MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
There is a proof (without using Whitney's embedding theorem) and some references at – Loop Space Mar 26 '13 at 10:45
Thank you for this link. Do you know some other reference where the proof is written down explicitly? – berl13 Mar 26 '13 at 10:57
up vote 5 down vote accepted

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).

share|cite|improve this answer
Thank you for your help! – berl13 Mar 26 '13 at 13:44

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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