I am revisiting some self-study I have done of manifolds. I often see statements such as "this holds for manifolds with or without boundary" or, "this only holds for manifolds without boundary", etc. This leads me to a general question:
What is the main issue/issues that arise when dealing with manifolds with boundary?
Is it simply the fact that in order to do arguments for points on the boundary, you have to consider smooth extensions of functions, while you do not have to do this for interior points?
Thanks!
One particular example:
"Cobordisms are studied both for the equivalence relation that they generate, and as objects in their own right. Cobordism is a much coarser equivalence relation than diffeomorphism or homeomorphism of manifolds, and is significantly easier to study and compute. It is not possible to classify manifolds up to diffeomorphism or homeomorphism in dimensions ≥ 4 – because the word problem for groups cannot be solved – but it is possible to classify manifolds up to cobordism. Cobordisms are central objects of study in geometric topology and algebraic topology."
https://en.wikipedia.org/wiki/Cobordism