On the Homotopy Type of Manifolds

Abstract

1. The object of this note is to call attention to certain theorems, which follow very easily from some results due to E. Stiefel,' H. Seifert,2 Hassler Whitney,3 and myself.4 They refer to a class of manifolds which we call the class H, and are intended to throw light on the question, raised by W. Hurewicz,5 whether two closed manifolds of the same homotopy type are necessarily homeomorphic. The theorems depend both on M. H. A. Newman's6 theory of combinatorial equivalence, as re-developed by J. W. Alexander7 and carried further in S. S., and on theorems concerning differentiable manifolds. Therefore it is necessary to give a precise meaning to the term 'manifold'. By an n-dimensional manifold, M , we shall mean a class of combinatorially equivalent, simplicial complexes covering the same space, each complex being a formal manifold, meaning that the complement7 of each vertex is combinatorially equivalent to A or to A n-, according as the vertex in question is inside Mn or on Mn, where Ak stands for a closed k-simplex and Mn is the boundary of Mn. These covering complexes will be called proper triangulations of Mn (of course any simplicial complex covering Mn is a proper triangulation if the 'Hauptvermutung' is true). The proper triangulations of an unbounded manifold of class C', or smooth manifold, are to be Cl-triangulations.7a By a smooth, bounded, n-dimensional manifold we shall mean the manifold of which a sub-complex Ko C Kn is a proper triangulation, where Kn is a C' triangulation of a smooth, unbounded n-dimensional manifold and Ko is a formal manifold. By the topological product Mn X A k we shall mean the manifold having a normal subdivision of the cell-complex K n X A k as a proper triangulation where Kn is a proper triangulation of Mn. We shall use =to indicate combinatorial equivalence, and Mn Mn will mean that K Kn , where Ki is a proper triangulation of Mn .