Conley–Morse–Forman Theory for Combinatorial Multivector Fields on Lefschetz Complexes

In principle, a Lefschetz complex may be viewed as a finitely generated free chain complex with a distinguished basis. However, in this paper we follow the original definition given by Lefschetz [19]. In this definition the elements of the basis are the primary objects and the algebraic structure is given on top of them. Such a reversed approach is natural in the algorithmic context, because a computer may store and process only finite sets. Thus, a Lefschetz complex consists of a finite collection of cells X graded by dimension and the incidence coefficient \(\kappa (x,y)\) encoding the incidence relation between cells \(x,y\in X\) (see Sect. 5.1 for a precise definition). A nonzero value of \(\kappa (x,y)\) indicates that the cell y is in the boundary of the cell x and the dimension of y is one less than the dimension of x. The cell y is then called a facet of x. The family K of all simplices of a simplicial complex [15, Definition 11.8], all elementary cubes in a cubical set [15, Definition 2.9] or, more generally, cells of a cellular complex (finite CW complex, see [20, Section IX.3]) are examples of Lefschetz complexes. In this case the incidence coefficient is obtained from the boundary homomorphism of the associated simplicial, cubical or cellular chain complex. A sample Lefschetz complex is presented in Fig. 2. It consists of eight vertices (0-cells or cells of dimension zero), ten edges (1-cells) and three squares (2-cells).

Condition (3) presented in Sect. 5.1 guarantees that the free group spanned by X together with the linear map given by \( \mathbf{\partial }x:=\sum _{y\in X}\kappa (x,y)y \) is a free chain complex with X as a basis. By the Lefschetz homology of X we mean the homology of this chain complex. We denote it by \(H^\kappa (X)\). In the case of a Lefschetz complex given as a cellular complex condition (3) is satisfied and the resulting chain complex and homology is precisely the cellular chain complex and the cellular homology. Given a Lefschetz complex X we denote by \(\beta _i(X):={\text {rank}}\,H^\kappa _i(X)\) the ith Betti number of X and write \(p_X\) for the respective Poincaré polynomial, that is,The closure of \(A\subseteq X\), denoted \({\text {cl}}\,A\), is obtained by recursively adding to A the facets of cells in A, the facets of the facets of cells in A and so on. The set A is closed if \({\text {cl}}\,A=A\), and it is open if \(X \setminus A\) is closed. The terminology is justified, because the open sets indeed form a \(T_0\) topology on X. We say that A is proper if \({\text {mo}}\,A:={\text {cl}}\,A \setminus A\), which we call the mouth of A, is closed. Proper sets are important for us, because every proper subset of a Lefschetz complex with incidence coefficients restricted to this subset is also a Lefschetz complex. Four Lefschetz complexes being proper subsets of a bigger Lefschetz complex are indicated in Fig. 2 by solid ovals. In the case of a proper subset X of a cellular complex K the Lefschetz homology of X is isomorphic to the relative cellular homology \( H( {\text {cl}}\,X, {\text {mo}}\,X). \) However, from the algorithmic point of view the direct use of Lefschetz homology is preferred, because it minimizes the amount of information which needs to be encoded.

A multivector is a proper \(V\subseteq K\) such that \(V\subseteq {\text {cl}}\,V^\star \) for a unique \(V^\star \in V\). Out of the four examples of Lefschetz complexes in Fig. 2 only the one enclosing vertex C is not a multivector. A multivector is critical if \(H^\kappa (V)\ne 0\). Otherwise it is regular. The only example of a regular multivector in Fig. 2 is the one enclosing vertex B. Roughly speaking, a regular multivector indicates that \({\text {cl}}\,V\), the closure of V, may be collapsed to \({\text {mo}}\,V\), the mouth of V. In the dynamical sense this means that one can set up a flow which eventually leaves V through \({\text {mo}}\,V\). A critical multivector indicates the contrary: \({\text {cl}}\,V\) may not be collapsed to \({\text {mo}}\,V\) and, in the dynamical sense, something must stay inside V.

A combinatorial multivector field is a partition \(\mathcal {V}\) of X into multivectors. We associate dynamics with \(\mathcal {V}\) via a directed graph \(G_\mathcal {V}\) with vertices in X and three types of arrows: up-arrows, down-arrows and loops. Up-arrows have heads in \(V^\star \) and tails in all the other cells of V. Down-arrows have tails in \(V^\star \) and heads in \({\text {mo}}\,V\). Loops join \(V^\star \) with itself for all critical multivectors V. A sample multivector field is presented in Fig. 3 (top) together with the associated directed graph \(G_\mathcal {V}\) (bottom). The terminology ‘up-arrows’ and ‘down-arrows’ comes from the fact that the dimensions of cells are increasing along up-arrows and decreasing along down-arrows. Notice that the up-arrows sharing the same head uniquely determine a multivector. Therefore, it is convenient to draw a multivector field not as a partition but by marking all up-arrows. For convenience, we also mark the loops, but the down-arrows are implicit and are usually omitted to keep the drawings simple.

A multivector may consist of one, two or more cells. If there are no more than two cells, we say that the multivector is a vector. Otherwise we call it a strict multivector. Note that the combinatorial multivector field in Fig. 3 has three strict multivectors:Observe that a combinatorial multivector field with no strict multivectors corresponds to the combinatorial vector field in the sense of Forman [10, 11].

A cell \(x\in X\) is critical with respect to \(\mathcal {V}\) if \(x=V^\star \) for a critical multivector \(V\in \mathcal {V}\). A critical cell x is non-degenerate if the Lefschetz homology of its multivector is zero in all dimensions except one in which it is isomorphic to the ring of coefficients. This dimension is then the Morse index of the critical cell. The combinatorial multivector field in Fig. 3 has three critical cells: F, C and BCGF. They are all non-degenerate. The cells F and C have Morse index equal zero. The cell BCGF has Morse index equal one.

A solution of \(\mathcal {V}\) (also called a trajectory or a walk) is a bi-infinite, backward infinite, forward infinite or finite sequence of cells such that any two consecutive cells in the sequence form an arrow in the graph \(G_\mathcal {V}\). The solution is full if it is bi-infinite. A finite solution is also called a path. The full solution is periodic if the sequence is periodic. It is stationary if the sequence is constant. By the dynamics of \(\mathcal {V}\) we mean the collection of all solutions. The dynamics is multivalued in the sense that there may be many different solutions going through a given cell.

