Function Repository Resource:

SimplicialHomologyGroup

Source Notebook

Compute the homology groups of simplicial complexes

Contributed by: Naman Taggar

ResourceFunction["SimplicialHomologyGroup"][sc,n]

computes the n^th homology group of sc.

ResourceFunction["SimplicialHomologyGroup"][sc]

computes all homology groups of sc.

ResourceFunction["SimplicialHomologyGroup"][sc,sub,n]

computes the n^th homology group of sc relative to sub.

ResourceFunction["SimplicialHomologyGroup"][sc,sub]

computes all homology groups of sc relative to sub.

Details and Options

SimplicialHomologyGroup depends upon the Taggar/SimplicialHomology paclet and installs it when used.

Both sc and sub should be specified by their facets, mesh regions, or using names.

The output is either a group or an association of groups indexed by dimension. The groups are specified as follows: is the additive group of integers; ℤ_n is the additive group of integers modulo n; ℚ is the additive group of rational numbers; and Gⁿ in general is the direct product of n copies of G.

The following named simplicial complexes are offered:

{"Simplex", n}

n-dimensional simplex

"Point"

point (0-dimensional simplex)

"Line"

line (1-dimensional simplex)

"Triangle"

triangle (2-dimensional simplex)

"Tetrahedron"

tetrahedron (3-dimensional simplex)

{"Circle", n}

n-dimensional circle

"Circle"

circle (1-dimensional circle)

"Sphere"

sphere (2-dimensional circle)

{"CircleWedge", n}

wedge product of n circles

"Torus"

1-holed (2-dimensional) torus

{"Torus", n}

n-dimensional torus

"KleinBottle"

Klein bottle

"MobiusStrip"

Mobius strip

"RealProjectivePlane"

real projective plane

{"RealProjectiveSpace", n}

n-dimensional real projective space

"ComplexProjectivePlane"

complex projective plane

"QuaternionicProjectivePlane"

quaternionic projective plane

"DunceHat"

dunce hat given by Hachimori (2016)

"PoincareHomologyThreeSphere"

Poincaré homology 3-sphere

"RudinBall"

non-shellable ball constructed by Rudin

"ZieglerBall"

non-shellable ball constructed by Ziegler

A direct product of groups in the output is represented as G₁⊕G₂⊕….

The following options can be given:

"Reduced"

whether to calculate reduced homology

"Coefficients"

coefficients over which to calculate homology

The value for "Coefficients" can be any of the following:

integers

rational numbers

p

elements of finite field p

Examples

Basic Examples (4)

Calculate a homology group of a space:

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0^th Homology group of two disjoint points demonstrating the number of connected components:

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Their reduced homology group:

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Calculate homology of a contractible complex:

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Examples of spaces whose homology have torsion:

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Scope (3)

Calculate all homology groups:

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Calculate the homology groups of D² relative to its boundary S¹:

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Compute all groups:

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Specify a simplicial complex using mesh region:

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Options (4)

Reduced (2)

By default, ordinary homology is calculated:

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Obtain reduced homology:

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Coefficients (2)

By default, homology is computed over the integers:

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Calculate over rationals or elements of a finite field:

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Applications (1)

Homology of 8 demonstrating the number of holes in 8:

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Properties and Relations (3)

The homology groups of a space relative to itself are always zero:

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The homology groups of a space relative to empty space are the same as absolute homology groups:

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Homology groups relative to a point are equal to reduced homology groups:

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Possible Issues (1)

A valid subcomplex must be provided:

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Publisher

Version History

1.0.0 – 05 August 2026

Related Resources

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License