I Simplicial complex geometric realization 1-manifold

[PsychonautQQ]

Prop 5.11 from John M. Lee's "Introduction to Topological Manifolds":If K is a simplicial complex whose geometric realization is a 1-manifold, each vertex of K lies one exactly two edges.

This proposition confuses me. If we look at the geometric realization of a simplex with two vertices, then this geometric realization could possibly 'look like' a curve in R^2, no? In this case, wouldn't each end point of the curve be a vertex laying on only one edge?

 

[lavinia]

I think by "manifold" he means a manifold without boundary. The boundary of a single 1 simplex is the two end points.

 

[lavinia]

I think this fact generalizes to higher dimensions. If the geometric realization of a simplicial complex is an n-manifold without boundary then each (n-1)-simplex is the face of exactly two n-simplices.