Let X be a Lefschetz complex and let \(\mathcal {V}\) be a combinatorial multivector field on X. Assume \(S\subseteq X\) is \(\mathcal {V}\)-compatible, that is, S equals the union of multivectors contained in it. We say that S is invariant if for every multivector \(V\subseteq S\) there is a full solution through \(V^\star \) in S. The invariant part of a subset \(A\subseteq X\) is the maximal, \(\mathcal {V}\)-compatible invariant subset of A. A path in \({\text {cl}}\,S\) is an internal tangency to S if the values at the endpoints of the path are in S but one of the values is not in S. The set S is isolated invariant if it is invariant and admits no internal tangencies.

An isolated invariant set is an attractor, respectively a repeller, if there is no full solution crossing it which goes away from it in forward, respectively backward, time. The attractors and repellers have the following topological characterization in terms of the \(T_0\) topology of X (see Sect. 8, Theorems 8.1 and 8.3).

Given a family \(\{M_r\}_{r\in {\mathbb {P}}}\) of mutually disjoint, non-empty, isolated invariant sets, we write \(r\le r'\) for \(r,r'\in \mathbb {P}\) if there exists a full solution such that all its sufficiently far terms belong to \(M_r\) and all sufficiently early terms belong to \(M_{r'}\). Such a full solution is called a connection running from \(M_{r'}\) to \(M_r\). The connection is heteroclinic if \(r\ne r'\). Otherwise it is called homoclinic. We say that \(\{M_r\}_{r\in {\mathbb {P}}}\) is a Morse decomposition of X if the relation \(\le \) induces a partial order in \(\mathbb {P}\). The Hasse diagram of this partial order with vertices labelled by the Poincaré polynomials \(p_{M_r}(t)\) is called the Conley–Morse graph of the Morse decomposition (comp. [7, Def. 2.11]).

The Poincaré polynomials \(p_{M_r}(t)\) are related to the Poincaré polynomial \(p_X(t)\) via the following theorem (see Sect. 9.3, Theorems 9.11 and 9.12).

We say that a combinatorial multivector field \(\mathcal {V}\) on X is gradient-like if there exists a real valued function f on X which is non-increasing along each solution of \(\mathcal {V}\) and constant only if the cells along the solution belong to the same multivector (see Sect. 9.5). The following theorem extends the results of Forman [10, Cor. 3.6] to the case of combinatorial multivector fields (see Sect. 9.5, Theorems 9.14 and 9.15).

Consider the planar regular cellular complex in Fig. 4 (left). It consists of 11 quadrilaterals and its faces. A proper subcollection of its 55 faces, marked by a circle in the centre of mass, forms a Lefschetz complex X. It consists of all cells of the cellular complex except vertices A, B, D and edges AB, AD. A combinatorial multivector field \(\mathcal {V}\) on X is marked by up-arrows and loops. The invariant part of X with respect to \(\mathcal {V}\) consists of all cells of X but the cells marked in white. The Lefschetz homology \(H^\kappa (X)\cong H(K,A)\) where A is the cellular complex consisting of vertices A, B, D and edges AB, AD. Thus, this is the homology of a pointed annulus. Therefore, \(p_X(t)=t\).

Consider the family of six isolated invariant setsmarked in Fig. 4 with the respective symbols. The family \(\mathcal {M}\) is a Morse decomposition of X. The respective Poincaré polynomials are: \(p_{\displaystyle {\bullet }}(t)=1\), \(p_{\ominus }(t)=t\), \(p_{\displaystyle {\circ }}(t)=t^2\), \(p_{\times }(t)=2t\), \(p_{\triangle }(t)=t^2+t\), \(p_{\Diamond }(t)=t+1\).

There are two attractors: stationary \(M_{\displaystyle {\bullet }}\) and periodic \(M_{\Diamond }\). There are also two repellers: stationary \(M_{\displaystyle {\circ }}\) and periodic \(M_{\triangle }\). The other two isolated invariant sets are neither attractors nor repellers. The Morse equation takes the form

A multivector field \(\mathcal {W}\) is a refinement of \(\mathcal {V}\) if each multivector in \(\mathcal {V}\) is \(\mathcal {W}\)-compatible. The refinement is proper if the invariant part with respect to \(\mathcal {W}\) of each regular multivector in \(\mathcal {V}\) is empty. A Forman refinement of a multivector field \(\mathcal {V}\) is a vector field \(\mathcal {W}\) which is a proper refinement of \(\mathcal {V}\) such that each multivector of \(\mathcal {V}\) contains at most one critical vector of \(\mathcal {W}\). Then \(\mathcal {W}\) has precisely one critical cell in any critical multivector of \(\mathcal {V}\).

The concept of Forman refinement raises two natural questions. The first question is whether a combinatorial multivector field always admits a Forman refinement. The second question is whether the study of the dynamics of a combinatorial multivector field which admits a Forman refinement may be reduced to the study of the dynamics of the refinement.

We do not know what the answer to the first question is. If the answer is negative, then there exists a multivector field \(\mathcal {V}\) on a Lefschetz complex X such that at least one multivector of \(\mathcal {V}\) cannot be partitioned into vectors with at most one critical vector in the partition. There are examples of zero spaces (Lefschetz complexes with zero homology) which do not admit a combinatorial vector field with empty invariant part. They may be constructed by adapting examples of contractible but not collapsible cellular complexes such as Bing’s house [4] or dunce hat [33]. One such example is presented in Fig. 5. This example fulfils all requirements of a multivector except the requirement that a multivector has precisely one top-dimensional cell, because it has five top-dimensional cells.

Regardless of what is the answer to the first question, even if a given combinatorial multivector field does have a Forman refinement, in general it is not unique. Figure 6 shows a combinatorial multivector field \(\mathcal {V}\) (left) and its two different Forman refinements: \(\mathcal {V}_1\) (middle) and \(\mathcal {V}_2\) (right). The critical cells of all three combinatorial multivector fields are the same: AB, B, C, DF and F. However, in the case of \(\mathcal {V}\) there are heteroclinic connections running from the critical cell DF to the critical cells AB, B and C. In the case of \(\mathcal {V}_1\) there is a heteroclinic connection running from DF to B but not to AB nor C. In the case of \(\mathcal {V}_2\) there is a heteroclinic connection running from DF to C but not to AB nor B. Our next example shows that the differences may be even deeper. Thus, the answer to the second question is clearly negative.

Figure 7 presents a combinatorial multivector field \(\mathcal {V}\) on a Lefschetz complex X (top) and one of its two Forman refinements \(\mathcal {V}_1\) (bottom). The combinatorial multivector field \(\mathcal {V}\) has homoclinic connections to the cell BEIF. Moreover, it admits chaotic dynamics in the sense that for each bi-infinite sequence of two symbols marking the two edges BF and EI, there is a full trajectory whose sequence of passing through the edges BF and EI is precisely the given one. The two Forman refinements of \(\mathcal {V}\) have neither homoclinic connections nor chaotic dynamics.

