Lefschetz Complexes

The fundamental structure underlying the functionality of ConleyDynamics.jl is a Lefschetz complex. It provides us with the basic model of phase space for combinatorial topological dynamics. In view of the combinatorial, and therefore discrete, character of the dynamical behavior, a Lefschetz complex is not a typical phase space in the sense of classical dynamics. While the latter one is usually a Euclidean space, a Lefschetz complex is basically a combinatorial model of it. In the following, we provide its precise mathematical definition, and explain how it can be created and modified within the package. We also discuss two important special cases, namely simplicial complexes and cubical complexes.

Basic Lefschetz Terminology

The original definition of a Lefschetz complex can be found in [Lef42], where it was simply referred to as a complex.

At first glance the above definition can seem daunting. However, it is based on a straightforward geometric idea. A Lefschetz complex is a structure that is built from elementary building blocks called cells. Each cell has a dimension associated with it, and it is topologically an open ball of this dimension. Thus, cells of dimension zero are points, also called vertices. Cells of dimension one are open curve segments, which we call edges, and two-dimensional cells are called faces and take the form of open two-dimensional membranes.

The incidence coefficient map encodes how these cells are glued together to form the Lefschetz complex $X$. In order to shed more light on this, consider the boundary map $\partial$ which is defined on cells via

\[ \partial x = \sum_{y \in X} \kappa(x,y) y \; .\]

This map sends a cell $x$ of dimension $k$ to a specific linear combination of cells of dimension $k-1$, called the boundary of $x$. By using ideas from linear algebra, the boundary map can be extended to map a general linear combination of $k$-dimensional cells to the corresponding linear combination of the separate boundaries. For example, if one chooses the field $F = \mathbb{Q}$ of rationals, one has $\partial (x_1 - 2x_2) = \partial x_1 - 2 \partial x_2$. Notice that using this extended definition of the boundary map, one can rewrite the summation condition in the definition of a Lefschetz complex in the equivalent form

\[ \partial( \partial x) = 0 \quad\text{ for all cells }\quad x \in X \; .\]

In other words, the boundary of any cell is itself boundaryless.

With the help of the boundary map, one can often infer the overall geometric structure of a Lefschetz complex $X$. For this, think of a Lefschetz complex as being build from the ground up in the following way. First, start by putting down all vertices of $X$ at different locations in some ambient space. Since the boundary of each one-dimensional cell is made up of a linear combination of vertices, one can then add a curve segment for each one-dimensional cell, which connects the vertices in its boundary. Note that in the general version of a Lefschetz complex it is possible that an edge has only one vertex in its boundary, or maybe even none, and in these cases the edge is either only connected to the one boundary vertex, or it is an open curve segment connected to no vertex at all, respectively. Continue in this fashion to add two-dimensional faces to fill in the space between the edges in its boundary, and so on for higher dimensions. Needless to say, in the case of a general complicated Lefschetz complex this procedure is of limited use, since the boundary of a cell can be an arbitrary linear combination of cells, with coefficients that can be any nonzero numbers in the field $F$. Yet, in many simple cases the above intuition is sufficient.

In addition to the Lefschetz complex definition, there are a handful of other concepts which will be important for our discussion of Lefschetz complexes. Specifically, the following notions are important:

• A facet of a cell $x \in X$ is any cell $y$ which satisfies $\kappa(x,y) \neq 0$.
• One can define a partial order on the cells of $X$ by letting $x \le y$ if and only if for some integer $n \in \mathbb{N}$ there exist cells $x = x_1, \ldots, x_n = y$ such that $x_k$ is a facet of $x_{k+1}$ for all $k = 1, \ldots, n-1$. It is not difficult to show that this defines a partial order on $X$, i.e., this relation is reflexive, antisymmetric, and transitive. We call this partial order the face relation. Moreover, if $x \le y$ then $x$ is called a face of $y$.
• A subset $C \subset X$ of a Lefschetz complex is called closed, if for every $x \in C$ all the faces of the cell $x$ are also contained in the subset $C$.
• The closure of a subset $C \subset X$ is the collection of all faces of all cells in $C$, and it is denoted by $\mathrm{cl}\, C$. Thus, a subset of a Lefschetz complex is closed if and only if it equals its closure.
• A subset $S \subset X$ is called locally closed, if its mouth $\mathrm{mo}\, S = \mathrm{cl}\, S \setminus S$ is closed. Note that every closed set is automatically locally closed, but the reverse implication is usually false.

