Comments on: 1-manifolds and curves http://blog.mikael.johanssons.org/archive/2009/05/1-manifolds-and-curves/ Because my LiveJournal is too silly Sat, 17 Sep 2011 02:28:35 +0000 hourly 1 http://wordpress.org/?v=3.2.1 By: Michi http://blog.mikael.johanssons.org/archive/2009/05/1-manifolds-and-curves/comment-page-1/#comment-164529 Michi Sun, 03 May 2009 10:33:42 +0000 http://blog.mikael.johanssons.org/?p=204#comment-164529 Oh, but I don't see the difficulty in the gluing. Taking the first link you gave, the proof of that classification theorem starts with pointing out that a 1-manifold if homeomorphic to a graph with each vertex of degree 2. By connectivity, this graph is actually path-connected, and so we can modify any countable basis we pick for the manifold into a basis with each open set covering two adjacent edges of the graph, excluding the furthermost endpoints. These, in turn, are easily glued together. A more interesting trouble point is the long line - gluing together intervals indexed by infinite ordinals - but that vanishes due to the second countability of manifolds. All this said, this is all on the geometric side of what I'm doing, and I do not consider myself an expert in this. Oh, but I don’t see the difficulty in the gluing. Taking the first link you gave, the proof of that classification theorem starts with pointing out that a 1-manifold if homeomorphic to a graph with each vertex of degree 2. By connectivity, this graph is actually path-connected, and so we can modify any countable basis we pick for the manifold into a basis with each open set covering two adjacent edges of the graph, excluding the furthermost endpoints. These, in turn, are easily glued together.

A more interesting trouble point is the long line – gluing together intervals indexed by infinite ordinals – but that vanishes due to the second countability of manifolds.

All this said, this is all on the geometric side of what I’m doing, and I do not consider myself an expert in this.

]]> By: Rodolfo Medina http://blog.mikael.johanssons.org/archive/2009/05/1-manifolds-and-curves/comment-page-1/#comment-164528 Rodolfo Medina Sun, 03 May 2009 09:42:29 +0000 http://blog.mikael.johanssons.org/?p=204#comment-164528 Hallo, and thanks indeed for taking care of the problem. It's just in glueing together those pieces the difficulty of the problem. We have demonstrations in Christenson and Voxman, `Aspects of Topology' and in John M. Lee's `Introduction to Topological Manifolds', see http://books.google.com/books?pg=PA115&lpg=PA91&id=wyuzE2lSPAgC#PPA118,M1 . In John Milnor's `Topology from the Differentiable Viewpoint' we have the demonstration in the general case (i.e. for 1-manifolds with boundary), but in a much too synthetic way, and here: www.math.ist.utl.pt/~ggranja/TD/08/classif1manifs.pdf we have Milnor's result demonstrated more in details. This latter seems to me the most exhaustive treatment of the subject. Bye Rodolfo Hallo, and thanks indeed for taking care of the problem.

It’s just in glueing together those pieces the difficulty of the problem.

We have demonstrations in Christenson and Voxman, `Aspects of
Topology’ and in John M. Lee’s `Introduction to Topological Manifolds’,
see

http://books.google.com/books?pg=PA115&lpg=PA91&id=wyuzE2lSPAgC#PPA118,M1

. In John Milnor’s `Topology from the Differentiable Viewpoint’ we
have the demonstration in the general case (i.e. for 1-manifolds with
boundary), but in a much too synthetic way, and here:

http://www.math.ist.utl.pt/~ggranja/TD/08/classif1manifs.pdf

we have Milnor’s result demonstrated more in details. This latter seems to me
the most exhaustive treatment of the subject.

Bye
Rodolfo

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