In Sect. 10 we present algorithm CMVF. Its input consists of a collection of classical vectors on an integer, planar lattice. These may be vectors of a smooth planar vector field evaluated at the lattice points. However, the algorithm accepts any collection of vectors, also vectors chosen randomly. It constructs a combinatorial multivector field based on the directions of the classical vectors, with varying number of strict multivectors: from many to none, depending on a control parameter.

As our first example consider the vector field of the differential equationrestricted to the \(10\times 10\) lattice of points in the square \([-3,3]\times [-3,3]\). The equation has three minimal invariant sets: a repelling stationary point at the origin and two invariant circles: an attracting periodic orbit of radius 1 and a repelling periodic orbit of radius 2. The outcome of algorithm CMVF maximizing the number of strict multivectors is presented in Fig. 8. It captures all three minimal invariant sets of (1). The variant forbidding strict multivectors is presented in Fig. 9. It captures only the attracting periodic orbit, whereas the repelling fixed point and repelling periodic trajectory degenerate into a collection of critical cells.

Figure 10 presents a combinatorial multivector field constructed by algorithm CMVF from a randomly selected collection of vectors at the lattice points. To ensure that the boundary of the selected region does not divide multivectors, all the vectors at the boundary are not random but point inwards. The resulting Morse decomposition consists of 102 isolated invariant sets out of which three consist of more than one multivector.

We denote the sets of reals, integers, non-negative integers and non-positive integers, respectively, by \(\mathbb {R}, \mathbb {Z}\), \(\mathbb {Z}^+\), \(\mathbb {Z}^-\). We also write \(\mathbb {Z}^{\ge n}\), \(\mathbb {Z}^{\le n}\), respectively, for integers greater or equal n and less or equal n. Given a set X, we write \({\text {card}}\,X\) for the number of elements of X and we denote by \(\mathcal {P}(X)\) the family of all subsets of X. We write \(f:X{\nrightarrow }Y\) for a partial map from X to Y, that is a map defined on a subset \({\text {dom}}\,f\subseteq X\), called the domain of f, and such that the set of values of f, denoted \({\text {im}}\,f\), is contained in Y.

Given a set X and a binary relation \(R\subseteq X\times X\), we use the shorthand xRy for \((x,y)\in R\). By the transitive closure of R we mean the relation \(\bar{R}\subseteq X\times X\) given by \(x\bar{R}y\) if there exists a sequence \(x=x_0,x_1,\ldots , x_n=y\) such that \(n\ge 1\) and \(x_{i-1}Rx_i\) for \(i=1,2,\ldots , n\). Note that \(\bar{R}\) is transitive but need not be reflexive. The relation \(\bar{R}\cup {\text {id}}\,_X\), where \({\text {id}}\,_X\) stands for the identity relation on X, is reflexive and transitive. Hence, it is a preorder, called the preorder induced by R. A \(y\in X\) covers an \(x\in X\) in the relation R if xRy but there is no \(z\in X\) such that \(x\ne z\ne y\) and xRz, zRy.

A multivalued map \(F:X{\,\overrightarrow{\rightarrow }\,}Y\) is a map \(F:X\rightarrow \mathcal {P}(Y)\). For \(A\subseteq X\) we define the image of A by \( F(A):=\bigcup \{\,F(x) \mid x\in A\,\} \) and for \(B\subseteq Y\) we define the preimage of B by \( F^{-1}(B):=\{\,x\in X\mid F(x)\cap B\ne \varnothing \,\}. \)

Given a relation R, we associate with it a multivalued map \(F_R:X{\,\overrightarrow{\rightarrow }\,}X\), by \(F_R(x):=R(x)\), where \( R(x):=\{\,y\in X\mid xRy\,\} \) is the image of \(x\in X\) in R. Obviously \(R\mapsto F_R\) is a one-to-one correspondence between binary relations in X and multivalued maps from X to X. Often, it will be convenient to interpret the relation R as a directed graph whose set of vertices is X and a directed arrow joins x with y whenever xRy. The three concepts relation, multivalued map and directed graph are equivalent on the formal level, and the distinction is used only to emphasize different directions of research. However, in this paper it will be convenient to use all these concepts interchangeably.

Assume \((X,\le )\) is a poset. Thus, \(\le \) is a partial order, that is a reflexive, antisymmetric and transitive relation in X. As usual, we denote the inverse of this relation by \(\ge \). We also write < and > for the associated strict partial orders, that is relations \(\le \) and \(\ge \) excluding identity. By an interval in X we mean a subset of X which has one of the following four formsIn the first case we speak about a closed interval. The elements x, y are the endpoints of the interval. We recall that \(A\subseteq X\) is convex if for any \(x,y\in A\) the closed interval [x, y] is contained in A. Note that every interval is convex but there may exist convex subsets of X which are not intervals. A set \(A\subseteq X\) is an upper set if for any \(x\in X\) we have \([x,\infty )\subseteq A\). Also, \(A\subseteq X\) is a lower set if for any \(x\in X\) we have \((-\infty ,x]\subseteq A\). Sometimes a lower set is called an attracting interval and an upper set a repelling interval. However, one has to be careful, because in general lower and upper sets need not be intervals at all. For \(A\subseteq X\) we also use the notation \(A^{\le }:=\{\,x\in X\mid \exists _{a\in A}\;x\le a\,\}\) and \(A^{<}:=A^{\le } \setminus A\).

For a topological space X and \(A\subseteq X\) we write \({\text {cl}}\,A\) for the closure of A. We also define the mouth of A byNote that A is closed if and only if its mouth is empty. We say that A is proper if \({\text {mo}}\,A\) is closed. Note that open and closed subsets of X are proper. In the case of finite topological spaces proper sets have a special structure. To explain it we first recall some properties of finite topological spaces based on the following fundamental result which goes back to Alexandroff [1].

Let \((X,\mathcal {T})\) be a finite topological space. For \(x\in X\) we write \( {\text {cl}}\,x:={\text {cl}}\,\{x\}\), \({\text {opn}}\,x:=\bigcap \{\,U\in \mathcal {T}\mid x\in U\,\}\). The following proposition may be easily verified.

In the sequel we will particularly often use property (iii) of Proposition 4.4. In particular, it is needed in the following characterization of proper sets in finite topological spaces.

