Atlas of Fundamental Concepts

© 2026 Advay Mengle Β· Terms Β· Edit me!

Graph centered on Homology group, showing the selected concept and its surrounding relations.Preparing the interactive atlas…
Curated starting points

Stories & Views

Open the stories and views directory
Relationship-aware analysis

Compare concepts

Choose two concepts to compare or connect.
Reading the graph

Guide to the Atlas

Open full user guide

Canonical static concept record

Homology group

Open this concept in the interactive graphRead the Markdown equivalent

Summary

The quotient Hn=kerβ‘βˆ‚n/imβ‘βˆ‚n+1H_n=\ker\partial_n/\operatorname{im}\partial_{n+1}, measuring cycles modulo boundaries in a chain complex.

Record metadata

Canonical name
Homology group
Concept ID
homology_group
Content version
2.3.0
Primary field
Mathematics
Fields
Mathematics
Primary domain
Algebraic topology
Domains
Algebraic topology, Homological algebra, Algebra
Alternate terminology
No alternate terminology is currently authored for this record.

Carrier(s)

Data

Canonically induces

Concept sources

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Homotopy equivalenceHomology group

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID
story_homotopy_equivalence_to_homology_group_theorem_implication
Relation type
Theorem implication theorem-implication
Direction
source β†’ target
Endpoint roles
source: Implies by theorem; target: Follows by theorem from
Authored annotation
induces isomorphisms

Authored explanation

Homotopy-equivalent spaces have isomorphic singular homology groups.

How to interpret this relation type

Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.

Relation sources

Singular chain complexHomology group

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID
story_singular_chain_complex_to_homology_group_quotient_construction
Relation type
Quotient construction quotient-construction
Direction
source β†’ target
Endpoint roles
source: Quotients to; target: Obtained as quotient of
Authored annotation
cycles modulo boundaries

Authored explanation

In degree nn, representatives are cycles z∈Zn=kerβ‘βˆ‚nz\in Z_n=\ker\partial_n; identify z∼zβ€²z\sim z' exactly when zβˆ’zβ€²βˆˆBn=imβ‘βˆ‚n+1z-z'\in B_n=\operatorname{im}\partial_{n+1}. The quotient is Hn=Zn/BnH_n=Z_n/B_n.

How to interpret this relation type

Identify elements by a stated equivalence relation and equip the quotient with the induced structure. The edge detail must state the representatives and equivalence relation.

Relation sources

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Homology groupAbelian group

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID
story_homology_group_to_abelian_group_induced_forgotten
Relation type
Induced / forgotten induced-forgotten
Direction
source β†’ target
Endpoint roles
source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation
is an abelian group

Authored explanation

Homology groups are abelian groups under addition induced from the chain groups.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources