PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 70, Number 1, June 1978 SHORTER NOTES The purpose of this department is to publish very short papers of an unusually elegant and polished character, for which there is no other outlet. PIECEWISELINEAR EMBEDDINGS OF BALLS INTO CONTRACTIBLE MANIFOLDS KENNETH C. MILLETT1 Abstract. The simplicial space of proper piecewise linear embeddings of /¿-balls into «-dimensional contractible manifolds which are equal to a fixed embedding along the boundary is contractible if n — k —3. As a consequence of the Alexander isotopy theorem the simplicial space of flat (which equals locally flat unless n — p = 2) piecewise linear embeddings of Dp into D" which are standard on the boundary, E(DP, D"; rel dD"), is contractible. The purpose of this note is to publicize the fact that this is often the case when Dn is replaced by any contractible p.l. manifold, N. Since aN need not be simply connected this result does not follow from Morlet's disjunction lemma [1], [3]. Proposition. If N" is a contractible p.l. manifold and n — p > 3, then U¡(E(DP, N; rel dD")) = Ofar all i. A direct proof of this fact employs Theorem 1.11 of my AMS Memoir, Piecewise linear concordances and isotopies [2], to show that U¡(E(DP, N; rel aD")) is isomorphic to ]\(E(DP X V, N X V; rel o(Dp X V))), where one has a fixed standard embedding of Dp c N, /, and takes the associated standard embedding / x 1: Dp X I1: -> N X I'. One then analyzes the iso- topy classes of embeddings of Dp X V in yV X I' as follows: Suppose /_, and/+1 are two such embeddings-there is an associated embedding /: 9 ((Dp X 7' ) X 7) -+d ((N X 7' ) X 7) given by f_x\D» X V X {-1}, f+x\Dp X F X { + 1}, and f_x=f+x\o(Dpxr)xl. Presented to the Society, January 5, 1978; received by the editors August 29, 1977. AMS (MOS) subject classifications (1970). Primary 57C35. Key words and phrases. Alexander isotopy,' contractible p.l. manifold, embedding space. 'Partially supported by NSF Grant MCS77-03529. © American Mathematical Society 1978 85 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use