While the first two points merely introduce notation for describing the combinatorial boundary of cells, the remaining three points establish important topological concepts. In fact, the above definition of closedness defines a topology on the Lefschetz complex $X$, which is the so-called Alexandrov topology from [Ale37]. As usual in the field of topology, a subset of a Lefschetz complex will be called open, if and only if its complement is closed.

We would like to point out that while the concept of local closedness is rarely considered in standard topology courses, it is of utmost important for the study of combinatorial topological dynamics. For the moment, we just mention the following result:

This result goes back to [MB09, Theorem 3.1]. In other words, in the category of Lefschetz complexes local closedness arises naturally. Due to its importance, we also mention the following two equivalent formulations:

• A subset $S \subset X$ is locally closed, if and only if it is the difference of two closed subsets of $X$.
• A subset $S \subset X$ is locally closed, if and only if it is an interval with respect to the face relation on $X$, i.e., whenever we have three cells with $S \ni x \le y \le z \in S$, then one has to have $y \in S$ as well.

The proof of these characterizations can be found in [MW23, Proposition 3.2] and [LKMW23, Proposition 3.10], respectively.

Lefschetz complexes are a very general mathematical concept, and they can be rather confusing at first sight. Nevertheless, they do encompass other complex types, which are more geometric in nature. As we already saw in the tutorial, every simplicial complex is automatically a Lefschetz complex, and we will further elaborate on this connection below. In addition, we will also demonstrate that cubical complexes are Lefschetz complexes. More general, any regular CW complex is a Lefschetz complex as well. For more details on this, we refer to the definition in [Mas91] and the discussion in [DKMW11].

Lefschetz Complex Data Structure

For the efficient and easy manipulation of Lefschetz complexes in ConleyDynamics.jl we make use of a specific composite data type:

LefschetzComplex

The fields of this struct relate to the mathematical definition of a Lefschetz complex $X$ in the following way:

• Internally, every cell of the Lefschetz complex is represented by an integer between 1 and the total number of cells. However, in order to make it easier to interpret the results of computations, each cell in a Lefschetz complex has to also be given a label. These labels are contained in the field labels::Vector{String}, where labels[k] gives the label of cell k.
• The vector dimensions is a Vector{Int} and collects the dimensions of the cells. In other words, the cell which is indexed by the integer k has dimension dimensions[k]. It is expected that the dimension vector is increasing, and the constructor method will verify this. Otherwise, an error is triggered.
• The incidence coefficient map $\kappa$ is encoded in the sparse matrix boundary. This matrix is a square matrix with ncells rows and columns. The $k$-th column contains the incidence coefficients $\kappa(k,\cdot)$ in the sense that the entry in row $m$ and column $k$ equals the value $\kappa(k,m)$. Since for most Lefschetz complexes the majority of the incidence coefficients is zero, the matrix is represented using the sparse format SparseMatrix, which is described in more detail in Sparse Matrices. An exception is raised if the square of the boundary matrix is not zero.

When creating a Lefschetz complex, only the above three items have to be specified, as they define a unique Lefschetz complex $X$. In other words, a Lefschetz complex is generally created via the command

   lc = LefschetzComplex(labels, dimensions, boundary)

During the construction of the Julia object, additional fields are initialized which simplify working with a Lefschetz complex:

• The integer ncells gives the total number of cells in $X$. Internally, these cells are numbered by integers ranging from 1 to ncells.
• The integer dim describes the overall dimension of the Lefschetz complex, which is the largest dimension of a cell.
• In order to easily determine the integer index for a cell with a specific label, the field indices contains a dictionary of type Dict{String,Int} which maps labels to indices. For example, if a cell has the label "124.010", then the associated integer index is given by indices["124.010"].

