Milnor–Wood inequality
E265520
The Milnor–Wood inequality is a result in differential geometry and topology that bounds the Euler class of flat circle bundles over surfaces, with important implications for foliations and group actions on the circle.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Milnor–Wood inequality canonical | 3 |
| Milnor–Wood inequality for flat circle bundles | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2418320 — 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: Milnor–Wood inequality Context triple: [John Milnor, notableWork, Milnor–Wood inequality]
-
A.
Teichmüller curve
A Teichmüller curve is a complex geodesic in the moduli space of Riemann surfaces that arises from flat surface structures and has rich connections to dynamics, geometry, and number theory.
-
B.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
C.
Hurwitz bound on automorphism groups of curves
The Hurwitz bound on automorphism groups of curves is a classical result in algebraic geometry stating that a compact Riemann surface of genus at least 2 has at most 84(g − 1) automorphisms.
-
D.
Atiyah–Bott fixed-point theorem
The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
-
E.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
- 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: Milnor–Wood inequality Target entity description: The Milnor–Wood inequality is a result in differential geometry and topology that bounds the Euler class of flat circle bundles over surfaces, with important implications for foliations and group actions on the circle.
-
A.
Teichmüller curve
A Teichmüller curve is a complex geodesic in the moduli space of Riemann surfaces that arises from flat surface structures and has rich connections to dynamics, geometry, and number theory.
-
B.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
C.
Hurwitz bound on automorphism groups of curves
The Hurwitz bound on automorphism groups of curves is a classical result in algebraic geometry stating that a compact Riemann surface of genus at least 2 has at most 84(g − 1) automorphisms.
-
D.
Atiyah–Bott fixed-point theorem
The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
-
E.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in differential geometry ⓘ result in geometric topology ⓘ |
| appliesTo |
flat S^1-bundles
ⓘ
oriented closed surfaces ⓘ representations of surface groups into Homeo^+(S^1) ⓘ |
| context |
dynamics of circle homeomorphisms
ⓘ
topology of surface bundles ⓘ |
| field |
differential geometry
ⓘ
geometric group theory ⓘ topology ⓘ |
| generalizedBy | Burger–Iozzi–Wienhard inequalities for higher rank groups ⓘ |
| givesBoundOn |
Euler class of flat circle bundles
ⓘ
Euler number of representations into Homeo^+(S^1) ⓘ |
| hasConsequence |
constraints on foliations of circle bundles
ⓘ
constraints on group actions on S^1 ⓘ rigidity of maximal Euler number representations ⓘ |
| hasForm | |e(ρ)| ≤ |χ(Σ)| for a representation ρ of π1(Σ) into Homeo^+(S^1) ⓘ |
| holdsFor | flat PSL(2,R)-bundles over surfaces ⓘ |
| implies | maximal Euler class corresponds to Fuchsian representations into PSL(2,R) ⓘ |
| involves |
Euler class in H^2
ⓘ
Euler number ⓘ bounded cohomology ⓘ characteristic classes ⓘ flat connection ⓘ foliations ⓘ |
| language |
characteristic classes of foliations
ⓘ
cohomology of groups ⓘ |
| mainSubject |
Euler class
ⓘ
flat circle bundles ⓘ group actions on the circle ⓘ surface group representations ⓘ |
| namedAfter |
John Milnor
ⓘ
John W. Wood ⓘ |
| originallyFormulatedFor | flat oriented 2-plane bundles over surfaces ⓘ |
| relatedTo |
Chern–Weil theory
ⓘ
Ghys’s work on group actions on the circle ⓘ Godbillon–Vey invariant ⓘ Toledo invariant ⓘ |
| relates |
Euler number
ⓘ
genus of the base surface ⓘ |
| strengthenedBy | John W. Wood ⓘ |
| usedIn |
bounded cohomology of groups
ⓘ
classification of surface group actions on S^1 ⓘ study of flat bundles ⓘ study of foliated bundles ⓘ |
| yearProvedByMilnor | 1958 ⓘ |
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: Milnor–Wood inequality Description of subject: The Milnor–Wood inequality is a result in differential geometry and topology that bounds the Euler class of flat circle bundles over surfaces, with important implications for foliations and group actions on the circle.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Milnor
Burger–Iozzi–Wienhard inequalities for higher rank groups
→
generalizes
→
Milnor–Wood inequality for flat circle bundles
ⓘ
linked to: Milnor–Wood inequality