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Homotopy theory, homological algebra, algebraic treatments of manifolds.

0 votes
1 answer
162 views

How many n-dimensional closed submanifolds of $R^n$ have Euler characteristic 1?

It is well-known that the closed $n$-ball has Euler characteristic $1$. Is it true that every closed (i.e., compact), connected $n$-dimensional submanifold (with boundary) of $\mathbb R^n$ having Eule …
1 vote
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219 views

Do closed hypersurfaces separate the euclidean space?

The following extension of the Jordan Curve Theorem is well known: every closed connected hypersurface of the sphere $\mathbb S^N$ separates $S^N$ into exactly two connected components. As a consequen …