# Homology Groups¶

This module defines a HomologyGroup() class which is an abelian group that prints itself in a way that is suitable for homology groups.

sage.homology.homology_group.HomologyGroup(n, base_ring, invfac=None)

Abelian group on $$n$$ generators which represents a homology group in a fixed degree.

INPUT:

• n – integer; the number of generators
• base_ring – ring; the base ring over which the homology is computed
• inv_fac – list of integers; the invariant factors – ignored if the base ring is a field

OUTPUT:

A class that can represent the homology group in a fixed homological degree.

EXAMPLES:

sage: from sage.homology.homology_group import HomologyGroup
sage: G = AbelianGroup(5, [5,5,7,8,9]); G
Multiplicative Abelian group isomorphic to C5 x C5 x C7 x C8 x C9
sage: H = HomologyGroup(5, ZZ, [5,5,7,8,9]); H
C5 x C5 x C7 x C8 x C9
sage: AbelianGroup(4)
Multiplicative Abelian group isomorphic to Z x Z x Z x Z
sage: HomologyGroup(4, ZZ)
Z x Z x Z x Z
sage: HomologyGroup(100, ZZ)
Z^100

class sage.homology.homology_group.HomologyGroup_class(n, invfac)

Discrete Abelian group on $$n$$ generators. This class inherits from AdditiveAbelianGroup_fixed_gens; see sage.groups.additive_abelian.additive_abelian_group for more documentation. The main difference between the classes is in the print representation.

EXAMPLES:

sage: from sage.homology.homology_group import HomologyGroup
sage: G = AbelianGroup(5, [5,5,7,8,9]); G
Multiplicative Abelian group isomorphic to C5 x C5 x C7 x C8 x C9
sage: H = HomologyGroup(5, ZZ, [5,5,7,8,9]); H
C5 x C5 x C7 x C8 x C9
sage: G == loads(dumps(G))
True
sage: AbelianGroup(4)
Multiplicative Abelian group isomorphic to Z x Z x Z x Z
sage: HomologyGroup(4, ZZ)
Z x Z x Z x Z
sage: HomologyGroup(100, ZZ)
Z^100


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