As mentioned above, note however that an object of type LefschetzComplex is created by passing only the first three the field items in the order given in LefschetzComplex. Consider for example the Lefschetz complex from Figure 4 in [MW23], see also the left complex in the next image. This complex consists of six cells with labels A, B, a, b, c, and alpha, and we initialize the vector of labels, the cell index dictionary, and the cell dimensions via the commands

ncL = 6
labelsL = Vector{String}(["A","B","a","b","c","alpha"])
cdimsL  = [0, 0, 1, 1, 1, 2]

The boundary matrix can then be defined using

bndmatrixL = zeros(Int, ncL, ncL)
bndmatrixL[[1,2],3] = [1; 1]     # a
bndmatrixL[[1,2],4] = [1; 1]     # b
bndmatrixL[[1,2],5] = [1; 1]     # c
bndmatrixL[[3,4],6] = [1; 1]     # alpha
bndsparseL = sparse_from_full(bndmatrixL, p=2)

Notice that we first create the matrix as a regular integer matrix, and then use the function sparse_from_full to turn it into sparse format over the field $GF(2)$ with characteristic p = 2. This is the most convenient method for small boundary matrices, yet for larger ones it is better to use the function sparse_from_lists. Finally, the Lefschetz complex is created using

lcL = LefschetzComplex(labelsL, cdimsL, bndsparseL)

Lefschetz complexes do not always have to contain cells of all dimensions. For example, the Lefschetz complex shown on the right side of the figure has no vertices, and it can be created using the commands

ncR = 4
labelsR  = Vector{String}(["a","b","c","alpha"])
cdimsR   = [1, 1, 1, 2]
bndmatrixR = zeros(Int, ncR, ncR)
bndmatrixR[[1,2,3],4] = [1; 1; 1]     # alpha
bndsparseR = sparse_from_full(bndmatrixR, p=2)
lcR = LefschetzComplex(labelsR, cdimsR, bndsparseR)

While Lefschetz complexes can always be created in ConleyDynamics.jl in this direct way, it is often more convenient to make use of special types, such as simplicial and cubical complexes, and then restrict the complex to a locally closed set using the function lefschetz_subcomplex.

Simplicial Complexes

One of the earliest types of complexes that have been studied in topology are simplicial complexes. As already mentioned in the tutorial, an abstract simplicial complex $X$ is a finite collection of finite sets, called simplices, which is closed under taking subsets. Each simplex $\sigma$ has a dimension $\dim\sigma$, which is one less than the number of its elements.

In order to see why every simplicial complex is automatically a Lefschetz complex, we need to be able to define the incidence coefficient map $\kappa$. For this, we make use of some notions from [Mun84]. Let $X_0$ denote the collection of all vertices of the simplicial complex $X$. Then we use the notation

\[ \sigma = \left[ v_0, v_1, \ldots, v_d \right] \quad\text{ with }\quad v_k \in X_0\]

to describe a $d$-dimensional simplex. Note that even though every simplex in $X$ is just the set of its vertices, in the above representation we pick an order of the vertices, called an orientation of the simplex. This orientation can be chosen arbitrarily, and there are two equivalence classes of orientations. To get from one orientation to the other, one just has to exchange two vertices, and we write

\[ \left[ \ldots, v_i, \ldots, v_j, \ldots \right] \; = \; -\left[ \ldots, v_j, \ldots, v_i, \ldots \right] \; .\]

For more complicated reorderings, one has to represent the corresponding vertex permutation as a sequence of such exchanges. Using these oriented simplices we can define the boundary operator

\[ \partial \sigma \; = \; \partial \left[ v_0, \ldots, v_d \right] \; = \; \sum_{i=0}^d (-1)^i \left[ v_0, \ldots, \hat{v}_i, \ldots, v_d \right] \; ,\]

where the notation $\hat{v}_i$ means that in the simplex behind the summation sign on the right-hand side the vertex $v_i$ is omitted. For example, for a two-dimensional simplex one obtains

\[ \partial \left[ v_0, v_1, v_2 \right] \; = \; \left[ v_1, v_2 \right] - \left[ v_0, v_2 \right] + \left[ v_0, v_1 \right] \; .\]

Thus, if one chooses a total order of all the vertices in the simplicial complex, and orients the individual simplices in such a way that their vertices are arranged using this overall order, then the incidence coefficient map is given by

\[ \kappa \left( \left[ v_0, \ldots, v_i, \ldots, v_d \right], \; \left[ v_0, \ldots, \hat{v}_i, \ldots, v_d \right] \right) \; = \; (-1)^i \; .\]

If some or all of the simplices are represented by different orientations, one simply has to multiply the value $(-1)^i$ by the sign of a suitable vertex permutation. In either case, one can show that the so-defined map $\kappa$ does indeed satisfy the definition of a Lefschetz complex. For more details, see [Mun84, Lemma 5.3].

