Hurewicz homomorphism
E911367
The Hurewicz homomorphism is a fundamental map in algebraic topology that relates the homotopy groups of a space to its homology groups, often serving as a bridge between geometric and algebraic invariants.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hurewicz homomorphism canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11219624 — 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: Hurewicz homomorphism Context triple: [Characteristic Classes, hasSubject, Hurewicz homomorphism]
-
A.
Hopf invariant
The Hopf invariant is a topological integer-valued invariant that classifies certain continuous maps between spheres, playing a central role in homotopy theory and the study of higher-dimensional linking.
-
B.
Whitehead product in homotopy theory
The Whitehead product in homotopy theory is a bilinear operation on homotopy groups that captures how spheres can be nontrivially linked or composed within a topological space.
-
C.
Whitehead product
The Whitehead product is a fundamental operation in algebraic topology that combines homotopy classes of maps to produce higher-order homotopy information, playing a key role in the structure of homotopy groups of spheres and related spaces.
-
D.
Eilenberg–MacLane spaces
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.
-
E.
Eilenberg–Zilber theorem
The Eilenberg–Zilber theorem is a fundamental result in algebraic topology that establishes a chain homotopy equivalence between the singular chain complex of a product space and the tensor product of the singular chain complexes of the factors.
- 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: Hurewicz homomorphism Target entity description: The Hurewicz homomorphism is a fundamental map in algebraic topology that relates the homotopy groups of a space to its homology groups, often serving as a bridge between geometric and algebraic invariants.
-
A.
Hopf invariant
The Hopf invariant is a topological integer-valued invariant that classifies certain continuous maps between spheres, playing a central role in homotopy theory and the study of higher-dimensional linking.
-
B.
Whitehead product in homotopy theory
The Whitehead product in homotopy theory is a bilinear operation on homotopy groups that captures how spheres can be nontrivially linked or composed within a topological space.
-
C.
Whitehead product
The Whitehead product is a fundamental operation in algebraic topology that combines homotopy classes of maps to produce higher-order homotopy information, playing a key role in the structure of homotopy groups of spheres and related spaces.
-
D.
Eilenberg–MacLane spaces
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.
-
E.
Eilenberg–Zilber theorem
The Eilenberg–Zilber theorem is a fundamental result in algebraic topology that establishes a chain homotopy equivalence between the singular chain complex of a product space and the tensor product of the singular chain complexes of the factors.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
concept in algebraic topology
ⓘ
homomorphism ⓘ |
| alsoKnownAs | Hurewicz map NERFINISHED ⓘ |
| appearsIn | Hurewicz theorem NERFINISHED ⓘ |
| assumes | basepoint choice for homotopy groups ⓘ |
| codomain | homology group H_n(X) of a topological space X ⓘ |
| construction | sends a homotopy class of maps S^n → X to the induced homology class of the fundamental cycle of S^n ⓘ |
| context |
CW-complexes
ⓘ
Postnikov towers NERFINISHED ⓘ stable homotopy theory ⓘ |
| definedBy | induced map on homology from representing sphere map ⓘ |
| domain | homotopy group π_n(X) of a topological space X ⓘ |
| field | algebraic topology ⓘ |
| generalizationOf | map from fundamental group to first homology group ⓘ |
| historicalPeriod | 20th-century mathematics ⓘ |
| isDefinedFor |
n ≥ 1
ⓘ
path-connected spaces ⓘ pointed topological spaces ⓘ |
| mathematicalDiscipline |
homological algebra
ⓘ
homotopy theory ⓘ |
| namedAfter | Witold Hurewicz NERFINISHED ⓘ |
| property |
for simply connected spaces, first nonzero Hurewicz homomorphism is an isomorphism under Hurewicz theorem hypotheses
ⓘ
is a group homomorphism ⓘ is compatible with long exact sequences of pairs ⓘ is natural with respect to continuous maps ⓘ |
| relatedTo |
Freudenthal suspension theorem
NERFINISHED
ⓘ
Whitehead theorem NERFINISHED ⓘ |
| relates |
homology groups
ⓘ
homotopy groups ⓘ |
| roleIn |
bridge between geometric and algebraic invariants of spaces
ⓘ
identifies first nontrivial homotopy group with corresponding homology group under connectivity assumptions ⓘ provides algebraic approximation to homotopy groups ⓘ |
| satisfies |
compatibility with suspension
ⓘ
naturality with respect to maps of spaces ⓘ |
| sourceStructure | homotopy group π_n(X) ⓘ |
| specialCase | for n = 1 coincides with abelianization map from π_1(X) to H_1(X) ⓘ |
| symbol | h_n ⓘ |
| targetStructure | abelian group H_n(X) ⓘ |
| usedIn |
classification of spaces up to homotopy type
ⓘ
computation of homotopy groups via homology ⓘ obstruction theory ⓘ spectral sequence arguments in homotopy theory ⓘ |
| usedToProve | relationships between connectivity and vanishing of homology groups ⓘ |
| uses |
homotopy classes of maps from spheres
ⓘ
singular homology ⓘ |
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
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Hurewicz homomorphism Description of subject: The Hurewicz homomorphism is a fundamental map in algebraic topology that relates the homotopy groups of a space to its homology groups, often serving as a bridge between geometric and algebraic invariants.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.