1 Sep 2005 13:53
Re: Preprint: A simple description of Thompson's group F
Peter Freyd <pjf <at> saul.cis.upenn.edu>
2005-09-01 11:53:54 GMT
2005-09-01 11:53:54 GMT
There's a good chance that the characterization of Thompson's group F
(not to mention its name) was set forth in the paper reviewed below
(the authors of which became aware of R.J.Thompson's priority via
this review).
Freyd, Peter; Heller, Alex
Splitting homotopy idempotents. II.
J. Pure Appl. Algebra 89 (1993), no. 1-2, 93--106.
A preliminary version of this paper was in the reviewer's hands in
1979 and was then of uncertain age. The authors have done a service in
publishing it (in somewhat revised form) belatedly.
The object of study is a free homotopy idempotent $f \colon X \to X$;
this means that $f$ is freely (base point not necessarily preserved
during the homotopy) homotopic to $f^2 \equiv f \circ f$. This $f$ is
said to split if there are maps $d \colon X \to Y$ and $u \colon Y \to
X$ such that $d \circ u \simeq \text{id}_Y$ and $u \circ d \simeq f$,
where $\simeq$ denotes free homotopy.
They construct a group $F$ and an endomorphism $\phi \colon F \to F$
such that, for a certain $\alpha_0 \in F$, $\phi^2(7) =
\alpha^{-1}_0\phi(7)\alpha_0$. The induced map $g \colon K(F,1) \to
K(F,1)$ is a homotopy idempotent which does not split; and it is
universal in the sense that it maps "canonically" into any homotopy
idempotent, and the corresponding homomorphism $F \to \pi_1(X)$ is
monic if and only if $f$ does not split.
This group $F$ is shown to be finitely presentable, has simple
commutator subgroup, is a totally ordered group and contains a copy of
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