In ConleyDynamics.jl there are three basic commands for defining a simplicial complex:

• create_simplicial_complex is the most general method, and it expects two input arguments. The first is usually called labels, and it has to have the data type Vector{String}. This vector lists the labels for each vertex. It is important that all of these labels have exactly the same number of characters. The second argument is usually called simplices, and it lists as many simplices as necessary for defining the underlying simplicial complex. This means that in practice one only needs to include the simplices which are not faces of higher-dimensional ones, see also the example below. The variable simplices can either be of type Vector{Vector{String}} or Vector{Vector{Int}}, depending on whether the vertices are identified via their labels or integer indices, respectively. Finally, the optional parameter p can be used to specify the underlying field for the boundary matrix. If p is a prime, then $F = GF(p)$, while for p = 0 the function uses $F = \mathbb{Q}$. If the argument p is ommitted, the function defaults to p = 2.
• create_simplicial_rectangle expects two integer arguments nx and ny, and then creates a triangulation of the square $[0,nx] \times [0,ny]$ by subdividing every unit square into four triangles which meet at the center of the square. As before, the optional parameter p specifies the underlying field.
• create_simplicial_delaunay creates a planar Delaunay triangulation inside a planar rectangle. The function selects a random sample of points inside the box, while either trying to maintain a minimum distance between the points, or just using a prespecified number of points. More details on these two options can be found in the documentation for the function.

To illustrate the first of these functions, consider the commands

labels = ["A","B","C","D","E","F","G","H"]
simplices = [["A","B"],["A","F"],["B","F"],["B","C","G"],["D","E","H"],["C","D"],["G","H"]]
sc = create_simplicial_complex(labels,simplices)

These create the simplicial complex sc, in the form of a Lefschetz complex. Note that the above commands only specify the labels for the vertices. The labels for simplices of dimension at least one are automatically generated by concatenating the labels for their vertices, sorted in lexicographic order. This can be seen in the following Julia output:

julia> sc.labels[end-4:end]
5-element Vector{String}:
 "DH"
 "EH"
 "GH"
 "BCG"
 "DEH"

The simplicial complex sc can be visualized using the commands

coords = [[0,0],[2,0],[4,0],[6,0],[8,0],[1,2],[4,2],[6,2]]
ldir   = [3,3,3,3,3,1,1,1]
fname  = "lefschetzex2.pdf"
plot_planar_simplicial(sc,coords,fname,labeldir=ldir,labeldis=10,hfac=2,vfac=1.5,sfac=50)

Similarly, the commands

sc2, coords2 = create_simplicial_rectangle(5,2)
fname2 = "lefschetzex3.pdf"
plot_planar_simplicial(sc2,coords2,fname2,hfac=2.0,vfac=1.2,sfac=75)

define and illustrate a second simplicial complex, which triangularizes a rectangle in the plane.

For a demonstration of the Delaunay triangulation approach, please see Analyzing Planar Vector Fields.

Cubical Complexes

The second important special case of a Lefschetz complex is called cubical complex, and it has been discussed in detail in [KMM04]. In the following, we only present the definitions that are essential for our purposes.

Loosely speaking, a cubical complex is a collection of cubes of varying dimensions in some Euclidean space $\mathbb{R}^d$. More precisely, we say that an interval $I \subset \mathbb{R}$ is an elementary interval if it is of the form

\[ I = [\ell, \ell+1] \quad\text{ or }\quad I = [\ell, \ell] \quad\text{ for some integer }\quad \ell \in \mathbb{Z} \; .\]

If the elementary interval $I$ consists of only one point, then it is called degenerate, and it is nondegenerate if it is of length one. Elementary intervals are the building blocks for the cubes in a cubical complex. For a complex in $\mathbb{R}^d$, an elementary cube $Q$ is of the form

\[ Q \; = \; I_1 \times I_2 \times \ldots \times I_d \; \subset \; \mathbb{R}^d \; ,\]

where $I_1, \ldots, I_d$ are elementary intervals. The dimension $\dim Q$ of an elementary cube is given by the number of nondegenerate intervals in its representation. For example, the cube $Q = [0,0] \times [1,1]$ is a zero-dimensional elementary cube in $\mathbb{R}^2$ which contains only the point $(0,1)$, while the elementary cube $R = [2,2] \times [3,4]$ is one-dimensional, and consists of the closed vertical line segment between the points $(2,3)$ and $(2,4)$.