Proposition 4.5 means that in the setting of finite topological spaces proper sets correspond to convex sets in the language of the associated partial order.

Let R be a fixed ring with unity. Given a set X we denote by R(X) the free module over R spanned by X. Given a graded, finitely generated module \(E=(E_k)_{k\in {\mathbb {Z^+}}}\) over R, we writefor the Poincaré formal power series of E. We have the following theorem (see [26]).

The following definition goes back to S. Lefschetz (see [19, Chpt. III, Sec. 1, Def. 1.1]).

The family of cells of a simplicial complex [15, Definition 11.8] and the family of elementary cubes of a cubical set [15, Definition 2.9] provide simple but important examples of Lefschetz complexes. In these two cases the respective formulas for the incident coefficients are explicit and elementary (see [23]). Also a general regular cellular complex (regular finite CW complex, see [20, Section IX.3]) is an example of a Lefschetz complex. In this case the incident coefficients may be obtained from a system of equations (see [20, Section IX.5]).

The Lefschetz complex \((X,\kappa )\) is called regular if for any \(x,y\in X\) the incidence coefficient \(\kappa (x,y)\) is either zero or invertible in R. One easily verifies that condition (3) implies that we have a free chain complex \((R(X),\mathbf{\partial }^\kappa )\) with \(\mathbf{\partial }^\kappa :R(X)\rightarrow R(X)\) defined on generators by \(\mathbf{\partial }^\kappa (x) := \sum _{y\in X}\kappa (x,y)y\). The Lefschetz homology of \((X,\kappa )\), denoted \(H^\kappa (X)\) is the homology of this chain complex. By a zero space we mean a Lefschetz complex whose Lefschetz homology is zero. Since X is finite, \((R(X),\mathbf{\partial }^\kappa )\) is finitely generated. In consequence, the Poincaré formal power series \(p_{H^{\kappa }(X)}(t)\) is a polynomial. We denote it briefly by \(p_X(t)\).

Given \(x,y\in X\) we say that y is a facet of x and write \(y\prec _{\kappa } x\) if \(\kappa (x,y)\ne 0\). It is easily seen that the relation \(\prec _{\kappa }\) extends uniquely to a minimal partial order. We denote this partial order by \(\le _\kappa \) and the associated strict order by \(<_\kappa \). We say that y is a face of x if \(y\le _\kappa x\). The \(T_0\) topology defined via Theorem 4.3 by the partial order \(\le _\kappa \) will be called the Lefschetz topology of \((X,\kappa )\). Observe that the closure of a set \(A\subseteq X\) in this topology consists of all faces of all cells in A. The Lefschetz complex via its Lefschetz topology is related to the abstract cell complex in the sense of [17] and [29, Section III]). In principle, the definitions in the sequel using Lefschetz topology could be restated in terms of the partial order \(\le _\kappa \). We prefer to use Lefschetz topology to emphasize that several definitions, in particular the definition of an index pair, are analogous to their counterparts in the classical Conley theory.

Proposition 5.2 shows that a Lefschetz complex consisting of just two cells may have zero Lefschetz homology. At the same time the singular homology of this two point space with Lefschetz topology is nonzero, because the singular homology of a non-empty space is never zero. Thus, the singular homology H(X) of a Lefschetz complex \((X,\kappa )\) considered as a topological space with its Lefschetz topology need not be the same as the Lefschetz homology \(H^\kappa (X)\). Some situations when the two homologies are isomorphic may be deduced from the results in [2]. However, this is irrelevant from the point of view of the needs of this paper.

A set \(A\subseteq X\) is a \(\kappa \)-subcomplex of X if \((A,\kappa _{|A\times A})\) is a Lefschetz complex. Lefschetz complexes, under the name of S-complexes, are discussed in [23]. In particular, the following proposition follows from the observation that a proper subset of a Lefschetz complex X satisfies the assumptions of [23, Theorem 3.1].

Note that a \(\kappa \)-subcomplex A of X does not guarantee that \((R(A),\mathbf{\partial }^\kappa _{|R(A)})\) is a chain subcomplex of \((R(X),\mathbf{\partial }^\kappa )\). However, we have the following theorem (see [23, Theorem 3.5]).

The following proposition is straightforward to verify.

We also need the following theorem which follows from [23, Theorems 3.3 and 3.4]

Let \((X,\kappa )\) be a fixed Lefschetz complex.

Note that we do not require the existence of a unique minimal element in a multivector but if such an element exists, we denote it by \(V_\star \). Multivectors admitting a unique minimal element are studied in [13] in the context of equivariant discrete Morse theory. A concept similar to our multivector appears also in [32].

A multivector V is regular if V is a zero space. Otherwise it is called critical. A combinatorial multivector V is a combinatorial vector or briefly a vector if \({\text {card}}\,V\le 2\). A vector always has a unique minimal element.

The following definition introduces the main new concept of this paper.

Proposition 5.9 implies that our concept of a vector field on the Lefschetz complex of a cellular complex is in one-to-one correspondence with Forman’s combinatorial vector field (see [11]). It also corresponds to the concept of partial matching [18, Definition 11.22]. Thus, the combinatorial multivector field is a generalization of the earlier definitions in which vectors were used instead of multivectors.

For each cell \(x\in X\) we denote by \([x]_{\mathcal {V}}\) the unique multivector in \(\mathcal {V}\) to which x belongs. If the multivector field \(\mathcal {V}\) is clear from the context, we write briefly \([x]:=[x]_{\mathcal {V}}\) and \(x^\star :=[x]_{\mathcal {V}}^\star \). We refer to a cell x as dominant with respect to \(\mathcal {V}\), or briefly as dominant, if \(x^\star =x\).

The map which sends x to \(x^\star \) determines the combinatorial multivector field. More precisely, we have the following theorem.

Given a combinatorial multivector field \(\mathcal {V}\) on X we associate with it the graph \(G_\mathcal {V}\) with vertices in X and an arrow from x to y if one of the following conditions is satisfied We write \(y \prec _{\mathcal {V}} x\) if there is an arrow from x to y in \(G_\mathcal {V}\). This lets us interpret \(\prec _{\mathcal {V}}\) as a relation in X. We denote by \(\le _{\mathcal {V}}\) the preorder induced by \(\prec _{\mathcal {V}}\). In order to study the dynamics of \(\mathcal {V}\), we interpret \(\prec _{\mathcal {V}}\) as a multivalued map \(\Pi _{\mathcal {V}}: X{\,\overrightarrow{\rightarrow }\,}X\), which sends a cell x to the set of cells covered by x in \(\prec _{\mathcal {V}}\), that isWe say that a cell \(x\in X\) is critical with respect to \(\mathcal {V}\) if the multivector \([x]_{\mathcal {V}}\) is critical and x is dominant in \([x]_{\mathcal {V}}\). A cell is regular if it is not critical. We denote by \(\langle x \rangle _{\mathcal {V}}\) the set of regular cells in \([x]_\mathcal {V}\). It is straightforward to observe thatAs an immediate consequence of the definition (8) and formula (9) we get the following proposition.

