On mutational deformation retracts

Abstract

In ANR theory, the following result is well known: Suppose that X ′ is an ANR and X is a subspace of X ′ . Then X is a strong (or stationary) deformation retract of X ′ if and only if X is a deformation retract of X ′ . In this paper, a generalization of this result is obtained in Fox shape theory: Let r:U(X ′ ,P)→U(X,P) be a deformation mutational retraction. Then r is stationary if and only if r is regular, where a mutational retraction r:U(X ′ ,P)→U(X,P) is regular if for every U ′ ∈U(X ′ ,P) and for every r,r ′ ∈r with range U ′ , there is V ′ ∈U(X ′ ,P) such that r∣ ∣ V ′ ≃r ′ ∣ ∣ V ′ (rel. X ) in U ′ .