# Difference of two-sided and oriented [closed]

I always encounter two definitions: two-sided and oriented (hypersurface or submanifold). What is the difference of them? Which one is stronger?

• I for one never encountered a definition of "two-sided". Could you provide me with a reference? – Johannes Hahn Dec 3 at 0:43
• Many of us can justify the "two-sided" notion as it was done in the past. I'd think that two-sideness is a theorem that follows from the definition of orientedness for codimension 1 surfaces. – Wlod AA Dec 3 at 1:14
• These notions are virtually equivalent (for codim 1) -- one is external while the other one is internal. The equivalence shows the invariance of the first one. – Wlod AA Dec 3 at 1:21
• (It's hard to formulate an MO-answer because this material is about as old as the whole combinatorial/algebraic topology). – Wlod AA Dec 3 at 1:28

If the submanifold $$M$$ of a manifold $$N$$ is co-dimension one, being two-sided typically means it has a trivial normal bundle, i.e. $$M$$ splits the normal bundle into two path components. These are the two sides.