2
$\begingroup$

The space of smooth embeddings of a manifold $M$ into $\mathbb{R}^\infty$ is contractible by the Whitney embedding argument.

Question: Is the space of topological embeddings of a manifold $M$ into $\mathbb{R}^\infty$ contractible? What about if $M$ is CW complex?

I suspect that the answer is no, but I am looking for a concrete counterexample.

$\endgroup$
2
  • $\begingroup$ This sounds reasonable. Did you check the book by Daverman and Venema on embeddings of manifolds? As a possible strategy for a proof (at least for compact $M$) I would start with a relative version of Menger's embedding theorem for finite dimensional metrizable compacts. $\endgroup$ Commented 6 hours ago
  • $\begingroup$ Yes, indeed, one has: For every $n$-dimensional compact $K$ the space of embeddings $K\to \mathbb R^{2n+1}$ is weakly contractible in the space of all embeddings $K\to \mathbb R^{\infty}$. This covers the case of compact manifolds and finite CW complexes. I am not completely sure about the noncompact case. $\endgroup$ Commented 1 hour ago

0

Reset to default

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.