After these preparations, the definition of a cubical complex is now straightforward. A cubical complex $X$ in $\mathbb{R}^d$ is a finite collection of elementary cubes in $\mathbb{R}^d$ which is closed under the inclusion of elementary subcubes. More precisely, if $Q \in X$ is an elementary cube in the cubical complex, and if $R \subset Q$ is any elementary cube contained in $Q$, then one also has $R \in X$.

The definition of a cubical complex is reminiscent of that of a simplicial complex. It is therefore not surprising that also in the cubical case one can describe the incidence coefficient map $\kappa$ explicitly, and thus recognize a cubical complex as a Lefschetz complex. For this, we need more notation.

Let $Q = I_1 \times I_2 \times \ldots \times I_d$ denote an elementary cube, and let the nondegenerate elementary intervals in this decomposition be given by $I_{i_1}, \ldots, I_{i_n}$, where $I_{i_j} = [k_j, k_j + 1]$ and $j = 1,\ldots,n = \dim Q$. For every index $j$, we further define the two $(n-1)$-dimensional elementary cubes

\[ \begin{array}{ccccccc} Q_j^- & = & I_1 \times \ldots \times I_{i_j - 1} & \times & [k_j, k_j] & \times & I_{i_j + 1} \times \ldots \times I_d \; , \\[2ex] Q_j^+ & = & I_1 \times \ldots \times I_{i_j - 1} & \times & [k_j + 1, k_j + 1] & \times & I_{i_j + 1} \times \ldots \times I_d \; . \end{array}\]

Geometrically, the two elementary cubes $Q_j^-$ and $Q_j^+$ are directly opposite sides of the elementary cube $Q$. Using them, one can define the algebraic boundary of the cube as

\[ \partial Q \; = \; \sum_{j=1}^n (-1)^{j-1} \left( Q_j^+ - Q_j^- \right) .\]

This formula is the cubical analogue of the boundary operator in a simplicial complex, and it allows us to define the incidence coefficient map via

\[ \kappa\left( Q, \, Q_j^+ \right) \; = \; (-1)^{j-1} \quad\text{ and }\quad \kappa\left( Q, \, Q_j^- \right) \; = \; (-1)^j \; , \quad\text{ for all }\quad j = 1, \ldots, n \; .\]

For all remaining pairs of elementary cubes in $X$ we let $\kappa = 0$. Then it was shown in [KMM04, Proposition 2.37] that the so-defined incidence coefficient map satisfies the summation condition in the definition of a Lefschetz complex, i.e., we have $\partial(\partial Q) = 0$ for every $Q \in X$. This in turn implies that every cubical complex is indeed a Lefschetz complex.

Cubical complexes in ConleyDynamics.jl are a little more restricted. Since a cubical complex in the above sense is always finite, one can assume without loss of generality that the left endpoints of all involved elementary intervals are nonnegative. In other words, we always assume that the cubical complex only contains elementary cubes from the set $(\mathbb{R}_0^+)^d$. This allows for a simple encoding of elementary cubes via labels of a fixed length, and without having to worry about the sign of an integer.

To describe this, fix a dimension $d$ of the ambient space. Then every elementary cube in $(\mathbb{R}_0^+)^d$ has the following label, which depends on a coordinate width $L$:

• The first $d \cdot L$ characters of the label encode the starting points of the elementary intervals $I_1, \ldots, I_d$ in the standard representation of the elementary cube. For this, the starting points, which are nonnegative integers, are concatenated without spaces, but with leading zeros. For example, with $L = 2$ the string "010203" would correspond to the starting points $1$, $2$, and $3$. Note that for given coordinate width $L$, one can only encode starting points between $0$ and $10^L-1$.
• The next entry in the label string is a period ..
• The remaining $d$ characters of the string are integers 0 or 1, which give the interval lengths of $I_1, \ldots, I_d$.

