3

I was wondering if there is a description of the mapping class group of a wedge of $n$ circles.

Are the only kinds of homeomorphism classes in the mapping glass group are compositions of permutations of the circles and maps which send a circle to its inverse?

What about maps which send a circle from the wedge to a multiple of another circle? By multiple, I mean concatenating a circle several times.

Or maps that send a circle to two other circles concatenated?

Felix Y.
  • 673
  • My guess is that it's the outer automorphism group of $F_n$. This is a pretty complicated group. – Qiaochu Yuan Jun 18 '14 at 06:13
  • 4
    The group of self-homotopy equivalences is indeed $\mathrm{Out}(F_n)$. The group of self-homeomorphisms up to homotopy is just a finite group permuting and reflecting the circles. "Mapping class group" means different things to different people. – Cheerful Parsnip Jun 18 '14 at 06:15
  • Ah, of course, most of the automorphisms of $F_n$ can't even act as bijections. Silly me. – Qiaochu Yuan Jun 18 '14 at 06:57
  • Thanks! This answered most of my questions. – Felix Y. Jun 18 '14 at 07:13
  • 1
    To follow up the answer of @GrumpyParsnip, the group you get is the signed permutation group on $n$ symbols. http://groupprops.subwiki.org/wiki/Signed_symmetric_group – Lee Mosher Jun 18 '14 at 14:49

0 Answers0