}, such that the long sequence

… → H(C_{2}) → H(C_{1} ⊕ C_{2}) → H(C_{1}) →^{δ^{∗}} H(C_{2}) → …

is exact. The proof of the existence of δ^{∗} is presented in [?].

Lemma 2.1 For any φ ∈ C_{1} ∩ ker∂_{11}

δ^{∗}[φ]_{im∂_{11}} = [∂_{21}φ]_{im∂_{22}}

Proof.

0 → C_{2} →^{α}  C_{1} ⊕ C_{2}  →^{β} C_{1} → 0

↓ ( \begin{matrix}∂_{11} & 0 \\ ∂_{21} & ∂_{22} \end{matrix} )

0 → C_{2} →^{α}  C_{1} ⊕ C_{2}  →^{β} C_{1} → 0

The operator δ^{∗} constructed in [?] has the following property. If β(φ_{1}⊕φ_{2}) = ψ and ∂(φ_{1} ⊕ φ_{2}) = αγ then δ^{∗}[ψ] = [γ].

Take φ_{1} := φ, φ_{2} := 0, ψ := φ and γ := ∂_{21}φ. We know that φ ∈ker∂_{11}. Clearly β(φ ⊕ 0) = φ and

∂(φ ⊕ 0) = ( \begin{matrix}∂_{11} & 0 \\ ∂_{21} & ∂_{22} \end{matrix} ) ( \begin{matrix}φ \\ 0 \end{matrix} ) = ( \begin{matrix}∂_{11}φ \\ ∂_{21}φ \end{matrix} ) = 0 ⊕ ∂_{21}φ = α(∂_{21}φ)

so δ^{∗}[φ] = [∂_{21}φ]. ■ For an index filtration {N_{i}}_{i∈P} for every l, j in P such that j < l,

C̄^{n}(N_{l}, N_{j−1}) ≅ ⨁_{k=j}^{l} C̄^{n}(N_{k}, N_{k−1}).

The coboundary operator ˆ∂ on ⨁_{k=j}^{l} C̄^{n}(N_{k}, N_{k−1})

ˆ∂([j, l]) = (ˆ∂_{ki} : C̄(N_{i}, N_{i−1}) → C̄(N_{k}, N_{k−1}) | k, i ∈ [j, l])

induced from ¯∂ takes the form of a lower triangular matrix. From now on for any Morse set M_{I}, by writing CH^{n}(M_{I}) we will have in mind the cohomology

H^{n}(⨁_{k∈I} C̄(N_{k}, N_{k−1}), ˆ∂(I)).

To simplify notation, for any two intervals L, J, we shall define ˆ∂_{LJ} := (ˆ∂_{lj})_{l∈L,j∈J} and Δ_{LJ} := (Δ_{lj})_{l∈L,j∈J}.

Theorem 2.2 Let K, I be two adjacent intervals. For Φ_{I} ∈ ⨁_{i∈I} C̄^{n}(N_{i}, N_{i−1})

pr_{ker ˆ∂_{KK}} ˆ∂_{KI}Φ_{I} = ˆ∂_{KI}pr_{ker ˆ∂_{II}}Φ_{I}.

The proof of this theorem as well as the proofs of the facts listed above will be presented in Section 3.

We will now present an inductive method of constructing a connection matrix. Let (M, P) be a Morse decomposition and {N_{J}} its index filtration. Let us assume that we already have a matrix Δ(I) for a given interval I = [1, k − 1], let us also assume that we have a family of cochain morphisms

{F_{J} : ⨁_{i∈J} C̄(N_{i}, N_{i−1}) → CΔ(J) | J is an interval in I}

which in cohomology induces the family of connecting isomorphisms {F_{J}^{∗} : CH(M_{J}) → HΔ(J)}. We will show how to construct the matrix Δ(Ik).

We already have the matrix Δ(I) : CΔ(I) → CΔ(I). We know that a connection matrix has to be a lower triangular matrix, so what we are looking for is a homomorphism

Δ_{kI} : CΔ(I) → CH(M_{k}).

We know that (M_{k}, M_{I}) is an attractor repeller decomposition of M_{Ik}. The cochain group on M_{k} is equal C̄(N_{k}, N_{k−1}) and the cochain group on M_{I} is isomorphic to ⨁_{i∈I} C̄(N_{i}, N_{i−1}). There exists a connecting homomorphism δ^{∗} such that

… → CH^{n}(M_{k}) → CH^{n}(M_{Ik}) → CH^{n}(M_{I}) →^{δ^{n}} CH^{n+1}(M_{k}) → …

is exact. The main idea of this paper is to use this homomorphism to construct Δ_{kI}. The idea for our construction is presented in the following diagram

CΔ(I) …^{Δ_{kI}} CH(M_{k})
↓  ↑ δ^{∗}
HΔ(I) →^{(F_{I}^{∗})^{−1}} CH(M_{I})

The operator Δ_{kI} will be defined as

Δ_{kI}(X_{I}) := δ^{∗} ∘ (F_{I}^{∗})^{−1}[pr_{kerΔ(I)}X_{I}]_{imΔ(I)}.

Since we shall be dealing with various kinds of classes of abstractions and with various cochain groups, before stating our main theorems let us somewhat clarify our notations. By letters X_{J}, Y_{J}, we shall denote the elements of CΔ(J), by x_{j}, y_{j}, z_{j} the elements of CH(M_{j}), by Φ_{J}, Ψ_{J}, the elements of ⨁_{j∈J} C̄(N_{j}, N_{j−1}) and by φ_{j}, ψ_{j}, ξ_{j} we will denote the elements of C̄(N_{j}, N_{j−1}). The elements of CH(M_{J}) will be denoted by [Φ_{J}], [Ψ_{J}], and the elements of HΔ(J) by [X_{J}], [Y_{J}].

