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Given a manifold M, there is an embedding A of M into the space of retractions of M, taking each point to the retraction of M to that point. Considering A as a map into the connected component containing its image, we prove that A is a weak homotopy equivalence for two choices of M, namely, the 2-sphere and the annulus. Introduction. In [3], Borsuk introduced the study of spaces of retractions and proved a theorem, which, when specialized to 2-manifolds, says that the space of retractions of a closed disk is contractible (in itself). Given a manifold M, there is a natural isometric embedding A of M into the space of retractions of M, defined by setting A(x) equal to the retraction taking M to x, for any xeM. Let £C(M) denote the component containing the image of A. Then ev o A is the identity map on M, where ev maps any retraction to the retraction evaluated at some basepoint of M. Hence the induced homomorphism A*: trn(M) -* 7rn(jSf(M)) is injective, for all «. In §2 and §3, we prove that A* is surjective for two choices of M, namely, the 2-sphere and the 2-dimensional annulus. Thus we prove that A in these two cases is a weak homotopy equivalence. We also show that for this pair of 2-manifolds, SE'(M) is just the space of nondeformation retractions of M. Hence these results show that the space of nondeformation retractions of the 2-sphere (resp. the annulus) has the same homotopy groups as the 2-sphere (resp. the annulus). The proofs rely heavily on a selection theorem of E. Michael, which is stated in §1. The proof in §3 requires certain results from conformai mapping theory which are also stated in §1. 1. Preliminaries. Throughout this paper we use certain standard notation, namely, En for Euclidean «-space, Sn for the unit «-sphere in En + 1, B" for the unit Presented to the Society January 23, 1970; received by the editors March 23, 1970 and, in revised form, November 3, 1970. AMS 1970 subject classifications. Primary 54C15, 54C35, 57A05; Secondary 54C65, 55D10.

The space of retractions of the 2-sphere and the annulus