For example, for $L = 2$ the string "030600.000" corresponds to the point $(3,6,0)$ in three dimensions. Similarly, the label "030600.101" represents the two-dimensional elementary cube $[3,4] \times [6,6] \times [0,1] \subset \mathbb{R}^3$. Note, however, that the label representation is not unique, since it depends on the coordinate width $L$. Thus, with $L = 1$ the latter cube could also be written as "360.101", or with $L = 3$ as "003006000.101". As we will see in a moment, though, within a given cubical complex all labels have to use the same coordinate width $L$! This implies in particular that for a given coordinate width $L$ one can only represent bounded cubical complexes which are contained in the $d$-dimensional box $[0,10^L-1]^d$.

The following three helper functions simplify the work with these types of cube labels:

• cube_field_size determines the field sizes of a given cube label. The first return value gives the dimension $d$ of the ambient space, while the second value returns the coordinate width $L$.
• cube_information returns all information encoded in the cube label. The function returns an integer vector of length $2d+1$, where $d$ is the dimension of the ambient space. The first $d$ entries give the vector of elementary interval starting points, while the next $d$ values yield the corresponding interval lengths. The last entry specifies the dimension of the cube.
• cube_label creates a label from a cube's coordinate information. As function parameters, one has to specify $d$ and $L$, and then pass an integer vector of length six which specifies the coordinates of the starting points and the interval lengths as in the previous item.

In ConleyDynamics.jl there are four basic commands for defining a cubical complex and working with it:

• create_cubical_complex creates a cubical complex in the Lefschetz complex data format. The complex is specified via a list of all the highest-dimensional cubes which are necessary to define the cubical complex. For this, every cube has to be given using the above-described special label format, with the same coordinate width $L$. In other words, all label strings have to be of the same length! If the optional parameter p is specified, the complex will be defined over a field with characteristic p, analogous to the case of a simplicial complex. If the characteristic is not specified, then the function defaults to the field $GF(2)$.
• get_cubical_coords determines the coordinates of all vertices of a given cubical complex from the cube labels. This vector can then be used for plotting purposes, see below.
• create_cubical_rectangle creates a cubical complex covering a rectangle in the plane. The rectangle is given by the subset $[0,nx] \times [0,ny]$ of the plane, where the nonnegative integers nx and ny have to be passed as arguments to the function. The function returns the cubical complex, and a vector of coordinates for the vertices. The latter can also be randomly perturbed as described in more detail in the function documentation.
• create_cubical_box creates a cubical complex covering a box in three-dimensional Euclidean space. The box is given by the subset $[0,nx] \times [0,ny] \times [0,nz]$ of space, where the nonnegative integers nx, ny, and nz have to be passed as arguments to the function. The optional parameters are the same as in the planar version.

To illustrate the first of these functions, consider the commands

cubes = ["00.11", "01.01", "02.10", "11.10", "11.01", "22.00", "20.11", "31.01"]
cc = create_cubical_complex(cubes)

These create the cubical complex cc, in the form of a Lefschetz complex. It can be visualized using the commands

coords = get_cubical_coords(cc)
fname  = "lefschetzex4.pdf"
plot_planar_cubical(cc,coords,fname,hfac=2.2,vfac=1.1,cubefac=60)

Similarly, the commands

cc2, coords2 = create_cubical_rectangle(5,2)
fname2 = "lefschetzex5.pdf"
plot_planar_cubical(cc2,coords2,fname2,hfac=1.7,vfac=1.2,cubefac=75)

define and illustrate a second cubical complex.

Finally, it is also possible to perturb the vertices in a cubical rectangle to obtain a Lefschetz complex consisting of quadrilaterals in the plane. This can be accomplied as follows:

cc3, coords3 = create_cubical_rectangle(5,2,randomize=0.2)
fname3 = "lefschetzex6.pdf"
plot_planar_cubical(cc3,coords3,fname3,hfac=1.7,vfac=1.2,cubefac=75)

The resulting Lefschetz complex is visualized in the last figure of this section.