Theorem 2.3 The function

Δ(Ik) := ( \begin{matrix}Δ(I) & 0 \\ Δ_{kI} & 0 \end{matrix} )

is a coboundary operator and HΔ(Ik) ≅ CH(M_{Ik}).

Proof. First we have to show that Δ(Ik) is a coboundary operator.

Δ(Ik)^{2} = ( \begin{matrix}Δ(I) & 0 \\ Δ_{kI} & 0 \end{matrix} )^{2} = ( \begin{matrix}Δ(I)^{2} & 0 \\ Δ_{kI}Δ(I) & 0 \end{matrix} )

Since Δ(I) is a coboundary operator, Δ(I)^{2} = 0. We need to show that Δ_{kI}Δ(I) = 0.

Δ_{kI}(Δ(I)X_{I}) = δ^{∗}(F_{I}^{∗})^{−1}[pr_{kerΔ(I)}Δ(I)X_{I}] = δ^{∗}(F_{I}^{∗})^{−1}[0] = 0

This means that Δ(Ik) is a coboundary. Let us consider the exact sequences

0 → C̄^{n}(N_{k}, N_{k−1}) →^{ρ} ⨁_{j∈Ik} C̄^{n}(N_{j}, N_{j−1}) →^{ι} ⨁_{i∈I} C̄^{n}(N_{i}, N_{i−1}) → 0

0 → CΔ(k) →^{ρ_{2}} CΔ(Ik) →^{ι_{2}} CΔ(I) → 0.

We already have a connecting homomorphism δ^{∗}. There exists a connecting homomorphism δ_{2}^{∗} such that the long sequences

… → CH(M_{k}) →^{ρ^{∗}} CH(M_{Ik}) →^{ι^{∗}} CH(M_{I}) →^{δ^{∗}} …

… → HΔ(k) →^{ρ_{2}^{∗}} HΔ(Ik) →^{ι_{2}^{∗}} HΔ(I) →^{δ_{2}^{∗}} …

are exact. Let us write the last two parts of both of these sequences in a slightly different form. We shall also add the functions which will appear later on, which might slightly clarify the later method of our proof.

→^{ρ^{∗}} kerι^{∗} ⊕ \frac{CH(M_{Ik})}{kerι^{∗}} →^{ι^{∗}} imι^{∗} ⊕ \frac{CH(M_{I})}{imι^{∗}} →^{δ^{∗}}
↓ G  ↓ F_{I}^{∗}
→^{ρ_{2}^{∗}} kerι_{2}^{∗} ⊕ \frac{HΔ(Ik)}{kerι_{2}^{∗}}  →^{ι_{2}^{∗}} imι_{2}^{∗} ⊕ \frac{HΔ(I)}{imι_{2}^{∗}}  →^{δ_{2}^{∗}}

To show that HΔ(Ik) ≅ CH(M_{Ik}), we will first show that kerι^{∗} ≅ kerι_{2}^{∗}. Since the sequences are exact

kerι^{∗} = imρ^{∗} = {[0 ⊕ φ_{k}] | 0 ∈ ⨁_{i∈I} C̄(N_{i}, N_{i−1}) and φ_{k} ∈ kerˆ∂_{kk}}

kerι_{2}^{∗} = imρ_{2}^{∗} = {[0 ⊕ x_{k}] | 0 ∈ CΔ(I) and ∃φ_{k} ∈ kerˆ∂_{kk} : x_{k} = [φ_{k}]}

Let us define a function G :kerι^{∗} →kerι_{2}^{∗}

G[0 ⊕ φ_{k}]_{im ˆ∂(Ik)} := [0 ⊕ [φ_{k}]]_{imΔ(Ik)}

We will show that G is an isomorphism. First let us show that G is well defined. Let us take any [0 ⊕ φ_{k}] = 0. We have to show that G[0 ⊕ φ_{k}] = 0.

If [0 ⊕ φ_{k}] = 0 then there exist Ψ_{I}, ψ_{k} such that 0 ⊕ φ_{k} = ˆ∂(Ik)(Ψ_{I} ⊕ ψ_{k}).

0 ⊕ φ_{k} = ˆ∂(Ik)(Ψ_{I} ⊕ ψ_{k}) = ˆ∂_{II}Ψ_{I} ⊕ (ˆ∂_{kI}Ψ_{I} + ˆ∂_{kk}ψ_{k})

We can see that ˆ∂(I)Ψ_{I} = 0. From Lemma ??, we know that δ^{∗}[Ψ_{I}]_{im ˆ∂(I)} = [ˆ∂_{kI}Ψ_{I}]_{im ˆ∂(k)}.

[φ_{k}]_{im ˆ∂(k)} = [ˆ∂_{kI}Ψ_{I} + ˆ∂_{kk}ψ_{k}]_{im ˆ∂(k)}
= [ˆ∂_{kI}Ψ_{I}]_{im ˆ∂(k)}
= δ^{∗}[Ψ_{I}]_{im ˆ∂(I)}
= δ^{∗}(F_{I}^{∗})^{−1}[F_{I}Ψ_{I}]_{imΔ(I)}

Since ˆ∂(I)Ψ_{I} = 0 we know that Δ_{I}F_{I}Ψ_{I} = 0, so [φ_{k}] = Δ_{kI}(F_{I}Ψ_{I}) and

0 ⊕ [φ_{k}] = Δ(Ik)(F_{I}Ψ_{I} ⊕ 0).

