CHomP -- Computational Homology Project software

Overview

This software is a tool for computational homology for both cellular complexes and induced homology on maps. It is based on discrete Morse theory. It is used to compute Conley indices in the Conley-Morse-Database project.

See "LICENSE" for license details. See "INSTALL" for more installation notes.

Installation

Dependencies

• boost
• CImg
• X11

General Instructions

• First install dependencies.
• Then, either:

Option 1. Install in default location.

To install into the system, i.e. /usr/local install with

./install.sh

Option 2. Install in a

Install via

./install.sh --prefix=/path/to/install/folder

Specific instructions for macOS:

For macOS the easiest way to install dependencies is with "homebrew" http://brew.sh.

Install homebrew via

/usr/bin/ruby -e "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/master/install)"

Install dependencies via

brew install boost
brew install cimg

To install X11, follow the instructions on https://www.xquartz.org

To install CHomP:

git clone https://github.com/shaunharker/CHomP.git
cd CHomP
./install.sh

To test an example:

chomp-simplicial ./examples/simplex.simp

Examples

Cubical complexes

Example:

chomp-cubical ./examples/square.cub

File format: a list of new-line separated d-tuples, where d is the dimension of the cubical complex, and the tuples represent d-dimensional cubes which will be added to the complex (along with all of their faces):

Example: (Annulus-like shape made out of squares)

(0, 1)
(0, 2)
(1, 0)
(1, 2)
(2, 0)
(2, 1)
(2, 2)

Simplicial complexes

chomp-simplicial ./examples/simplex.simp

File format: each line represents a simplex in the complex given by its vertices. For each simplex encountered, the simplex and all of its faces are generated. For instance 0 1 2 constructs a 2-cell, 3 1-cells, and 3 0-cells. The line 4 would create a single 0-cell corresponding to the vertex 4. Because subfaces are automatically created, it is sufficient to include maximal simplices (though allowed to include all)

Example: (Tetrahedron along with an isolated vertex)

0 1 2
0 1 3
0 2 3
1 2 3
4

General chain complexes (given by matrices for boundary map)

chomp-matrix ./examples/torus.mat

File format: The format is in terms of a sparse matrix representation of the boundary maps $d_n : C_{n+1} \to C_{n}$ for $n = 0 \cdots D-1$. The file has $D$ sections, corresponding to each map $d_0$, $d_1$, in turn. Each section begins with a line n, where $n$ is the index of the boundary map. Then follow lines indicating the non-zero entries for the incidence matrix for $d_n$. In particular each line is a triple i j k indicating that $d_{n}{i,j} = k$, i.e. the jth cell in $C{n+1}$ has in its boundary $k$ times the cell $i$ in $C_{n}$. It is allowed to include zero entries in the matrix explicitly, and it necessary for the software to learn of any cells which are not in the image of a boundary operator.

Example: (Torus; 1 2d-cell, 2 1d-cells, and 1 1d cell)

0
0 0 0
0 1 0
1
0 0 0
1 0 0

More information

Documentation can be found here.

Please e-mail Shaun Harker sharker81@gmail.com about problems or feature requests.