We extend the relation \(\le _{\mathcal {V}}\) to multivectors \(V,W\in \mathcal {V}\) by assuming that \(V\le _{\mathcal {V}}W\) if and only if \(V^\star \le _{\mathcal {V}}W^\star \).

We say that \(\mathcal {V}\) is acyclic if \(\le _{\mathcal {V}}\) is a partial order on X.

A partial map \(\varphi :\mathbb {Z}{\nrightarrow }X\) is a solution of \( \mathcal {V}\) if \({\text {dom}}\,\varphi \) is an interval in \(\mathbb {Z}\) and \( \varphi (i+1)\in \Pi _\mathcal {V}(\varphi (i)) \text { for } i,i+1\in {\text {dom}}\,\varphi . \) A solution \(\varphi \) is in \(A\subseteq X\) if \({\text {im}}\,\varphi \subseteq A\). We call \(\varphi \) a full (respectively forward or backward) solution if \({\text {dom}}\,\varphi \) is \(\mathbb {Z}\) (respectively \(\mathbb {Z}^{\ge n}\) or \(\mathbb {Z}^{\le n}\) for some \(n\in \mathbb {Z}\)). We say that \(\varphi \) is a solution through \(x\in X\) if \(x\in {\text {im}}\,\varphi \). We denote the set of full (respectively forward, backward) solutions in A through x by \({\text {Sol}}\,(x,A)\) (respectively \({\text {Sol}}\,^+(x,A)\), \({\text {Sol}}\,^-(x,A)\)). We drop A in this notation if A is the whole Lefschetz complex X. As an immediate consequence of Proposition 5.12 we get the following proposition.

Given \(n\in \mathbb {Z}\), let \( \tau _n:\mathbb {Z}\ni i\mapsto i+n\in \mathbb {Z}\) denote the n-translation map. Let \(\varphi \) be a solution in A such that \(n\in \mathbb {Z}\) is the right endpoint of \({\text {dom}}\,\varphi \) and let \(\psi \) be a solution in A such that \(m\in \mathbb {Z}\) is the left endpoint of \({\text {dom}}\,\psi \). We define \( \varphi \cdot \psi :\tau _n^{-1}({\text {dom}}\,\varphi )\cup \tau _{m-1}^{-1}({\text {dom}}\,\psi )\rightarrow A, \) the concatenation of \(\varphi \) and \(\psi \) byIt is straightforward to observe that if \(\psi (m)\in \Pi _\mathcal {V}(\varphi (n))\) then the concatenation \(\varphi \cdot \psi \) is also a solution in A.

Let \(\varphi \in {\text {Sol}}\,^+(x,A)\) and let \({\text {dom}}\,\varphi =\mathbb {Z}^{\ge n}\) for some \(n\in \mathbb {Z}\). We define \(\sigma _+\varphi :\mathbb {Z}^{\ge n}\rightarrow A\), the right shift of \(\varphi \), by \(\sigma _+\varphi (i):=\varphi (i+1)\). Let \(\psi \in {\text {Sol}}\,^-(x,A)\) and let \({\text {dom}}\,\psi =\mathbb {Z}^{\le n}\) for some \(n\in \mathbb {Z}\). We define \(\sigma _-\psi :\mathbb {Z}^{\le n}\rightarrow A\), the left shift of \(\psi \), by \(\sigma _-\psi (i):=\psi (i-1)\). It is easily seen that for \(k\in \mathbb {Z}^+\) we have \(\sigma _+^k\varphi \in {\text {Sol}}\,^+(\varphi (n+k),A)\) and \(\sigma _-^k\psi \in {\text {Sol}}\,^-(\psi (n-k),A)\). For a full solution \(\varphi \) we denote respectively by \(\varphi ^+\), \(\varphi ^-\) the restrictions \(\varphi _{|{\mathbb {Z}^+}}\), \(\varphi _{|{\mathbb {Z}^-}}\).

Obviously, if \(\varphi \) is a solution in A then \(\varphi \circ \tau _n\) is also a solution in A. We say that solutions \(\varphi \) and \(\varphi '\) are equivalent if \(\varphi '=\varphi \circ \tau _n\) or \(\varphi =\varphi '\circ \tau _n\) for some \(n\in \mathbb {Z}\). It is straightforward to verify that this is indeed an equivalence relation. It preserves forward, backward and full solutions through x. Moreover, it is not difficult to verify that the concatenation extends to an associative operation on equivalence classes of solutions. In the sequel we identify solutions in the same equivalence class. This allows us to treat the solutions as finite or infinite words over the alphabet consisting of cells in X. In the sequel, whenever we pick up a representative of a forward (backward) solution, we assume its domain is respectively \(\mathbb {Z}^+\), (\(\mathbb {Z}^-\)).

A solution \(\varphi \) such that \({\text {dom}}\,\varphi \) is a finite interval is called a path joining the value of \(\varphi \) at the left end of the domain with the value at the right end. The cardinality of \({\text {dom}}\,\varphi \) is called the length of the path. In the special case when \(\varphi \) has length one we identify \(\varphi \) with its unique value. We also admit the trivial path of length zero (empty set). In particular, it acts as the neutral element of concatenation. For every \(x\in X\) there is a unique path joining x with \(x^\star \), denoted \(\nu (x)\) and given byNote that the concatenation \(x\cdot x^\star \) is a solution, because \(x^\star \in \Pi _\mathcal {V}(x)\). We also defineNote that \(\nu (x)=\nu ^-(x)\cdot x^\star \).

We say that \(A\subseteq X\) is \( \mathcal {V}\)-compatible if \(x\in A\) implies \([x]_{\mathcal {V}}\subseteq A\) for \(x\in X\). We denote by \([A]_\mathcal {V}^-\) the maximal \(\mathcal {V}\)-compatible subset of A and by \([A]_\mathcal {V}^+\) the minimal \(\mathcal {V}\)-compatible superset of A. The following proposition is straightforward.

