Research Announcements

It has been conjectured that a manifold which is a homotopy sphere is topologically a sphere. This conjecture has implications, for example, in the theory of differentiate structures on spheres (see, e.g., [3, p. 33]). Here I shall sketch a proof of the following theorem: Let M be a piecewise-linear manifold of dimension n^7, which has the same homotopy-type as the n-sphere S. Then there is a piecewiselinear equivalence of M— {point} with euclidean n-space\ in particular, M is topologically equivalent to S. This theorem is not the best possible, for C. Zeeman has been able to refine the method presented here so as to prove the same theorem for n^5. A piecewise-linear n-manifold is a polyhedron with a linear triangulation satisfying the condition that the link of each vertex is combinatorially equivalent to the standard (n —1)-sphere; all the manifolds with which I am concerned here have no boundary. In general, all the spaces in this paper will be polyhedra, finite or infinite, and each map will be polyhedral, i.e., induced by a simplicial map of linear triangulations. Let K be a finite subpolyhedron of the finite polyhedron L; let K' be a finite subpolyhedron of the finite polyhedron Z/; l e t / : L—>L' be a polyhedral map. ƒ is called a relative equivalence (L, K)=$(L'', K'), iîf(K) C.K' and L — K is mapped by ƒ in a 1-1 manner onto L' — K'. Recall J. H. C. Whitehead's definition of contraction [7, p. 247]: If the simplicial complex A has a simplex a which is the face of just one simplex r, and B is the simplicial complex obtained from A by removing the open simplexes a and T, then A—>B is called an elementary contraction dit a. A finite sequence of elementary contractions is a contraction. If K is a finite subpolyhedron of the finite polyhedron Z, then it is said tha t L contracts onto K, if there is a linear triangulation A of L, 1 The author has a fellowship of the National Science Foundation, U.S.A.