This means that G[0 ⊕ φ_{k}]_{im ˆ∂(Ik)} = [0 ⊕ [φ_{k}]]_{imΔ(Ik)} = 0. G is well defined.

It is obvious that G is an epimorphism. To show that it is a monomorphism let us take [0 ⊕ φ_{k}] such that G[0 ⊕ φ_{k}] = 0. [0 ⊕ [φ_{k}]] = 0 which means that there exist X_{I}, x_{k} such that 0 ⊕ [φ_{k}] = Δ(Ik)(X_{I} ⊕ x_{k}).

0 ⊕ [φ_{k}] = Δ(Ik)(X_{I} ⊕ x_{k}) = Δ(I)X_{I} ⊕ Δ_{kI}X_{I}

We can see that Δ(I)X_{I} = 0.

Δ_{kI}X_{I} = δ^{∗}(F_{I}^{∗})^{−1}[pr_{kerΔ(I)}X_{I}] = δ^{∗}(F_{I}^{∗})^{−1}[X_{I}]

There exists a Φ_{I} ∈ ⨁_{i∈I} C̄(N_{i}, N_{i−1}) ∩ kerˆ∂(I) such that [Φ_{I}] = (F_{I}^{∗})^{−1}[X_{I}].

[φ_{k}] = Δ_{kI}X_{I} = δ^{∗}(F_{I}^{∗})^{−1}[X_{I}] = δ^{∗}[Φ_{I}] = [ˆ∂_{kI}Φ_{I}]

This means that there exists a ψ_{k} ∈ C̄(N_{k}, N_{k−1}) such that

φ_{k} = ˆ∂_{kI}Φ_{I} + ˆ∂_{kk}ψ_{k}.

Since Φ_{I} ∈ kerˆ∂(I), it is clear that 0 ⊕ φ_{k} = ˆ∂(Ik)(Φ_{I} ⊕ ψ_{k}), which means that [0 ⊕ φ_{k}] = 0. G is a monomorphism.

To show that HΔ(Ik) ≅ CH(M_{Ik}) we now only need to see wether

\frac{CH(M_{Ik})}{kerι^{∗}} ≅ \frac{HΔ(Ik)}{kerι_{2}^{∗}}.

First let us show that imι^{∗} ≅ imι_{2}^{∗}. We will show that F_{I|imι^{∗}}^{∗} : imι^{∗} → imι_{2}^{∗} is an isomorphism. From the exactness of the sequences

… → CH(M_{k}) →^{ρ^{∗}} CH^{n}(M_{Ik}) →^{ι^{∗}} CH^{n}(M_{I}) →^{δ^{∗}} …

… → HΔ(k) →^{ρ_{2}^{∗}} HΔ(Ik) →^{ι_{2}^{∗}} HΔ(I) →^{δ_{2}^{∗}} …

we know that imι^{∗} =kerδ^{∗} and imι_{2}^{∗} =kerδ_{2}^{∗}. We know that F_{I}^{∗} is an isomorphism, so what we shall need is the fact that

F_{I}^{∗}(kerδ^{∗}) = kerδ_{2}^{∗}.

From Lemma ?? we know that δ_{2}^{∗}[X_{I}] = [Δ_{kI}X_{I}]. We also know that Δ_{kk} = 0. From these two facts we get

δ_{2}^{∗}F_{I}^{∗}[Φ_{I}]_{imΔ(I)} = [Δ_{kI}F_{I}Φ_{I}]_{imΔ_{kk}} = Δ_{kI}F_{I}Φ_{I} = δ^{∗}(F_{I}^{∗})^{−1}[F_{I}Φ_{I}] = δ^{∗}[Φ_{I}]

which means that F_{I}^{∗}[Φ_{I}] ∈ kerδ_{2}^{∗} if and only if [Φ_{I}] ∈kerδ^{∗}. We have shown

that imι^{∗} ≅^{F_{I}^{∗}}imι_{2}^{∗}. We know that

ι_{|CH(M_{Ik})/kerι^{∗}}^{∗} : \frac{CH(M_{Ik})}{kerι^{∗}} → imι^{∗}

ν := ι_{2}^{∗}_{|HΔ(Ik)/kerι_{2}^{∗}} : \frac{HΔ(Ik)}{kerι_{2}^{∗}} → imι_{2}^{∗}

are isomorphisms so

ν^{−1} ∘ F_{I}^{∗} ∘ ι^{∗} : \frac{CH(M_{Ik})}{kerι^{∗}} → \frac{HΔ(Ik)}{kerι_{2}^{∗}}

is also an isomorphism, which concludes the proof of our theorem. For any [Φ_{Ik}] in CH(M_{Ik}) there exists a [α] in kerι^{∗} and a [β] in \frac{CH(M_{Ik})}{kerι^{∗}} such that [Φ_{Ik}] = [α] + [β]. We have constructed an isomorphism

F : CH(M_{Ik}) → HΔ(Ik)
F [Φ_{Ik}] = G[α] + ν^{−1} ∘ F_{I}^{∗} ∘ ι^{∗}[β]  ■

Theorem 2.4 The functions F_{J}^{∗} : CH(M_{J}) → HΔ(J) from our construction are generated by cochain morphisms F (J) : ⨁_{i∈J} C̄(N_{i}, N_{i−1}) → CΔ(J).

F (J)(⨁_{i∈J} φ_{i}) := ⨁_{i∈J}[pr_{ker ˆ∂_{ii}}φ_{i}]_{im ˆ∂_{ii}}

Proof. Let us first show that the functions {F (J)}, are indeed cochain morphisms. We shall prove this fact by induction. For an interval I = [1] we have Δ([1]) = 0 so for φ_{1} ∈ C̄(N_{1}, N_{0})

Δ([1])F ([1])φ_{1} = 0 = [ˆ∂_{11}φ_{1}]_{im ˆ∂_{11}} = [pr_{ker ˆ∂_{11}} ˆ∂_{11}φ_{1}]_{im ˆ∂_{11}} = F ([1])ˆ∂_{11}φ_{1}.

