Can Non-Smooth C^1 Knots be Orientable?

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WWGD said:
Can you find some embedded n-submanifold with non-zero homology in the range {1,2,..,n-1}?

Consider the annulus A = {x in Rn : 1/2 < ||x|| < 1} and note that A is an n-dimensional submanifold of Rn. Then A deformation retracts onto the (n-1)-sphere and therefore has non-zero homology in dimension n-1.