New documentation

Examples and usage documentation is here (under construction).

Usage guide

After compiling the library, several executables will be placed in capd/capdRedHom/bin subdirectory. The main executables are:

• redHom - computes Betti numbers and torsion coefficients of cubical sets, simplicial complexes and general, regular CW complexes using geometric reductions/coreductions and discrete Morse theory
• cubRedHom - a version of redHom specialized for cubical sets. Uses reductions via aycyclic subspace construction as well as geometric reductions/coreductions. Slightly faster than redHom in low dimensions.
• CrbHomMD - computes homology generators of cubical sets using geometric reductions/coreductions
• CrbMapHomMD - computes homology of acyclic valued, upper semicontinuous cubical maps

Below we provide basic instructions how to use the main executables.

redHom

The simplest use is
redHom fileName
where fileName is the name of the input file containing a cubical set or simplicial complex in one of the acceptable formats. This will execute in turn: shaving, coreduction algorithm, discrete Morse theory reduction and finally algebraic reductions. Output will contain homology groups (Betti numbers and torsion coefficients).

Command line options

• -h
  displays help information
• --filetype ext or briefly -f ext
  disregards the file extension in determining the format and forces the format provided by ext, which must be one of (fcub, pcub, dat, obj) described in details in section formats.
• --shave
  performs only shaving
• --coreduce
  performs only coreductions
• --morse
  performs only discrete Morse theory reductions

cubRedHom

The simplest use is
cubRedHom fileName
where fileName is the name of the input file containing a cubical set in one of the acceptable formats. This will execute reductions via acyclic subspace for 2 or 3 dimensional full cubical sets or coreductions for other inputs followed by algebraic reductions. Output will contain homology groups (Betti numbers and torsion coefficients).

Command line options

• mod=p
  computes Betti numbers over Z_p coefficients with p a prime
• silent=1
  outputs homology only, all auxiliary information is excluded from output.

CrbHomMD

The simplest use is
CrbHomMD fileName
where fileName is the name of the input file containing a cubical set in one of the acceptable formats.

This will execute coreductions followed by algebraic reductions. Output will contain homology generators.

CrbMapHomMD

The simplest use is
CrbMapHomMD fileName
where fileName is the name of the input file containing an acyclic valued, upper semicontinuous cubical map in the prescribed format. This will execute in turn

• the construction of the graph
• the coreductions of the domain, range and graph
• the construction of the chain maps of the projections of the reduced graph onto reduced domain and range
• the algebraic reductions of the domain, range, graph and the projections.

Output will contain the matrices of the maps induced in homology as well as homology generators of the domain, range and graph.