Vol. 45, No. 1, 1973

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Equivariant extensions of maps

Jan W. Jaworowski

Vol. 45 (1973), No. 1, 229–244

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.

Mathematical Subject Classification 2000
Primary: 54C20
Secondary: 54C55
Received: 23 December 1971
Revised: 17 March 1972
Published: 1 March 1973
Jan W. Jaworowski