Atlas of Fundamental Concepts

© 2026 Advay Mengle · Terms · Edit me!

Graph centered on Cohomology 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

Cohomology group

Open this concept in the interactive graphRead the Markdown equivalent

Summary

The quotient Hn=ker⁡δn/im⁡δn−1H^n=\ker\delta^n/\operatorname{im}\delta^{n-1} of cocycles by coboundaries.

Record metadata

Canonical name
Cohomology group
Concept ID
cohomology_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 equivalenceCohomology group

Permalink to relation

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

Relation ID
story_homotopy_equivalence_to_cohomology_group_theorem_implication
Relation type
Theorem implication theorem-implication
Direction
source → target
Endpoint roles
source: Implies by theorem; target: Follows by theorem from
Authored annotation
induces isomorphisms

Authored explanation

Homotopy-equivalent spaces have isomorphic singular cohomology 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 cochain complexCohomology group

Permalink to relation

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

Relation ID
story_singular_cochain_complex_to_cohomology_group_quotient_construction
Relation type
Quotient construction quotient-construction
Direction
source → target
Endpoint roles
source: Quotients to; target: Obtained as quotient of
Authored annotation
cocycles modulo coboundaries

Authored explanation

In degree nn, representatives are cocycles α∈Zn=ker⁡δn\alpha\in Z^n=\ker\delta^n; identify α∼α′\alpha\sim\alpha' exactly when α−α′∈Bn=im⁡δn−1\alpha-\alpha'\in B^n=\operatorname{im}\delta^{n-1}. The quotient is Hn(X;A)=Zn/BnH^n(X;A)=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.

Cohomology groupAbelian group

Permalink to relation

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

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

Authored explanation

Cohomology groups are abelian groups under addition induced from the cochain 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

Cohomology groupCohomology ring

Permalink to relation

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

Relation ID
story_cohomology_group_to_cohomology_ring_add_data
Relation type
Add data add-data
Direction
source → target
Endpoint roles
source: Builds toward; target: Built from
Authored annotation
collect all degrees and add the cup product

Authored explanation

With ring coefficients, collect the groups Hn(X;R)H^n(X;R) over all degrees and equip their graded direct sum with the cup product.

How to interpret this relation type

Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.

Relation sources