On the construction FK

Abstract

The next paper is an unpublished one by Milnor, to which many other authors have referred. The work on semi-simplicial loop-spaces has been carried further by Kan; however, Milnor's work remains the standard reference for the generalisation of Hilton's theorem. The prerequisite is a knowledge of semisimplicial complexes; see the remarks in §3 and the book by May. In particular, for the ‘geometrical realization’ appearing in Lemma 1, see May chap. Ill or Milnor, ‘The geometrical realization of a semi-simplicial complex’, Annals of Math. 65 (1957), 357-362. Introduction The reduced product construction of loan James assigns to each CW-complex a new CW-complex having the same homotopy type as the loops in the suspension of the original. This paper will describe an analogous construction proceeding from the category of semi-simplicial complexes to the category of group complexes. The properties of this construction FK are studied in §2. A theorem of Peter Hilton asserts that the space of loops in a union S 1 …S r of spheres splits into an infinite direct product of loops spaces of spheres. In §3 the construction of FK is applied to prove a generalization (Theorem 4) of Hilton's theorem in which the spheres may be replaced by the suspensions of arbitrary connected (semi-simplicial) complexes. The author is indebted to many helpful discussions with John Moore. The construction It will be understood that with every semi-simplicial complex there is to be associated a specified base point.