This is certainly not a reasearch question, but I got no answers on math.stackexchange so I duplicate the question here.

Siebenmann on page 1 of his manuscript gives an example of an ENR space (euclidean neighborhood retract) whose group of homeomorphisms is not locally contactable. Take the sphere $S^3=R^3\cup\{\infty\}$ and a sequence of wild non-cellular arcs $A_n$ which are copies of an arc in a unit ball in $R^3$ translated by $(4n,0,0)$. Then factorize by sending each arc to a point. The resulting space is said not to be locally contactable since there are homeomorphisms arbitrary close to the identity which permute images of $A_n$ and none of these homeomorphisms is isotopic to the identity. The nonexistence of isotopy is not clear. Could anybody point where (and how) are exactly the wildness and non-cellularity used?

  • $\begingroup$ I wonder if you would be willing to explain a little more how these wild arcs work -- I don't like opening links on the web, and I'm having a little trouble understanding your question without reading the Siebenmann manuscript. Thanks! Anyway, perhaps you'll get more answers to your question if you explain the question a little more. $\endgroup$ – tweetie-bird Oct 19 '12 at 21:05
  • $\begingroup$ I would rephrase the question as "Why the space described is not locally contractable". Wild arc is an embedding of an interval which is not ambient isotopic to the standard one. Cellular subset of R^n is a compact subset of X such that each neighbourhood U of X contains an n-cell B, s.t. X\subset Int B\subset U\subset U (i.e. X has arbitrary small neighborhoods homeomorphic to R^n) $\endgroup$ – Dmitry K Oct 19 '12 at 22:41

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