Let $X$ be a compact surface of genus $g \geq 1$. Then it is a well known fact that the space of embeddings into $\mathbb{R}^{\infty}$ is contractible. The proof uses Whitney's embedding theorem. Moreover, this space a CW-complex by simplicial approximation on embedding spaces $X \rightarrow \mathbb{R}^n$.

Is there an article or text book that I may refer to. All papers I read just mention it without proof.


See page 86 of:

Peter W. Michor: Gauge theory for fiber bundles. Monographs and Textbooks in Physical Sciences, Lecture Notes 19, Bibliopolis, Napoli, (1991), 107 pp. MR 94a:53056. Zbl 953.53001 (pdf)

It is for $\ell^2$ there also with a universal Ehresmann connection. The proof adapts to $\mathbb R^{(\infty)}$ as used in section 47 of (here).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.