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 generatorsbase_ring
– ring; the base ring over which the homology is computedinv_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 # needs sage.groups 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) # needs sage.groups 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)#
Bases:
AdditiveAbelianGroup_fixed_gens
Discrete Abelian group on \(n\) generators. This class inherits from
AdditiveAbelianGroup_fixed_gens
; seesage.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 # needs sage.groups 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)) # needs sage.groups True sage: AbelianGroup(4) # needs sage.groups 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