Let us assume that for I = [1, k − 1], F (I) is a cochain morphism, we will show that F (Ik) is also a cochain morphism. What we have to show is that Δ(Ik)F (Ik) = F (Ik)ˆ∂(Ik).

Δ(Ik)F (Ik) = ( \begin{matrix}Δ(I) & 0 \\ Δ_{kI} & 0 \end{matrix} ) ( \begin{matrix}F (I) & 0 \\ 0 & F (k) \end{matrix} ) = ( \begin{matrix}Δ(I)F (I) & 0 \\ Δ_{kI}F (I) & 0 \end{matrix} )

F (Ik)ˆ∂(Ik) = ( \begin{matrix}F (I) & 0 \\ 0 & F (k) \end{matrix} ) ( \begin{matrix}ˆ∂(I) & 0 \\ ˆ∂_{kI} & ˆ∂_{kk} \end{matrix} ) = ( \begin{matrix}F (I)ˆ∂(I) & 0 \\ F (k)ˆ∂_{kI} & F (k)ˆ∂_{kk} \end{matrix} )

We know that Δ(I)F (I) = F (I)ˆ∂(I). For any φ_{k} ∈ C̄(N_{k}, N_{k−1})

F (k)ˆ∂_{kk}φ_{k} = [pr_{ker ˆ∂_{kk}} ˆ∂_{kk}φ_{k}]_{im ˆ∂_{kk}} = [ˆ∂_{kk}φ_{k}]_{im ˆ∂_{kk}} = 0

We need to show that Δ_{kI}F (I) = F (k)ˆ∂_{kI}. From Theorem ?? we know that

pr_{ker ˆ∂_{kk}} ˆ∂_{kI}Φ_{I} = ˆ∂_{kI}pr_{ker ˆ∂_{II}}Φ_{I}.

F (k)ˆ∂_{kI}Φ_{I} = [pr_{ker ˆ∂_{kk}} ˆ∂_{kI}Φ_{I}] = [ˆ∂_{kI}pr_{ker ˆ∂_{II}}Φ_{I}] = δ^{∗}[pr_{ker ˆ∂_{II}}Φ_{I}] =

= δ^{∗}(F_{I}^{∗})^{−1}[F (I)pr_{ker ˆ∂_{II}}Φ_{I}] = δ^{∗}(F_{I}^{∗})^{−1}[pr_{kerΔ(I)}F (I)Φ_{I}] = Δ_{kI}F (I)Φ_{I}

We have shown that F (Ik) is a cochain morphism. To complete the proof we still need to show that the function F constructed in the proof of the previous theorem is equal F (Ik)^{∗}.

Let us take any ⨁_{i∈Ik} φ_{i} ∈ ⨁_{i∈IK} C̄(N_{i}, N_{i−1})∩kerˆ∂(Ik). It is clear that [0 ⊕ pr_{ker ˆ∂_{kk}}φ_{k}] ∈ kerι^{∗} and [⨁_{i∈I} φ_{i} ⊕ (φ_{k} − pr_{ker ˆ∂_{kk}}φ_{k})] ∈ CH(M_{Ik})/kerι^{∗}.

F [⨁_{i∈Ik} φ_{i}] = G[0 ⊕ pr_{ker ˆ∂_{kk}}φ_{k}] + ν^{−1} ∘ F_{I}^{∗} ∘ ι^{∗}[⨁_{i∈I} φ_{i} ⊕ (φ_{k} − pr_{ker ˆ∂_{kk}}φ_{k})]
= [0 ⊕ [pr_{ker ˆ∂_{kk}}φ_{k}]] + ν^{−1} ∘ F_{I}^{∗}[⨁_{i∈I} φ_{i}]
= [0 ⊕ [pr_{ker ˆ∂_{kk}}φ_{k}]] + ν^{−1}[⨁_{i∈I}[pr_{ker ˆ∂_{ii}}φ_{i}]]
= [0 ⊕ [pr_{ker ˆ∂_{kk}}φ_{k}]] + [⨁_{i∈I}[pr_{ker ˆ∂_{ii}}φ_{i}] ⊕ 0]
= [F (Ik) ⨁_{i∈Ik} φ_{i}]

■

So far we have constructed our matrix Δ(Ik) and have shown that the connecting isomorphism for this matrix are generated by a family of cochain morphisms {F (J)} defined in the previous theorem. The functions F (J) are simply diagonal matrices so it is quite obvious that for any two adjacent intervals K, L the diagram

0 → ⨁_{i∈L} C̄(N_{i}, N_{i−1}) → ⨁_{i∈KL} C̄(N_{i}, N_{i−1}) → ⨁_{i∈K} C̄(N_{i}, N_{i−1}) → 0

↓ F_{L}  ↓ F_{KL}  ↓ F_{K}

0 → CΔ(L) → CΔ(KL) → CΔ(K) → 0

commutes. Passing to cohomology we obtain the following commuting diagram

… → CH(M_{L}) → CH(M_{KL}) → CH(M_{K}) → …
↓ F_{L}^{∗} ↓ F_{KL}^{∗} ↓ F_{K}^{∗}
… → HΔ(L) → HΔ(KL) → HΔ(K) → …

Which means that we have proofed the following theorem.

