1. CMB 2007 (vol 50 pp. 206)
 Golasiński, Marek; Gonçalves, Daciberg Lima

Spherical Space Forms: Homotopy Types and SelfEquivalences for the Group $({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times SL_2\,(\mathbb{F}_p)$
Let $G=({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times
\SL_2(\mathbb{F}_p)$, and let $X(n)$ be an $n$dimensional
$CW$complex of the homotopy type of an $n$sphere. We study the
automorphism group $\Aut (G)$ in order to compute the number of
distinct homotopy types of spherical space forms with respect to free
and cellular $G$actions on all $CW$complexes $X(2dn1)$, where $2d$
is the period of $G$. The groups ${\mathcal E}(X(2dn1)/\mu)$ of self
homotopy equivalences of space forms $X(2dn1)/\mu$ associated with
free and cellular $G$actions $\mu$ on $X(2dn1)$ are determined as
well.
Keywords:automorphism group, $CW$complex, free and cellular $G$action, group of self homotopy equivalences, LyndonHochschildSerre spectral sequence, special (linear) group, spherical space form Categories:55M35, 55P15, 20E22, 20F28, 57S17 

2. CMB 1997 (vol 40 pp. 108)
 Schaer, J.

Continuous Selfmaps of the Circle
Given a continuous map $\delta$ from the circle $S$ to itself we
want to find all selfmaps $\sigma\colon S\to S$ for which
$\delta\circ\sigma = \delta$. If the degree $r$ of $\delta$ is not
zero, the transformations $\sigma$ form a subgroup of the cyclic
group $C_r$. If $r=0$, all such invertible transformations form a
group isomorphic either to a cyclic group $C_n$ or to a dihedral
group $D_n$ depending on whether all such transformations are
orientation preserving or not. Applied to the tangent image of
planar closed curves, this generalizes a result of Bisztriczky and
Rival [1]. The proof rests on the theorem: {\it Let
$\Delta\colon\bbd R\to\bbd R$ be continuous, nowhere constant, and
$\lim_{x\to \infty}\Delta(x)=\infty$, $ \lim_{x\to+\infty}\Delta
(x)=+\infty$; then the only continuous map $\Sigma\colon\bbd R\to\bbd
R$ such that $\Delta\circ\Sigma=\Delta$ is the identity
$\Sigma=\id_{\bbd R}$.
Categories:53A04, 55M25, 55M35 
