Taggar/ SimplicialHomology

Abstract simplicial complex object in Wolfram Language

Contributed by: Naman T.

SimplicialHomology is a Wolfram language paclet (package) for constructing, manipulating, and analysing abstract simplicial complexes. It provides tools for computing simplicial homology, performing common topological constructions, and studying combinatorial invariants of finite simplicial complexes. So far, Mathematica lacked in-built or third party support for simplicial homology or discrete topology in general. Primarily, that motivated the development of this package. I have largely taken both inspiration and reference from the implementation of simplicial complexes in sagemath (Python).

Installation Instructions

To install this paclet in your Wolfram Language environment, evaluate this code:
PacletInstall["Taggar/SimplicialHomology"]

To load the code after installation, evaluate this code:
Needs["Taggar`SimplicialHomology`"]

Details

Features include construction of simplicial complexes from facets or cells, computation of reduced and unreduced homology groups, Euler characteristic and f-vectors, joins, cones, suspensions, and access to a collection of standard and enumerated simplicial complexes for testing and experimentation. The paclet is designed for research, education, and computational topology workflows in the Wolfram Language.

If you use the package for research or otherwise, please cite it as follows: Taggar, N. (2026). SimplicialHomology: implementation of abstract simplicial complex in Mathematica. arXiv:2608.14240[math.AT] https://arxiv.org/abs/2608.14240

Paclet Guide

Examples

Basic Examples (2)

Construct a simplicial complex from a given list of simplices as facets:

In[1]:=

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307.0436840057373`], List[153.98200225830078`, 235.99870109558105`], List[195.`, 212.31669425964355`], List[195.`, 22.863691329956055`], List[30.928985595703125`, 117.59068870544434`], List[30.928985595703125`, 307.0436840057373`]], List[List[200.`, 410.42970085144043`], List[364.0710144042969`, 315.7036876678467`], List[241.01799774169922`, 244.65868949890137`], List[200.`, 220.97669792175293`], List[158.98200225830078`, 244.65868949890137`], List[35.928985595703125`, 315.7036876678467`], List[200.`, 410.42970085144043`]], List[List[376.5710144042969`, 320.03370475769043`], List[202.5`, 420.53370475769043`], List[200.95300006866455`, 421.42667961120605`], List[199.04699993133545`, 421.42667961120605`], List[197.5`, 420.53370475769043`], List[23.428985595703125`, 320.03370475769043`], List[21.882003784179688`, 319.1406993865967`], List[20.928985595703125`, 317.4896984100342`], List[20.928985595703125`, 315.7036876678467`], List[20.928985595703125`, 114.70369529724121`], 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Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Taggar/SimplicialHomology", "Taggar`SimplicialHomology`SimplicialComplex"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{Simplex[{1, 2, 3}], Simplex[{2, 3, 4}]}]$

Out[1]=

Obtain information about the complex by clicking on + or viewing its properties:

In[2]:=

Out[2]=

In[3]:=

Out[3]=

Construct a complex from one of the named examples:

In[4]:=

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Out[4]=

Calculate its homology groups:

In[5]:=

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Out[5]=

Scope (8)

Construct a random simplicial complex on 12 vertices and of dimension 4:

In[6]:=

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Out[6]=

Calculate the reduced homology of a space:

In[7]:=

Out[7]=

Or relative homology:

In[8]:=

Out[8]=

Or cohomology:

In[9]:=

Out[9]=

Check whether two simplicial complexes are isomorphic:

In[10]:=

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Out[10]=

Get the associated chain complex of a simplicial complex:

In[11]:=

Out[11]=

Obtain the boundary matrices:

In[12]:=

Out[12]=

Find the automorphism group of a simplicial complex:

In[13]:=

Out[13]=

Verify that it is isomorphic to AlternatingGroup[5]:

In[14]:=

Out[14]=

Construct cones or suspensions of complexes:

In[15]:=

Out[15]=

In[16]:=

Out[16]=

In[17]:=

Out[17]=

Construct the stars of links of simplices:

In[18]:=

Out[18]=

In[19]:=

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Out[19]=

In[20]:=

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Out[20]=

Collect facets of all 180 abstract simplicial complexes made up of upto 5 vertices:

In[21]:=

Out[22]=

Create simplicial complexes out of them:

In[23]:=

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Perform analysis on them:

In[24]:=

Out[24]=

Find out the most common homology group:

In[25]:=

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"Taggar`SimplicialHomology`HomologyGroup"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Taggar/SimplicialHomology", "Taggar`SimplicialHomology`HomologyGroup"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]] /@ complexes], 1]$

Out[25]=

Publisher

Naman T.

Disclosures

Wolfram Language built-in symbols
Learn More »

Compatibility

Wolfram Language Version 15

External Links

Version History

1.2.1 – 26 August 2026
1.2.0 – 23 August 2026
1.1.1 – 08 August 2026
1.1.0 – 01 August 2026
1.0.0 – 10 July 2026

License Information

MIT License

Paclet Source

Source Metadata

Citation:
- Munkres, J. R. Elements of Algebraic Topology, Addison Wesley Publishing Company, 1984.
- Hatcher A., Algebraic Topology, Cambridge University Press, Cambridge, 2002.
- W. Kühnel and T. F. Banchoff, The 9-vertex complex projective plane, Math. Intelligencer 5 (1983), no. 3, 11-22. doi:10.1007/BF03026567
- M. Hachimori. http://infoshako.sk.tsukuba.ac.jp/~hachi/math/library/dunce_hat_eng.html.
- Anders Björner and Frank H. Lutz, Simplicial manifolds, bistellar flips and a 16-vertex triangulation of the Poincaré homology 3-sphere, Experiment. Math. 9 (2000), no. 2, 275-289.
- U. Brehm and W. Kuhnel, 15-vertex triangulations of an 8-manifold, Math. Annalen 294 (1992), no. 1, 167-193.
- M. E. Rudin. An unshellable triangulation of a tetrahedron. Bull. Amer. Math. Soc. 64 (1958), 90-91.
- Frank H. Lutz, Császár's Torus, Electronic Geometry Model No. 2001.02.069 (2002). http://www.eg-models.de/models/Classical_Models/2001.02.069/_direct_link.html.
- G. M. Ziegler. Shelling polyhedral 3-balls and 4-polytopes. Discrete Comput. Geom. 19 (1998), 159-174. doi:10.1007/PL00009339

Wolfram Language Paclet Repository