Theorem 2.5 For a Morse decomposition M(P, <), the inductively constructed matrix Δ(P) is a connection matrix. The family of connecting isomorphisms {F_{J}^{∗}} is generated by {F (J) | J is an interval in P }

F (J)(⨁_{i∈J} φ_{i}) := ⨁_{i∈J}[pr_{ker ˆ∂_{ii}}φ_{i}]_{im ˆ∂_{ii}}.

3 Properties of cochains on index filtrations

In this section we will prove some technical facts concerning the properties of cochains on index triples and on index filtrations. The main task of this section is to present the proof of Theorem ??.

We have defined C̄^{n}(X, A) as a kernel of ī^{#} : C̄^{n}(X) → C̄^{n}(A), where ī^{#} was induced by an inclusion i : A → X.

Lemma 3.1 For every φ̄ ∈ C̄^{n}(X, A) there exists a ψ ∈ keri^{#} such that [ψ]_{C_{0}^{n}(X)} = φ̄.

Proof. Let us remind that for ξ ∈ C^{n}(X) and (x_{0}, …, x_{n}) ∈ A^{n+1}

i^{#}ξ(x_{0}, …, x_{n}) = ξ(i(x_{0}), …, i(x_{n})) = ξ(x_{0}, …, x_{n}).

We can see that keri^{#} = { ξ ∈ C^{n}(X) : ξ_{|A^{n+1}} = 0}. Let us take any φ ∈ C^{n}(X) such that φ̄ = [φ]_{C_{0}^{n}(X)}. We can define our function ψ ∈ keri^{#} as

ψ(x) := \{ \begin{matrix}0 & x ∈ A^{n+1} \\ φ(x) & otherwise \end{matrix}

What we need to show is that [φ]_{C_{0}^{n}(X)} = [ψ]_{C_{0}^{n}(X)}. We know that [φ]_{C_{0}^{n}(X)} ∈ kerī^{#} so there exists a covering U of A such that φ_{|U^{n+1}} = 0. The difference

(φ − ψ)_{|(X^{n+1}\A^{n+1})∪U^{n+1}} = 0,

which means that [φ − ψ]_{C_{0}^{n}(X)} = 0. ■

Lemma 3.2 If φ ∈ keri^{#} then ∂φ ∈ keri^{#}.

Proof. keri^{#} = {φ ∈ C^{n}(X) : φ_{|A^{n+1}} = 0} so we have to check wether (∂φ)_{|A^{n+2}} = 0. Let (x_{0}, …, x_{n+1}) ∈ A^{n+2}

∂φ(x_{0}, …, x_{n+1}) = ∑_{k=0}^{n+1}(−1)^{k}φ(x_{0}, …, x̂_{k}, …, x_{n+1})

For any k (x_{0}, …, x̂_{k}, …, x_{n+1}) is in A^{n+1}, so φ(x_{0}, …, x̂_{k}, …, x_{n+1}) = 0 which means that ∂φ(x_{0}, …, x_{n+1}) = 0. ■

From now on, for any topological pair (X, A), if we have [φ] ∈ C̄^{n}(X, A) we will assume that φ ∈ keri^{#}.

Let S be an isolated invariant set for a flow on a locally compact metric space. Let (A, R) be an attractor repeller decomposition of S. A sequence of compact sets (N_{2}, N_{1}, N_{0}) is called an index triple for (A, R) if (N_{2}, N_{0}), (N_{2}, N_{1}) and (N_{1}, N_{0}) are index pairs for S, R, A respectively. The attractor repeller decomposition is the simplest example of a Morse decomposition. The index triple is the index filtration for this decomposition. By

i_{01}^{#} : C^{n}(N_{1}) → C^{n}(N_{0}),
i_{02}^{#} : C^{n}(N_{2}) → C^{n}(N_{0}),
i_{12}^{#} : C^{n}(N_{2}) → C^{n}(N_{1}),

we shall denote the natural inclusions.

Lemma 3.3

keri_{01}^{#} ⊕ keri_{12}^{#} ≅ keri_{02}^{#}.

Proof. Let us define a function

Ψ : keri_{01}^{#} ⊕ keri_{12}^{#} → keri_{02}^{#},

Ψ(φ_{1} ⊕ φ_{2})(x) := \{ \begin{matrix}φ_{1}(x) & x ∈ N_{1}^{n+1} \\ φ_{2}(x) & otherwise. \end{matrix}

We will show that Ψ is an isomorphism. Since φ_{2} ∈ keri_{12}^{#} we know that (φ_{2})_{N_{1}^{n+1}} = 0. If ψ(φ_{1} ⊕ φ_{2}) = 0 then it is clear that φ_{1} = 0 and φ_{2} = 0. To show that Ψ is an epimorphism let us take any φ ∈keri_{02}^{#} and define

φ_{1} := φ_{|N_{1}^{n+1}} φ_{2}(x) := \{ \begin{matrix}0 & x ∈ N_{1}^{n+1} \\ φ(x) & otherwise. \end{matrix}

Since φ_{|N_{0}^{n+1}} = 0 we can see that (φ_{1})_{|N_{0}^{n+1}} = 0 so φ_{1} ∈ keri_{01}^{#} and from the definition φ_{2} ∈ keri_{12}^{#}. It is clear that Ψ(φ_{1}, φ_{2}) = φ. ■

It is easy to show that the inverse function Ψ^{−1} is

(Ψ^{−1})(φ) := i_{12}^{#}(φ) ⊕ pr_{keri_{12}^{#}}(φ) = φ_{1} ⊕ φ_{2},

where φ_{1} and φ_{2} are the functions defined at the end of the proof.

Lemma 3.4

C̄^{n}(N_{1}, N_{0}) ⊕ C̄^{n}(N_{2}, N_{1}) ≅ C̄^{n}(N_{2}, N_{0}).

