This paper treats extension and
retraction properties in the category p of compact metric spaces with periodic maps
of a prime period p; the subspaces and maps in 𝒜p are called equivariant subspaces
and maps, respectively. The motivation of the paper is the following question: Let E
be a Euclidean space and a : E × E → E × E be the involution (x,y) → (y,x), i.e.,
the symmetry with respect to the diagonal. Suppose that Z is a symmetric (i.e.,
equivariant) closed subset of E × E which is an absolute retract; that is, Z is
a retract of E × E. When does there exist a symmetric (i.e., equivariant)
retraction E × E → Z? This is an extension problem in the category 𝒜p. If
X and Y are spaces in 𝒜p,A is a closed equivariant subspace of X and
f : A → Y is an equivariant map, then the existence of an extension of f does
not, in general, imply the existence of an equivariant extension. It is shown,
however, that if A contains all the fixed points of the periodic map and
dim(X −A) < ∞, then a condition for the existence of an extension is also sufficient
for the existence of an equivariant extension. In particular, it follows that a
finite dimensional space X in 𝒜p is an equivariant ANR (i.e., an absolute
neighborhood retract in the category 𝒜p) if and only if it is an ANR and the
fixed point set of the periodic map on X is an ANR. Generally speaking,
the paper deals with the question of symmetry in extension and retraction
problems.