Observe that if \(A\subseteq X\) is a \(\mathcal {V}\)-compatible \(\kappa \)-subcomplex of X then \(\mathcal {V}':=\{\,V\in \mathcal {V}\mid V\subseteq A\,\}\) is a multivector field on A. We call it the restriction of \(\mathcal {V}\) to A and denote it \(\mathcal {V}_{|A}\).

We define the invariant part of A, the positive invariant part of A and the negative invariant part of A, respectively, byNote that by replacing \(x^\star \) by x or \([A]_\mathcal {V}^-\) by A in the definition of the invariant part of A we may not obtain the invariant part of A. Indeed, consider the set \(\{AB,AC,ABC,B,BC\}\) marked in black in Fig. 11. Its invariant part is \(\{AB,AC,ABC,B\}\). But, by replacing \(x^\star \) by x in the definition of the invariant part we obtain \(\{ABC,B\}\). And by replacing \([A]_\mathcal {V}^-\) by A we obtain \(\{AB,AC,ABC,B,BC\}\).

We say that A is invariant with respect to \(\mathcal {V}\) if \({\text {Inv}}\,A=A\).

Assume \(A\subseteq X\) is invariant. We say that a path \(\varphi \) from \(x\in A\) to \(y\in A\) is an internal tangency of A, if \({\text {im}}\,\varphi \subseteq {\text {cl}}\,A\) and \({\text {im}}\,\varphi \cap {\text {mo}}\,A\ne \varnothing \). We say that \(S\subseteq X\) is an isolated invariant set if it is invariant and admits no internal tangencies.

A pair \(P=(P_1, P_2)\) of closed subsets of X such that \(P_2\subseteq P_1\) is an index pair for S if the following three conditions are satisfied We say that the index pair P is saturated if \(P_1 \setminus P_2=S\).

A sample isolated invariant set together with an index pair is presented in Fig. 12.

The following proposition follows immediately from the definition of index pair.

Given an index pair P, consider the setof cells in \(P_1\) whose every forward solution intersects \(P_2\).

In the following lemma, given an index pair P we first shrink \(P_1\) and then grow \(P_2\) to saturate P.

We write \(P\subseteq Q\) for index pairs P, Q meaning \(P_i\subseteq Q_i\) for \(i=1,2\). We say that index pairs P, Q of S are semi-equal if \(P\subseteq Q\) and either \(P_1=Q_1\) or \(P_2=Q_2\). For semi-equal pairs P, Q, we write

The following theorem, an analogue of a classical results in the Conley index theory, allows us to define the homology Conley index of an isolated invariant set S as \(H^\kappa (P_1,P_2)=H^\kappa (P_1\setminus P_2)\) for any index pair P of S. We denote it \({\text {Con}}\,(S)\).

In the case of an isolated invariant set S we call the polynomial \(p_S(t)\) the Conley polynomial of S and \(\beta _i(S)\) the ith Conley coefficient of S.

We call the index pair \(({\text {cl}}\,S,{\text {mo}}\,S)\) canonical, because it minimizes both \(P_1\) and \(P_1\setminus P_2\). Indeed, for any index pair \({\text {cl}}\,S\subseteq P_1\), because \(P_1\) is closed and by (17) \(S\subseteq P_1\setminus P_2\). In the case of the canonical index pair both inclusions are equalities. Note that in the case of the classical Conley index there is no natural choice of a canonical index pair. Since \(S={\text {cl}}\,S\setminus {\text {mo}}\,S\), we see that \({\text {Con}}\,(S)\cong H^\kappa (S)\). Thus, in our combinatorial setting Theorem 7.11 is actually not needed to define the Conley index. However, the importance of Theorem 7.11 will become clear in the proof of Morse inequalities via the following corollary.

Let S be an isolated invariant set and assume \(S_1,S_2\subseteq S\) are also isolated invariant sets. We say that S decomposes into \(S_1\) and \(S_2\) if \(S_1\cap S_2=\varnothing \) and every full solution \(\varrho \) in S is a solution either in \(S_1\) or in \(S_2\).

We say that a \(\mathcal {V}\)-compatible \(N\subseteq X\) is a trapping region if \(\Pi _{\mathcal {V}}(N)\subseteq N\). This is easily seen to be equivalent to the requirement that N is \(\mathcal {V}\)-compatible and for any \(x\in N\) We say that A is an attractor if there exists a trapping region N such that \( A={\text {Inv}}\,N\).

We say that a \(\mathcal {V}\)-compatible \(N\subseteq X\) is a backward trapping region if \(\Pi _{\mathcal {V}}^{-1}(N)\subseteq N\). This is easily seen to be equivalent to the requirement that N is \(\mathcal {V}\)-compatible and for any \(x\in N\) We say that R is a repeller if there exists a backward trapping region N such that \( R={\text {Inv}}\,N\). The following proposition is straightforward.

Let \(A\subseteq X\) be \(\mathcal {V}\)-compatible. We write \(x\mathop {\rightarrow }\limits ^{A}_{\mathcal {V}} y\) if there exists a path of \(\mathcal {V}\) in A from \(x^\star \) to \(y^\star \) of length at least two. We write \(x\mathop {\leftrightarrow }\limits ^{A}_{\mathcal {V}} y\) if \(x\mathop {\rightarrow }\limits ^{A}_{\mathcal {V}} y\) and \(y\mathop {\rightarrow }\limits ^{A}_{\mathcal {V}} x\). We drop \(\mathcal {V}\) in this notation if \(\mathcal {V}\) is clear from the context. Also, we drop A if \(A=X\). We say that A is weakly recurrent if for every \(x\in A\) we have \(x\mathop {\leftrightarrow }\limits ^{A} x\). It is strongly recurrent if for any \(x,y\in A\) we have \(x\mathop {\rightarrow }\limits ^{A} y\), or equivalently for any \(x,y\in A\) there is \(x\mathop {\leftrightarrow }\limits ^{A} y\). Obviously, every strongly recurrent set is also weakly recurrent.

The chain recurrent set of X, denoted \({\text {CR}}\,(X)\), is the maximal weakly recurrent subset of X. It is straightforward to verify thatObviously, the relation \(\leftrightarrow _{\mathcal {V}}\) restricted to \({\text {CR}}\,(X)\) is an equivalence relation. By a basic set of \(\mathcal {V}\) we mean an equivalence class of \(\leftrightarrow _{\mathcal {V}}\) restricted to \({\text {CR}}\,(X)\).