Proof. We will show that the function

Ψ^{∗} : C̄^{n}(N_{1}, N_{0}) ⊕ C̄^{n}(N_{2}, N_{1}) → C̄^{n}(N_{2}, N_{0}),
Ψ^{∗}([φ_{1}]_{C_{0}^{n}(N_{1})} ⊕ [φ_{2}]_{C_{0}^{n}(N_{2})}) := [Ψ(φ_{1} ⊕ φ_{2})]_{C_{0}^{n}(N_{2})}

is an isomorphism. From Lemma ?? and the fact that Ψ is an epimorphism we know that Ψ^{∗} is also an epimorphism. If [Ψ(φ_{1} ⊕ φ_{2})] = 0 then there exists a covering U of N_{2} such that Ψ(φ_{1} ⊕ φ_{2})_{|U^{n+1}} = 0. Since φ_{2} ∈ keri_{12}^{#} we know that (φ_{2})_{|N_{1}^{n+1}} = 0 , so

(φ_{1})_{|U^{n+1}∩N_{1}} = 0 and (φ_{2})_{|U^{n+1}} = 0.

This means that [φ_{1}] = 0 and [φ_{2}] = 0. ■

We will now define a coboundary ˜∂ on keri_{01}^{#}⊕keri_{12}^{#} induced from the operator ∂,

˜∂:= Ψ^{−1} ∘ ∂ ∘ Ψ.

We can look at ˜∂ as a two by two matrix

˜∂ = (˜∂_{lk} : keri_{k−1 k}^{#} → keri_{l−1 l}^{#} | k, l ∈ {1, 2}).

This coboundary induces a coboundary ˆ∂ on C̄^{n}(N_{1}, N_{0}) ⊕ C̄^{n}(N_{2}, N_{1})

ˆ∂([φ_{1}] ⊕ [φ_{2}]) := [˜∂_{11}φ_{1} + ˜∂_{12}φ_{2}] ⊕ [˜∂_{21}φ_{1} + ˜∂_{22}φ_{2}],

where φ_{1} ∈keri_{01}^{#} and φ_{2} ∈keri_{12}^{#}. We have two isomorphic cochain complexes, (C̄^{n}(N_{2}, N_{0}), ¯∂^{n}) and (C̄^{n}(N_{1}, N_{0}) ⊕ C̄^{n}(N_{2}, N_{1}), ˆ∂^{n}). The coboundary ˆ∂^{n} is a two by two matrix

ˆ∂^{n} = (ˆ∂_{lk}^{n} : C̄^{n}(N_{k}, N_{k−1}) → C̄^{n+1}(N_{l}, N_{l−1}) | l, k ∈ {1, 2}).

Lemma 3.5 (ˆ∂_{kl})_{l,k∈{1,2}} is a lower triangular matrix.

Proof. We have to show that ˆ∂_{12} = 0. We will first show that ˜∂_{12} = 0. From Lemma ?? we know that if φ_{2} ∈ keri_{12}^{#} then ∂φ_{2} ∈ keri_{12}^{#} so

˜∂(0 ⊕ φ_{2}) = Ψ^{−1} ∘ ∂ ∘ Ψ(0 ⊕ φ_{2})
= Ψ^{−1} ∘ ∂φ_{2}
= i_{12}^{#}(∂φ_{2}) ⊕ pr_{keri_{12}^{#}}(∂φ_{2})
= 0 ⊕ ∂φ_{2}

which means that ˜∂_{12} = 0 (and also ˜∂_{22}φ_{2} = ∂φ_{2}). We can now write the following equation

ˆ∂([0] ⊕ [φ_{2}]) = [˜∂_{11}0 + ˜∂_{12}φ_{2}] ⊕ [˜∂_{21}0 + ˜∂_{22}φ_{2}]
= [˜∂_{12}φ_{2}] ⊕ [˜∂_{22}φ_{2}]
= [0] ⊕ [˜∂_{22}φ_{2}]

which gives us ˆ∂_{12}[φ_{2}] = 0. ■

Lemma 3.6 Let (x_{1}, …, x_{n+2}) ∈ N_{1}^{n+2} and φ_{1} ∈ keri_{01}^{#}. If there exists x_{0} ∈ N_{2} \ N_{1} such that ˜∂_{22} ˜∂_{21}φ_{1}(x_{0}, …, x_{n+2}) = 0 then ˜∂_{11}φ_{1}(x_{1}, …, x_{n+2}) = 0.

Proof. At the beginning of our last proof we have shown that

(1) if φ_{2} ∈ keri_{12}^{#} then ˜∂_{22}φ_{2} = ∂φ_{2}.

We also know that ˜∂_{21}φ_{1} ∈ keri_{12}^{#} which means that

(2) if x ∈ N_{1}^{n+2} then ˜∂_{21}φ_{1}(x) = 0.

keri_{12}^{#} = {ξ ∈ C^{n}(N_{2}) : ξ_{|N_{1}^{n+1}} = 0} so for any φ ∈ C^{n}(N_{2})

(3) if x ∉ N_{1}^{n+1} then (pr_{keri_{12}^{#}}φ)(x) = φ(x).

We know that x_{0} ∈ N_{2} \ N_{1}. Let us take any k ≠ 0.

(∂ ∘ Ψ)(φ_{1} ⊕ 0)(x_{0}, …, x̂_{k}, …, x_{n+2})
= ∑_{l<k}(−1)^{l}Ψ(φ_{1} ⊕ 0)(x_{0}, …, x̂_{l}, …, x̂_{k}, …, x_{n+2})

+ ∑_{l>k}(−1)^{l−1}Ψ(φ_{1} ⊕ 0)(x_{0}, …, x̂_{k}, …, x̂_{l}, …, x_{n+2})

We know that for l ≠ 0 (x_{0}, …, x̂_{l}, …, x̂_{k}, …, x_{n}) ∉ N_{1}^{n+1} so

Ψ(φ_{1} ⊕ 0)(x_{0}, …, x̂_{l}, …, x̂_{k}, …, x_{n+2}) = 0

This means that for k ≠ 0 we have

(4) \begin{matrix}(∂ ∘ Ψ)(φ_{1} ⊕ 0)(x_{0}, …, x̂_{k}, …, x_{n+2}) \\ = & Ψ(φ_{1} ⊕ 0)(x̂_{0}, x_{1}, …, x̂_{k}, …, x_{n+2}) \\ = & φ_{1}(x̂_{0}, x_{1}, …, x̂_{k}, …, x_{n+2}) \end{matrix}

From the points (1), (2), (3) and (4) we get

0 = ˜∂_{22} ˜∂_{21}φ_{1}(x_{0}, …, x_{n+2})
{(1) \atop =} ∂ ˜∂_{21}φ_{1}(x_{0}, …, x_{n+2})
= ∑_{k=0}^{n+2}(−1)^{k} ˜∂_{21}φ_{1}(x_{0}, …, x̂_{k}, …, x_{n+2})

{(2) \atop =} ∑_{k≠0}(−1)^{k} ˜∂_{21}φ_{1}(x_{0}, …, x̂_{k}, …, x_{n+2})

= ∑_{k≠0}(−1)^{k}(pr_{keri_{12}^{#}} ∘ ∂ ∘ Ψ)(φ_{1} ⊕ 0)(x_{0}, …, x̂_{k}, …, x_{n+2})

{(3) \atop =} ∑_{k≠0}(−1)^{k}(∂ ∘ Ψ)(φ_{1} ⊕ 0)(x_{0}, …, x̂_{k}, …, x_{n+2})

{(4) \atop =} ∑_{k≠0}(−1)^{k}φ_{1}(x̂_{0}, x_{1}, …, x̂_{k}, …, x_{n+2})
= −˜∂_{11}φ_{1}(x̂_{0}, x_{1}, …, x_{n+2}).
■

Lemma 3.7 For any φ̄ ∈ C̄^{n}(N_{1}, N_{0}), if ˆ∂_{22} ˆ∂_{21}φ̄ = 0 then

ˆ∂_{21} ∘ pr_{ker ˆ∂_{11}} φ̄ = ˆ∂_{21}φ̄.

Proof. By Lemma ?? there exists a φ ∈ keri_{01}^{#} such that φ̄ = [φ]. We know that ˆ∂_{22} ˆ∂_{21}[φ] = 0 so there exists a covering U of N_{2} such that ˜∂_{22} ˜∂_{21}φ = 0 on U^{n+3}. Let us define a family of sets

A := {V ∩ N_{1} : V ∈ U and V ∩ N_{2} ≠ ∅}.

For any (x_{1}, …, x_{n+2}) ∈ A^{n+2} there exists an x_{0} ∉ N_{1} such that

(x_{0}, x_{1}, …, x_{n+2}) ∈ U^{n+3}.

We know that ˜∂_{22} ˜∂_{21}φ(x_{0}, …, x_{n+2}) = 0 so from Lemma ?? we know that ˜∂_{11}φ(x_{1}, …, x_{n+2}) = 0. We have shown that ˜∂_{11}φ = 0 on A^{n+2} which means that for x ∈ A^{n+1}

(pr_{ker ˜∂_{11}}φ)(x) = φ(x).

Let us now show that for any ψ ∈ keri_{01}^{#} such that ψ = 0 on A^{n+1}

ˆ∂_{21}[ψ] = 0.

We will show that for any (x_{0}, …, x_{n+1}) ∈ U^{n+2}

˜∂_{21}ψ(x_{0}, …, x_{n+1}) = 0.

If (x_{0}, …, x_{n+1}) ∈ N_{1}^{n+2} ∩ U^{n+2} then since ˜∂_{21}ψ ∈ keri_{12}^{#} we have

˜∂_{21}ψ(x_{0}, …, x_{n+1}) = 0.

Let us now take (x_{0}, …, x_{n+1}) ∉ N_{1}^{n+2}.

˜∂_{21}ψ(x_{0}, …, x_{n+1}) = ∑_{k=0}^{n+1}(−1)^{k}Ψ(ψ ⊕ 0)(x_{0}, …, x̂_{k}, …, x_{n+1}).

If for every k (x_{0}, …, x̂_{k}, …, x_{n+1}) ∉ N_{1}^{n+1} then

Ψ(ψ ⊕ 0)(x_{0}, …, x̂_{k}, …, x_{n+1}) = 0,

which means that ˜∂_{21}ψ(x_{0}, …, x_{n+1}) = 0. If there exists a k such that x_{k} ∉ N_{1} and (x_{0}, …, x̂_{k}, …, x_{n+1}) ∈ N_{1}^{n+1} then since (x_{0}, …, x_{n+1}) ∈ U^{n+2}

(x_{0}, …, x̂_{k}, …, x_{n+1}) ∈ A^{n+1}.

We have assumed that ψ = 0 on A^{n+1} so

Ψ(ψ ⊕ 0)(x_{0}, …, x̂_{k}, …, x_{n+1}) = ψ(x_{0}, …, x̂_{k}, …, x_{n+1}) = 0.

