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

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    $\begingroup$ I for one never encountered a definition of "two-sided". Could you provide me with a reference? $\endgroup$ – Johannes Hahn Dec 3 at 0:43
  • $\begingroup$ 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. $\endgroup$ – Wlod AA Dec 3 at 1:14
  • $\begingroup$ 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. $\endgroup$ – Wlod AA Dec 3 at 1:21
  • $\begingroup$ (It's hard to formulate an MO-answer because this material is about as old as the whole combinatorial/algebraic topology). $\endgroup$ – Wlod AA Dec 3 at 1:28
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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.

Technically, being two-sided is unrelated to being oriented, but there are cases where they are related.

For example, take the central circle in the Moebius band. The circle is orientable, the Moebius band is not, the normal-bundle is not trivial. This is in the one-sided case.

Or take the Moebius band as the 0-section in the 1-dimensional bundle (with orientable total space) that contains it. The Moebius band is not orientable. The total space of the 1-dimensional bundle over the Moebius band is orientable. This is again a 1-sided case.

If the ambient manifold is orientable (or really, just the total space of the tubular neighbourhood), then the triviality of the normal bundle is equivalent to the orientability of the co-dimension one submanifold.

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  • $\begingroup$ Ryan Budney, Thank you very much. I can understand two-sidedness now. When we say oriented sounds like we measure from internal. We need to use coordinate charts or homology to define it. For co-dimension 1, the two definition are equivalent in orientable ambient manifold. $\endgroup$ – Slm2004 Dec 4 at 13:25

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