Let \(\varrho :\mathbb {Z}\rightarrow X\) be a full solution. Recall that \([A]_\mathcal {V}^+\) denotes the intersection of \(\mathcal {V}\)-compatible sets containing A. We define the \(\alpha \) and \(\omega \) limit sets of \(\varrho \) by

Since by Theorem 8.1 an attractor is in particular a trapping region, it follows from Proposition 8.2 that if A is an attractor, then \({\text {Inv}}\,(X\setminus A)\) is a repeller. It is called the dual repeller of A and is denoted by \(A^\star \). Similarly, if R is a repeller, then \({\text {Inv}}\,(X\setminus R)\) is an attractor, called the dual attractor of R and denoted by \(R^\star \).

For \(A,A'\subseteq X\) define

The pair ( A, R) of subsets of X is said to be an attractor–repeller pair in X if A is an attractor, R is a repeller, \(A=R^\star \) and \(R=A^\star \).

The following corollary is an immediate consequence of Theorem 8.7.

Let \(\mathbb {P}\) be a finite set. The collection \(\mathcal {M}=\{\,M_r\mid r\in \mathbb {P}\,\}\) is called a Morse decomposition of X if there exists a partial order \(\le \) on \(\mathbb {P}\) such that the following three conditions are satisfied:Note that since the sets in \(\mathcal {M}\) are mutually disjoint, the indices \(r,r'\) in (ii) are determined uniquely. A partial order on \(\mathbb {P}\) which makes \(\mathcal {M}=\{\,M_r\mid r\in \mathbb {P}\,\}\) a Morse decomposition of X is called an admissible order. Observe that the intersection of admissible orders is an admissible order and any extension of an admissible order is an admissible order. In particular, every Morse decomposition has a unique minimal admissible order as well as an admissible linear order.

The following proposition is straightforward.

Let \(\mathcal {B}\) be the family of all basic sets of \(\mathcal {M}\). For \(B_1,B_2\in \mathcal {B}\) we write \(B_1\le _{\mathcal {V}}B_2\) if there exists a solution \(\varrho \) such that \(\alpha (\varrho )\subseteq B_2\) and \(\omega (\varrho )\subseteq B_1\). It follows easily from the definition of the basic set that \(\le _{\mathcal {V}}\) is a partial order on \(\mathcal {B}\).

Given two Morse decompositions \(\mathcal {M}\), \(\mathcal {M}'\) of X, we write \(\mathcal {M}'\sqsubset \mathcal {M}\) if for each \(M'\in \mathcal {M}'\) there exists an \(M\in \mathcal {M}\) such that \(M'\subseteq M\). Then, we say that \(\mathcal {M}'\) is finer than \(\mathcal {M}\). The relation \(\sqsubset \) is easily seen to be a partial order on the collection of all Morse decompositions of X.

Given \(I\subseteq \mathbb {P}\) define the Morse set of I by

We also have a dual statement. We skip the proof, because it is similar to the proof of Theorem 9.5.

For a lower set I in \(\mathbb {P}\) setObserve that by Corollary 8.8 we have

The following corollary is an immediate consequence of Theorem 9.8 and the definition of the lower set.

We begin by the observation that an isolated invariant set S as a proper and \(\mathcal {V}\)-compatible subset of X in itself may be considered a Lefschetz complex with a combinatorial multivector field being the restriction \(\mathcal {V}_{|S}\). In particular, it makes sense to consider attractors, repellers and Morse decompositions of S with respect to \(\mathcal {V}_{|S}\) and the results of the preceding sections apply.

Recall that for a proper \(S\subseteq X\) we write \(\beta _k(S):={\text {rank}}\,H^\kappa _k(S)\).

We say that \(\mathcal {V}\) is gradient-like if there exists an \(f:X\rightarrow \mathbb {R}\) such that for any solution \(\rho :\mathbb {Z}{\nrightarrow }X\) of \(\mathcal {V}\) and \(n,n+1\in {\text {dom}}\,\rho \) the following two conditions hold

Let c be a critical cell of \(\mathcal {V}\). It follows from Proposition 7.2 that the critical vector \([c]_\mathcal {V}\) is an isolated invariant set. We say that c is non-degenerate if there is a \(k\in \mathbb {Z}^+\) such that the ith Conley coefficient of \([c]_\mathcal {V}\) satisfiesWe say that the number k is the Morse index of c.

The requirement that a multivector has a unique maximal element with respect to the partial order \(\le _\kappa \) is convenient. It simplifies the analysis of combinatorial multivector fields, because every non-dominant cell x admits precisely one arrow in \(G_\mathcal {V}\) originating in x, namely the up-arrow from x to \(x^\star \). It is also not very restrictive in combinatorial modelling of a dynamical system. But, in some situations the requirement may be inconvenient. For instance, if sampling is performed near a hyperbolic or repelling fixed point, it may happen that the sampled vectors point into different top-dimensional cells. Or, when a parameterized family of dynamical systems is studied (see Sect. 11.2), it may be natural to model a collection of nearby dynamical systems by one combinatorial multivector field. In such a situation the uniqueness requirement will not be satisfied in places where neighbouring systems point into different cells.

Taking these limitations into account it is reasonable to consider the following extension of the concept of a combinatorial multivector field. A generalized multivector in a Lefschetz complex \((X,\kappa )\) is a proper collection of cells in X. A generalized multivector V is critical if \(H^\kappa (V)\ne 0\). A generalized multivector field \(\mathcal {V}\) is a partition of X into generalized multivectors. For a cell \(x\in X\) we denote by [x] the unique generalized multivector in \(\mathcal {V}\) to which x belongs. The associated generalized directed graph \(G_\mathcal {V}\) has vertices in X and an arrow from x to y if one of the following conditions is satisfied We write \(y \prec _{\mathcal {V}} x\) if there is an arrow from x to y in \(G_\mathcal {V}\), denote by \(\le _{\mathcal {V}}\) the preorder induced by \(\prec _{\mathcal {V}}\), interpret \(\prec _{\mathcal {V}}\) as a multivalued map \(\Pi _{\mathcal {V}}: X{\,\overrightarrow{\rightarrow }\,}X\) and study the associated dynamics. An example of a generalized multivector field together with the associated up-arrows is presented in Fig. 14.

The main ideas of the theory presented in this paper extend to this generalized case. The proofs get more complicated, because there are more cases to analyse. This is caused by the fact that more than one up-arrow may originate from a given cell.

It is tempting to simplify the analysis by gluing vertices of \(G_\mathcal {V}\) belonging to the same multivector. But, when proceeding this way, the original phase space is lost. And, the concept of index pair makes sense only in the original space, because in general the sets \(P_1\), \(P_2\) in an index pair \(P=(P_1,P_2)\) are not \(\mathcal {V}\)-compatible. Also, since in the gluing process two different combinatorial multivector fields would modify the space in a different way, it would not be possible to compare their dynamics. In consequence, the concepts of perturbation and continuation briefly explained in the following section would not make sense.

