Which Fields Are Naturally Manifolds Beyond ##\mathbb{R}## and ##\mathbb{C}##?

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Discussion Overview

The discussion revolves around identifying fields, beyond the real numbers ##\mathbb{R}## and complex numbers ##\mathbb{C}##, that can be considered "naturally" as manifolds. Participants explore the relationship between fields, Lie groups, and manifold structures, as well as the implications of these relationships in the context of finite-dimensional vector spaces and topological fields.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants inquire about fields that can be viewed as manifolds, suggesting that ##\mathbb{R}## and ##\mathbb{C}## are the primary examples.
  • There is a proposal to consider these fields as groups and to investigate whether they qualify as Lie groups, with the assertion that every Lie group is a differentiable manifold.
  • One participant suggests that finite-dimensional vector spaces over a field ##\mathbb{F}## can be viewed as manifolds, as they are isomorphic to ##\mathbb{F}^n##.
  • Another participant mentions that every "Lie field" must also be a commutative Lie group, noting that connected commutative Lie groups take a specific form involving products of circles and Euclidean spaces.
  • There is a discussion about the nature of the additive group of a Lie field and whether certain spheres, such as ##S^3## and ##S^7##, can be classified as connected commutative Lie groups, with a later clarification that these spheres are not commutative.
  • A participant provides a link to a proof regarding which spheres can be classified as Lie groups, specifically mentioning the circle and the 3-sphere.

Areas of Agreement / Disagreement

Participants express differing views on the classification of fields and their manifold structures, particularly regarding the nature of Lie groups and the conditions under which certain fields can be considered manifolds. The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

There are limitations regarding the definitions and properties of fields and groups discussed, as well as the assumptions underlying the classification of certain mathematical structures as manifolds. The complexity of proving the properties of topological fields is acknowledged but not fully explored.

WWGD
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Does anyone know of fields (with additional structure/properties) other than either ##\mathbb R, \mathbb C ## that are "naturally" manifolds?

Thanks.
 
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Just a comment that we could look at the two as groups and then the issue is whether these are Lie groups.
 
WWGD said:
Just a comment that we could look at the two as groups and then the issue is whether these are Lie groups.
Every Lie group is a manifold - in fact a differentiable manifold. I don't think there are any other fields that are a Euclidean space.
 
Yes, thanks, that is where I was going with the comment. I don't know if there are similar results for fields; I was actually thinking of finite-dimensional vector spaces as manifolds, given that an f.d vector space over a field ##\mathbb F ## is isomorphic to ##\mathbb F^n ##
 
Hint: every "Lie field" must also be a commutative Lie group. And the connected commutative Lie groups are exactly of the form ##S^1\times ... \times S^1\times \mathbb{R}^n##.

More generally, you could be interested in topological fields. It turns out that the only connected, locally compact fields are ##\mathbb{R}## and ##\mathbb{C}##, but this is a tad more difficult to prove than the Lie case.
 
Last edited:
Do you mean that the restriction of the Lie field to the additive (Abelian) group is a Lie group? And aren't ##S^3, S^7 ## also connected, commutative Lie groups? EDIT: Never mind, these are not Abelian.
 
WWGD said:
Do you mean that the restriction of the Lie field to the additive (Abelian) group is a Lie group? And aren't ##S^3, S^7 ## also connected, commutative Lie groups?
they are not commutative
 
Yes, I just remembered and edited, sorry.
 

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