Orientability of 1-Dimensional Manifolds: A Closer Look

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    Manifold Orientation
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Discussion Overview

The discussion centers on the orientability of 1-dimensional smooth manifolds, specifically addressing the conditions under which such manifolds can be considered orientable. Participants explore the relationship between the topology of 1-dimensional manifolds, namely the real line and the circle, and the existence of nowhere vanishing 1-forms as a criterion for orientability.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant asserts that any 1-dimensional topological manifold is either R or S1 and claims that every such manifold is orientable in the sense of orientation on simplices.
  • Another participant explains that while the argument for orientability works for R, it does not hold for S1 due to its inability to be covered by a single coordinate chart, suggesting the need for multiple charts to define a well-defined nowhere vanishing 1-form.
  • A participant reiterates the need for a nowhere vanishing 1-form, noting that while dx works on the real line, it does not remain non-zero on the circle, where dtheta would be applicable.
  • One participant points out that every connected one-dimensional topological manifold without boundary is either R or S1, adding a clarification to the discussion.
  • Another participant comments on the terminology, stating that "manifold" refers to a specific kind of "manifold with boundary," and emphasizes the importance of connectedness and second-countability in the definition of manifolds.

Areas of Agreement / Disagreement

Participants generally agree on the classification of 1-dimensional manifolds as either R or S1, but there is contention regarding the conditions for orientability, particularly in relation to the circle and the use of coordinate charts. The discussion remains unresolved regarding the implications of these conditions on orientability.

Contextual Notes

Limitations include the dependence on definitions of manifolds and the requirement for connectedness and second-countability, which some participants note as relevant but not universally agreed upon.

jem05
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I have the result that any 1 dim topological manifold is either R or S1. And I have the fact that every 1-dim topological manifold is orientable in the sense of orientation on simplices.

i want to get that any 1-dim manifold (smooth) is orientable, where orientability is given by the existence of a nowhere vanishing 1-form. Since i know my manifold is either the real line or the circle, does the section s:M --> T∗M that takes each point p to the differential dx at p work as the 1-form i need?
 
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For the real line, this argument works, because R is covered by a single coordinate chart. But the circle cannot be covered by a single coordinate chart (why?), so dx only defines a 1-form locally on some open subset of the circle.

So what you try to do for the circle is to find a collection of charts U_1,...,U_n such that dx^i coincides with dx^j on U_i n U_j. This way, the 1-form a defined by "a=dx^i (on U_i)", is a well defined nowhere vanishing 1-form.

Hint: use two angular coordinate charts.
 
jem05 said:
I have the result that any 1 dim topological manifold is either R or S1. And I have the fact that every 1-dim topological manifold is orientable in the sense of orientation on simplices.

i want to get that any 1-dim manifold (smooth) is orientable, where orientability is given by the existence of a nowhere vanishing 1-form. Since i know my manifold is either the real line or the circle, does the section s:M --> T∗M that takes each point p to the differential dx at p work as the 1-form i need?

dx works on the line

dtheta works on the circle.

dx is not everywhere non-zero on the circle.
 
Minor nitpick: every connected one-dimensional topological manifold without boundary is R or S1.
 
"Manifold" is a special kind of "manifold with boundary", not the other way around. A "manifold with boundary" that has a non-empty boundary is, in fact, not a manifold at all. (Of course, if you are considering "manifolds with boundary", you might decide to rename "manifold" to "manifold without boundary" to avoid this trap of language)

The connected bit is relevant though. (Also, we need second-countable, although some people opt to include that in their definition of manifold)
 

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