Among the fundamental features of the classical Conley index theory is the continuation property [9, Chapter IV]. An analogous property may be formulated in the combinatorial case. Let \(\mathcal {V}\), \(\bar{\mathcal {V}}\) be two combinatorial multivector fields on a Lefschetz complex X. We say that \(\bar{\mathcal {V}}\) is a perturbation of \(\mathcal {V}\) if \(\bar{\mathcal {V}}\) is a refinement of \(\mathcal {V}\) or \(\mathcal {V}\) is a refinement of \(\bar{\mathcal {V}}\). A parameterized family of combinatorial multivector fields is a sequence \(\mathcal {V}_1,\mathcal {V}_2,\ldots ,\mathcal {V}_n\) of combinatorial multivector fields on X such that \(\mathcal {V}_{i+1}\) is a perturbation of \(\mathcal {V}_i\). Assume S is an isolated invariant set of \(\mathcal {V}\) and \(\bar{S}\) is an isolated invariant set of \(\bar{\mathcal {V}}\). We say that S and \(\bar{S}\) are related by a direct continuation if there is a pair \(P=(P_1,P_2)\) of closed sets in X such that P is an index pair for S with respect to \(\mathcal {V}\) and for \(\bar{S}\) with respect to \(\bar{\mathcal {V}}\). We say that S and \(\bar{S}\) are related by continuation along a parameterized family \(\mathcal {V}=\mathcal {V}_1,\mathcal {V}_2,\ldots ,\mathcal {V}_n=\bar{\mathcal {V}}\) if there exist isolated invariant sets \(S_i\) of \(\mathcal {V}_i\) such that \(S_1=S\), \(S_n=\bar{S}\) and \(S_i\), \(S_{i+1}\) are related by a direct continuation. In a similar way one can define continuation of Morse decompositions of isolated invariant sets.

Figure 15 presents two parameterized families of combinatorial multivector fields: \(\mathcal {V}_1\), \(\mathcal {V}_2\), \(\mathcal {V}_3\), \(\mathcal {V}_4\), \(\mathcal {V}_5\) in the top row and \(\mathcal {W}_1\), \(\mathcal {W}_2\), \(\mathcal {W}_3\), \(\mathcal {W}_4\), \(\mathcal {W}_5\), \(\mathcal {W}_6\), \(\mathcal {W}_7\) in the bottom row. Note that \(\mathcal {V}_1=\mathcal {W}_1\) and \(\mathcal {V}_5=\mathcal {W}_7\). The singleton \(\{AB\}\) is an isolated invariant set of \(\mathcal {V}_1\). The set \(\{A,AB,AC\}\) is an isolated invariant set of \(\mathcal {V}_2\). These two sets are related by a direct continuation, because the pair \((\{A,B,C,AB,AC\},\{B,C\})\) is an index pair for \(\{AB\}\) with respect to \(\mathcal {V}_1\) and for \(\{A,AB,AC\}\) with respect to \(\mathcal {V}_2\). Note that \(\{AB\}\) is also an isolated invariant set of \(\mathcal {V}_5\) but it is not related by continuation along \(\mathcal {V}_1\), \(\mathcal {V}_2\), \(\mathcal {V}_3\), \(\mathcal {V}_4\), \(\mathcal {V}_5\) to \(\{AB\}\) as an isolated invariant set of \(\mathcal {V}_1\). But \(\{AB\}\) as an isolated invariant set for \(\mathcal {W}_1\) is related by continuation along \(\mathcal {W}_1\), \(\mathcal {W}_2\), \(\mathcal {W}_3\), \(\mathcal {W}_4\), \(\mathcal {W}_5\), \(\mathcal {W}_6\), \(\mathcal {W}_7\) to \(\{CD\}\) as an isolated invariant set for \(\mathcal {W}_7\). Actually, the Morse decomposition of \(\mathcal {W}_1\) consisting of Morse sets \(\{ABCD\}\), \(\{AB\}\), \(\{CD\}\), \(\{B\}\), \(\{C\}\) is related by continuation along \(\mathcal {W}_1\), \(\mathcal {W}_2\), \(\mathcal {W}_3\), \(\mathcal {W}_4\), \(\mathcal {W}_5\), \(\mathcal {W}_6\), \(\mathcal {W}_7\) to the Morse decomposition of \(\mathcal {W}_7\) consisting of Morse sets \(\{ABCD\}\), \(\{CD\}\), \(\{AB\}\), \(\{C\}\), \(\{B\}\). In particular, the isolated invariant sets \(\{ABCD\}\), \(\{AB\}\), \(\{CD\}\), \(\{B\}\), \(\{C\}\) of \(\mathcal {W}_1\) are related by continuation respectively to the isolated invariant sets \(\{ABCD\}\), \(\{CD\}\), \(\{AB\}\), \(\{C\}\), \(\{B\}\) of \(\mathcal {W}_7\).

Consider a finite regular CW complex X with the collection of cells K. We say that \(V\subseteq K\) is a multivector if |V|, the union of all cells in V, is a proper subset of X. We say that a multivector V is regular if \({\text {mo}}\,|V|:={\text {cl}}\,|V|\setminus |V|\) is a deformation retract of \({\text {cl}}\,|V|\subseteq X\). Otherwise we call V critical. We define the generalized multivector field \(\mathcal {V}\) on X as a partition of K into generalized multivectors. Using the modified concepts of regular and critical multivectors as in the case of Lefschetz complexes we introduce the directed graph \(G_\mathcal {V}\) and the associated dynamics. This lets us introduce the concept of an isolated invariant set S and the associated Conley index defined as the homotopy type of \(P_1/P_2\) for an index pair \((P_1,P_2)\) isolating S. The proofs rely on gluing the homotopies provided by the deformation retractions of the individual regular multivectors.

The main result of discrete Morse theory [10, Cor 3.5] establishes a homotopy equivalence between the original complex X and a complex \(X'\) with the number of k-dimensional cells equal to the number of critical points of the Morse function whose Morse index is k. A natural question is whether one can use multivectors instead of vectors to obtain a similar result under the additional assumption that all the critical cells are non-degenerate. It seems that the collapsing approach to discrete Morse theory presented in [6] extends to the case of combinatorial multivector fields on CW complexes defined in the preceding section. In the algebraic case of Lefschetz complexes one needs an extra assumption that the closure of every regular multivector is chain homotopy equivalent to its mouth.