Eilenberg–MacLane spaces
E634842
Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of cohomology.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Eilenberg–MacLane spaces canonical | 6 |
| Eilenberg–MacLane space | 1 |
| K(G,n) | 1 |
| Postnikov systems | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7011078 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Eilenberg–MacLane spaces Context triple: [Samuel Eilenberg, notableWork, Eilenberg–MacLane spaces]
-
A.
"Algebraic Topology"
"Algebraic Topology" is a foundational mathematical text that develops topological concepts using algebraic methods such as homology and cohomology theories.
-
B.
Alexandrov–Čech cohomology
Alexandrov–Čech cohomology is a topological cohomology theory that computes invariants of spaces using inverse limits over open covers, closely related to and often coinciding with sheaf cohomology.
-
C.
Atiyah–Hirzebruch spectral sequence
The Atiyah–Hirzebruch spectral sequence is a fundamental computational tool in algebraic topology that relates generalized cohomology theories, such as K-theory, to ordinary cohomology, enabling the step-by-step calculation of these invariants from simpler data.
-
D.
Atiyah–Segal axioms
The Atiyah–Segal axioms are a set of mathematical conditions that rigorously define topological quantum field theories as functorial assignments from geometric data to algebraic structures.
-
E.
Grothendieck spectral sequence
The Grothendieck spectral sequence is a fundamental tool in homological algebra that relates the derived functors of a composite functor to the derived functors of its components, enabling efficient computation of cohomology.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Eilenberg–MacLane spaces Target entity description: Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of cohomology.
-
A.
"Algebraic Topology"
"Algebraic Topology" is a foundational mathematical text that develops topological concepts using algebraic methods such as homology and cohomology theories.
-
B.
Alexandrov–Čech cohomology
Alexandrov–Čech cohomology is a topological cohomology theory that computes invariants of spaces using inverse limits over open covers, closely related to and often coinciding with sheaf cohomology.
-
C.
Atiyah–Hirzebruch spectral sequence
The Atiyah–Hirzebruch spectral sequence is a fundamental computational tool in algebraic topology that relates generalized cohomology theories, such as K-theory, to ordinary cohomology, enabling the step-by-step calculation of these invariants from simpler data.
-
D.
Atiyah–Segal axioms
The Atiyah–Segal axioms are a set of mathematical conditions that rigorously define topological quantum field theories as functorial assignments from geometric data to algebraic structures.
-
E.
Grothendieck spectral sequence
The Grothendieck spectral sequence is a fundamental tool in homological algebra that relates the derived functors of a composite functor to the derived functors of its components, enabling efficient computation of cohomology.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
homotopy-theoretic construction
ⓘ
mathematical object ⓘ topological space ⓘ |
| appearsIn |
homological algebra
ⓘ
spectra representing ordinary cohomology ⓘ stable homotopy theory ⓘ |
| characterizedBy | having exactly one nontrivial homotopy group ⓘ |
| construction |
can be constructed as CW complexes with prescribed homotopy groups
ⓘ
can be obtained via bar constructions in some cases ⓘ |
| definedBy | π_n(X) ≅ G and π_k(X) = 0 for k ≠ n ⓘ |
| field |
algebraic topology
ⓘ
homotopy theory ⓘ |
| generalizationOf |
circle as K(ℤ,1)
ⓘ
infinite projective spaces as K(G,n) for suitable G and n ⓘ |
| hasHomotopyGroup |
π_k(K(G,n)) = 0 for k ≠ n
ⓘ
π_n(K(G,n)) ≅ G ⓘ |
| hasProperty |
can be taken as CW complexes
ⓘ
corepresent cohomology functors in the homotopy category ⓘ homotopy type determined by group G and integer n ⓘ path-connected when n ≥ 1 ⓘ serves as a classifying space for cohomology with coefficients in G ⓘ serves as building block for decomposing spaces via Postnikov towers ⓘ unique up to homotopy equivalence for fixed G and n ⓘ |
| introducedIn | mid 20th century ⓘ |
| namedAfter |
Samuel Eilenberg
ⓘ
Saunders Mac Lane ⓘ |
| notation |
K(G,n)
ⓘ
linked to:
Eilenberg–MacLane spaces
|
| parameter |
group G
ⓘ
integer n ≥ 0 ⓘ |
| relatedConcept |
Postnikov system
ⓘ
classifying space ⓘ cohomology group ⓘ homotopy group ⓘ |
| relatedTo |
Brown representability theorem
ⓘ
ordinary cohomology theories ⓘ |
| representsFunctor | [X,K(G,n)] ≅ H^n(X;G) ⓘ |
| specialCase |
K(0,n) is contractible for all n
ⓘ
K(ℤ,1) is homotopy equivalent to the circle S^1 ⓘ K(ℤ,2) is homotopy equivalent to the infinite complex projective space ℂP^∞ ⓘ K(ℤ/2ℤ,1) is homotopy equivalent to ℝP^∞ ⓘ |
| usedFor |
classifying cohomology classes
ⓘ
constructing Postnikov towers ⓘ constructing spectral sequences in algebraic topology ⓘ defining cohomology operations ⓘ defining singular cohomology ⓘ representing cohomology theories ⓘ studying homotopy types ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: Eilenberg–MacLane spaces Description of subject: Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of cohomology.
Referenced by (9)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Eilenberg–MacLane spaces
linked to: Eilenberg–MacLane spaces
subject linked to:
Eilenberg–MacLane spaces
linked to: Eilenberg–MacLane spaces
subject linked to:
CW complexes