For j ≠ k (x_{0}, …, x̂_{j}, …, x_{n+1}) ∉ N_{1}^{n+1} so Ψ(ψ⊕0)(x_{0}, …, x̂_{j}, …, x_{n+1}) = 0. This means that

˜∂_{21}ψ(x_{0}, …, x_{n+1}) = ∑_{i=0}^{n+1}(−1)^{i}Ψ(ψ ⊕ 0)(x_{0}, …, x̂_{i}, …, x_{n+1}) = 0.

We have shown that for all x ∈ U^{n+2} ˜∂_{21}ψ(x) = 0 which gives us ˆ∂_{21}[ψ] = 0. From the beginning of the proof we know that

(pr_{ker ˜∂_{11}}φ)(x) = φ(x) for x ∈ A^{n+1}.

This means that

ˆ∂_{21}[pr_{ker ˜∂_{11}}φ − φ] = 0
ˆ∂_{21}pr_{ker ˆ∂_{11}}[φ] − ˆ∂_{21}[φ] = 0  ■

Lemma 3.8 For φ̄ ∈ C̄^{n}(N_{1}, N_{0}) if φ̄ ∈ kerˆ∂_{11} then ˆ∂_{21}φ̄ ∈ kerˆ∂_{22}.

Proof. We know that ˆ∂ is a coboundary operator so ˆ∂^{2} = 0.

0 = ( \begin{matrix}ˆ∂_{11} & 0 \\ ˆ∂_{21} & ˆ∂_{22} \end{matrix} )^{2} = ( \begin{matrix}ˆ∂_{11}^{2} & 0 \\ ˆ∂_{21} ˆ∂_{11} + ˆ∂_{22} ˆ∂_{21} & ˆ∂_{22}^{2} \end{matrix} )

ˆ∂_{22} ˆ∂_{21}φ̄ = −ˆ∂_{21} ˆ∂_{11}φ̄ = −ˆ∂_{21}0 = 0  ■

Since C̄^{n}(N_{1}, N_{0}) is a vector space we know that there exists a subspace (kerˆ∂_{11})′ of C̄^{n}(N_{1}, N_{0}) such that C̄^{n}(N_{1}, N_{0}) = kerˆ∂_{11} ⊕ (kerˆ∂_{11})′.

Lemma 3.9 If φ̄ ∈ (kerˆ∂_{11})′ then pr_{ker ˆ∂_{22}} ∘ ˆ∂_{21}φ̄ = 0.

Proof. Let us take ψ̄ ∈ (ˆ∂_{21})^{−1}({pr_{ker ˆ∂_{22}} ∘ ˆ∂_{21}φ̄}). Since φ̄ ∈ (kerˆ∂_{11})′ we can chose ψ̄ so that

ψ̄ ∈ (kerˆ∂_{11})′.

It is clear that ˆ∂_{22} ˆ∂_{21}ψ̄ = 0, so from Lemma ?? we know that ˆ∂_{21}∘pr_{ker ˆ∂_{11}} ψ̄ = ˆ∂_{21}ψ̄. Since ψ̄ ∈ (kerˆ∂_{11})′

pr_{ker ˆ∂_{22}} ∘ ˆ∂_{21}φ̄ = ˆ∂_{21}ψ̄ = ˆ∂_{21} ∘ pr_{ker ˆ∂_{11}} ψ̄ = ˆ∂_{21}0 = 0.  ■

The lemmas proved above have been a preparation to the proof of Theorem ??. The following lemma is a restriction of the theorem for index filtrations to the case of an index triple.

Lemma 3.10 For any φ̄ ∈ C̄^{n}(N_{1}, N_{0})

pr_{ker ˆ∂_{22}}(ˆ∂_{21}φ̄) = ˆ∂_{21}(pr_{ker ˆ∂_{11}} φ̄)

Proof. If φ̄ ∈kerˆ∂_{11} then from Lemma ??

ˆ∂_{21}(pr_{ker ˆ∂_{11}} φ̄) = ˆ∂_{21}φ̄ = pr_{ker ˆ∂_{22}}(ˆ∂_{21}φ̄)

If φ̄ ∈ (kerˆ∂_{11})′ then from Lemma ?? we get

pr_{ker ˆ∂_{22}}(ˆ∂_{21}φ̄) = 0 = ˆ∂_{21}(pr_{ker ˆ∂_{11}} φ̄)  ■

We know that for any two adjacent intervals K, I the pair (M_{K}, M_{I}) is a atractor repeller decomposition of M_{KI}. By using this fact, Lemma ?? and induction we can easyly show that for every l, j in P such that j < l,

C̄^{n}(N_{l}, N_{j−1}) ≅ ⨁_{k=j}^{l} C̄^{n}(N_{k}, N_{k−1}).

By using Lemma ?? and induction we can show that the coboundary ˆ∂ on ⨁_{k=j}^{l} C̄^{n}(N_{k}, N_{k−1}) is a lower triangular matrix. From Lemma ?? we also get the proof of Theorem ??.

References

[1] E.Spanier, Algebraic topology, McGraw-Hill, New York, 1966

[2] C.Conley, Isolated invariant sets and the Morse index, CMBS Lecture notes no.38, AMS 1978

[3] R.Franzosa, Index Filtrations and Connection Matrices For Partially Ordered Morse Decompositions, Thesis, Univ. of Wisconsin

[4] - Index Filtrations and the homology Index Braid for Partially Ordered Morse decompositions. Trans. AMS 298 (1986) pp. 193-213.

[5] - The Continuation Theory for Morse Decompositions and Connection Matrices, Trans. AMS 310 (1988) pp. 781-803

[6] - The Connection Matrix Theory for Morse Decompositions, Trans. AMS (1989) pp. 561-592

[7] D.Richeson, Connection Matrix Pairs, Banach Center Publ. , 47.