I recall having heard somewhere that the only 1-manifolds (second countable, Hausdorff, connected spaces locally homeomorphic to $\mathbb R$) are $\mathbb R$ and $S^1$. Is this true? If so, is there a reasonably elementary proof?
The only 1-manifolds are $\mathbb R$ and $S^1$
12
$\begingroup$
general-topology
manifolds
-
1@draks: and a link which is not a knot is not connected. – 2012-02-28
1 Answers
11
There's a proof outlined in Problems 17-5 and 17-7 of John Lee's "Introduction to Smooth Manifolds" that uses a basic classification of integral curves of vector fields, specifically that a nonconstant maximally defined integral curve is either injective or periodic, which implies (after a small amount of work) that the image of any nonconstant integral curve is diffeomorphic to either $\mathbb{R}$ or $\mathbb{S}^1$. The problem is finished by showing that any 1-manifold is orientable, and thus admits a nonvanishing global vector field, of which you consider a maximally defined integral curve.
I don't think this is the same proof as given in Guillemin and Pollack or in Milnor, and for my money it's quite a bit simpler than both.
-
0And when trying to ask if there shouldn't be a simpler, more intuitive proof, I did come up with idea of "choosing a global direction on our manifold (that will be our vector field) and then gluing together integral curves as long as we can. Finishing it as @youler outlined. – 2016-08-13