Lefschetz Complex Operations

Once a Lefschetz complex has been created, there are a number of manipulations and queries that one has to be able to perform on the complex. At the moment, ConleyDynamics.jl supplies a number of functions for this. The following three functions provide basic information:

• lefschetz_field returns the field $F$ over which the Lefschetz complex is defined as a String.
• lefschetz_is_closed determines whether a given Lefschetz complex cell subset is closed or not.
• lefschetz_is_locally_closed checks whether a given Lefschetz complex cell subset is locally closed or not.

The next set of functions can be used to extract certain topological features from a Lefschetz complex:

• lefschetz_boundary computes the support of the boundary $\partial\sigma$ of a Lefschetz complex cell $\sigma$. In other words, it returns the vector of all facets of $\sigma$. The cell can either be specified via its index or its label, and the return format corresponds to the input format.
• lefschetz_coboundary returns all cells which lie in the coboundary of the specified cell $\sigma$, i.e., it returns all cells which have $\sigma$ as a facet.
• lefschetz_closure determines the closure of a given cell subset, i.e., the union of all faces of cells in the cell subset.
• lefschetz_openhull computes the open hull of a cell subset, i.e., the smallest open set which contains the given cell subset.
• lefschetz_lchull finds the locally closed hull of a Lefschetz complex subset. This is the smallest locally closed set which contains the given cell subset. One can show that it is the intersection of the closure and the open hull of the cell subset.
• lefschetz_clomo_pair determines the closure-mouth-pair associated with a Lefschetz complex subset.
• lefschetz_skeleton computes the $k$-dimensional skeleton of a Lefschetz complex or of a given Lefschetz complex subset. While in the first case the $k$-skeleton of the full Lefschetz complex is returned, in the second case it returns the $k$-skeleton of the closure of the given subset.
• manifold_boundary returns a list of cells which form the "manifold boundary" of the given Lefschetz complex. More precisely, if the complex has dimension $d$, then it determines all cells of dimension $d-1$ which have at most one cell in their coboundary, as well as all cells of dimensions less than $d-1$ which have no cell in their coboundary, and finally returns the closure of the resulting cell subset.

The following functions create Lefschetz subcomplexes from a Lefschetz complex:

• lefschetz_subcomplex determines a Lefschetz subcomplex from a given Lefschetz complex. The subcomplex has to be locally closed, and it is given by a collection of cells.
• lefschetz_closed_subcomplex extracts a closed Lefschetz subcomplex from the given Lefschetz complex. The subcomplex is the closure of the specified collection of cells.
• permute_lefschetz_complex determines a new Lefschetz complex which is obtained from the original one by a permutation of the cells. Note that the permutation has to respect the ordering of the cells by dimension, otherwise an error is raised. In other words, the permutation has to decompose into permutations within each dimension. This is automatically done if no permutation is explicitly specified and the function creates a random one.

There are also two helper functions which can sometimes be useful:

• lefschetz_gfp_conversion changes the base field of the given Lefschetz complex from the rationals $\mathbb{Q}$ to a finite field $GF(p)$. Note that it is not possible to perform the reverse conversion.
• lefschetz_filtration computes a filtration on a Lefschetz subset. Based on integer filtration values assigned to some cells of the given Lefschetz complex, it determines the smallest closed subcomplex lcsub which contains all cells with nonzero filtration values, as well as filtration values fvalsub on this subcomplex, which give rise to a filtration of closed subcomplexes, and which can be used to compute persistent homology.

In addition, ConleyDynamics.jl provides the following helper functions for the fundamental objects of cells and cell subsets, which can be represented either by integer cell indices or by cell labels:

• convert_cells converts a vector of cells from integer to label format, or vice versa.
• convert_cellsubsets converts a vector of cell subsets from integer to label format, and vice versa.

Finally, there are a couple of ccordinate helper functions which allow for the transformation of vertex coordinates in a Lefschetz complex:

• convert_planar_coordinates transforms a given collection of planar coordinates in such a way that the extreme coordinates fit precisely in a given rectangle in the plane.
• convert_spatial_coordinates transforms a given collection of spatial coordinates in such a way that the extreme coordinates fit precisely in a given rectangular box in space.

For more details on the usage of any of these functions, please see their documentation in the API section of the manual.

References

See the full bibliography for a complete list of references cited throughout this documentation. This section cites the following references: