Spectral splittings in the Conley index theory

We introduce spectral splittings and spectral graphs. We explain the relation between spectral splittings, graphs and sequences.


Introduction
One of the main ideas behind the Conley index theory is to apply the tools from algebraic topology in studying dynamical systems, especially the structure of invariant sets (see [5,6,10,14]). This approach, motivated by Morse theory, focuses on decomposing isolated invariant sets into invariant subsets (Morse sets) and connecting orbits between them. This structure is called a Morse decomposition of an isolated invariant set. A filtration of index pairs associated with a Morse decomposition can be used to find connections between different Morse sets. The principal tools for this purpose are connection matrices (see [3,4,9,13]), connection graphs (see [2,8]) and spectral sequences (see [1,4,7]).
The main goal of this paper is to explain and clarify the basic relations between connection matrices (generalized here to spectral splittings), connection graphs (called here spectral graphs) and spectral sequences. Although the comparison is done mainly on algebraic level, we hope that it sheds some new light on important aspects of the Conley index theory. This paper is mainly intended as a brief survey summarizing results from [1,2,3,4].
The organization of the paper is as follows. Section 2 contains some preliminaries. In Section 3 we introduce spectral splittings and study their basic properties. In Section 4 we briefly explain the relation between spectral splittings and connection matrices. Section 5 contains a brief description of spectral graphs. Finally, in Section 6 we use spectral splittings and graphs 404 P. Bart lomiejczyk JFPTA 2 P. Bart lomiejczyk to study the structure of the Morse decomposition of a flow on a compact metric space.

Preliminaries
Recall that a filtered vector space is a vector space A equipped with a finite increasing filtration, that is, a sequence {A p } n 0 of subspaces of A such that We will use the following convenient notation: Definition 2.1. A filtered differential vector space (f-d space for short) is a filtered vector space A together with an endomorphism d such that d 2 = 0 and d preserves the filtration, i.e., dA p ⊂ A p .
Remark 2.2. The main advantage of using filtered differential vector spaces instead od filtered differential modules is that we can apply standard methods of linear algebra, but this approach has also the disadvantage that we obtain homology with coefficients in a field, which is blind to torsion.
Observe that we have two natural finite filtrations associated with the f-d space. Namely, Let us briefly recall the definition of the spectral sequences of the f-d space (see [15] for more details). We introduce the following notation. Let Remark 2.4. It is easily seen that if the filtration of the f-d space A has length n, then the spectral sequence stabilizes at the nth term; i.e., E r E r+1 for r ≥ n.
In the next section we will make use of the following two easy-to-check facts from linear algebra.

P. Bart lomiejczyk
Let us introduce the following notation: Remark 3.3. By Definition 3.1, Of course, an arbitrary complement to U p,q in V p,q may not satisfy condition (3.3) of Definition 3.1.

Remark 3.4. From Definition 3.1 and Lemma 2.5 applied to
The following two observations are essential for applications in the Conley index theory.

Proposition 3.5. The total homology of the f-d space A is given by
Proposition 3.6. Using the notation Proof. By Definition 3.1 and Remark 3.4, we have Let d denote the differential on quotients d : which completes the proof.
Remark 3.7. Similar reasoning shows even more general result. Namely, if p > q, then Let us introduce the notation for some "blocks" in the spectral splitting. Set min = min{r, n + 1 − p}, for every p, r = 1, . . . , n. Here and subsequently, we will use the symbol A r p to denote the direct sum max q=min A p,q . (3.5) Vol. 17 (2015) Spectral splittings 407 Spectral splittings 5 the following diagram commutes for every p, r = 1, . . . , n.
it is easy to see that Φ r p are isomorphisms and the diagram (3.6) commutes for every p, r = 1, . . . , n. Theorem 3.9. There is a spectral splitting for any f-d space.
Proof. The construction of a spectral splitting proceeds as follows. If p + q ≤ n + 1, we choose A p,q to be any complement to U p,q in V p,q . On the other hand, if p + q > n + 1, we apply Lemma 2.6 to V = V p,q , U = U p,q and B = A p+q−n−1,n−q+1 to get such complement A p,q to U p,q in V p,q that dA p,q ⊂ A p+q−n−1,n−q+1 .

Spectral splittings and connection matrices
There is an easy way to obtain connection matrices from spectral splittings as a part of it.
p=0 has the following properties: Recall that any f-d space satisfying the above three conditions is called a connection matrix. As we can see a connection matrix is simply a filtration of some "subblocks" of the spectral splitting. Moreover, if we delete from the spectral splitting all direct summands A p,0 and A p,n+1 for p = 1, . . . , n, we obtain a detailed connection matrix, which is a more refined version of a usual connection matrix (see [4] for definition).

Spectral graphs
each vertex has 0 or 1 edge, Remark 5.2. The spectral graph exhibits the following duality: • edges of the graph give full description of the differential in the f-d space, • the set of vertices without edges determines the homology of the f-d space A. In other words, the vertices from C p ≈ H(A p /A p−1 ) which do not contribute to the global homology H(A) are "cancelled" by joining pairs of them with edges.
The following result is a natural consequence of the previous results.

Theorem 5.3. There is a spectral graph for any f-d space.
Proof. First, by Theorem 3.9, we get a spectral splitting. Then we choose any bases in the spaces A p,q for p+q ≥ n+1. Finally bases in A p,q for p+q < n+1 are determined by the isomorphisms d : A p+q,n−q+1 → A p,q .
Remark 5.4. The proof of Theorem 5.3 shows how to obtain a spectral graph from a spectral splitting. However, the reverse reconstruction is also possible. Namely, condition (2) of Definition 5.1 guarantees that using the formula we can recover a spectral splitting from a spectral graph.
The vertices of a spectral graph may be labeled as follows. Each vertex v is labeled by an integer l(v) = p := max{i | v ∈ A i }. We use v p to denote a vertex labeled by p. Of course, two different vertices may have the same label. Moreover, let W = {v ∈ V | dv ∈ V and l(v) = l(dv)}. A full subgraph of a spectral graph spanned by the vertices V \ (W ∪ dW ) will be called a connection graph.

Morse decompositions
Spectral splittings and graphs may be used for studying the structure of Morse decompositions. Let X be a compact metric space. Assume we have a flow on X. Recall that a collection {M i } n 1 of mutually disjoint compact invariant subsets of X is called a Morse decomposition if for every The sets M i are called Morse sets. Moreover, generalized Morse sets for i ≤ j are defined as Spectral splittings and graphs may be naturally related to a filtration of index pairs associated with the Morse decomposition. Recall that an increasing filtration of compact sets {N i } n 0 is called an index filtration if (1) N 0 = ∅ and N n = X, (2) Such filtrations, even regular, exist (see [11,14] for more details). Recall that if the index filtration is regular, then the Conley index of each Morse set M ji is isomorphic to the singular homology of the pair (N j , N i−1 ).
Let {N i } n 0 be an index filtration for the Morse decomposition {M i } n 1 . Let C(N k ) be the vector space of singular chains in N k and let i : C(N k ) → C(N ) be a homomorphism induced by the inclusion N k ⊂ N . It is evident that a filtration i C(N k ) n 0 , equipped with the boundary map on singular chains, is an f-d space. A spectral splitting (graph) for the Morse decomposition {M i } n 1 is simply a spectral splitting (graph) for the above f-d space.
Repeating the reasoning from the proof of [1, Theorem 7.2] we can easily show the following result. Proposition 6.1. Any two spectral splittings (spectral graphs) for a given Morse decomposition are isomorphic; i.e., they are independent (up to isomorphism) of the choice of an index filtration.
In our next theorem we will make use of the following definition. Our last result shows how to use spectral graphs in analyzing the structure of the Morse decomposition. It is worth pointing out that even an easy comparison of local Conley indices of Morse sets with the total homology of the whole space by means of the spectral graph may provide information on how different Morse sets are linked.  N k ) , This gives v s = dv p = dv + dv and consequently because the differential on singular chains operates component by component.
We thus get v ∈ ker d = D n+1 and so contrary to (6.1).
We end this section with two simple examples illustrating the computation of connection graphs. The second example is adapted from [12].
Example (A gradient flow on a deformed 2-sphere). Consider the gradient flow of the height function on a manifold M diffeomorphic to S 2 with phase portrait as in Figure 1. Observe that two minima (1 and 2), two maxima (4 and 5) and a "circle" consisting of two saddle points and two connecting orbits between them (3) form a Morse decomposition of M . Moreover, the respective sublevel sets on M form a natural index filtration. It will be convenient to represent a connection graph for the Morse decomposition in a tabular form. The columns are labeled by Morse sets and the rows correspond  As we have mentioned before, the connection graph shows the relationship between the Conley indices of Morse sets (columns) and the homology groups of the manifold M (vertices without edges). Besides, the edges describe the action of the differential of the respective f-d space.
Example (A flow on the 2-sphere). Consider the local flow in R 2 with phase portrait as in Figure 2 (see [12] for more details). If we consider the flow as a flow on the 2-sphere, then the point at infinity becomes an attractor. Observe that three repelling rest points (2, 6, 7), two saddle rest points (4, 5), one attracting rest point (3) and one attracting periodic orbit (1) form a Morse decomposition of S 2 . We can easily construct an index filtration for this Morse It may be worth pointing out that the phenomenon of nonuniqueness of connection matrices (graphs) cannot be observed here (see Proposition 6.1).