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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.

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    $\begingroup$ There is a proof (without using Whitney's embedding theorem) and some references at nlab.mathforge.org/nlab/show/embedding+of+smooth+manifolds $\endgroup$
    – Andrew Stacey
    Commented Mar 26, 2013 at 10:45
  • $\begingroup$ Thank you for this link. Do you know some other reference where the proof is written down explicitly? $\endgroup$
    – berl13
    Commented Mar 26, 2013 at 10:57

1 Answer 1

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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).

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