Homology circles and knot complements

Abstract

BY A lzonzology circle we mean a CW complex C (or a Kan complex) for which there is an isomorphism H,(C. 2) = H,(S’, 2) where S’ denotes the usual circle. Such spaces are always equipped with a map C + S’ which induces that isomorphism. The best known examples of homology circles are knot complements C = S”+‘N(K) where N(K) is an open tubular neighborhood of a submanifold K homeomorphic to the n-sphere S”. Here we are interested only in the homotop_v type of homology circles. Indeed it was proven by Browder, Lashof and Shaneson [9] that given the homotopy type of the pair (C, aC) it corresponds to, at most, two isotopy types of knots-if a,C = Z. From this point of view the homotopy type of (compact) homology circles is of major importance. Kervaire has done some fundamental work on the first non-trivial homotopy group ai(S”, C)[8] and we shall be interested in the higher homotopy groups. Kervaire also notes that “almost” arbitrary compact homology circles can be realized as knot complements, up to the middle dimension [see 5.12 below]. More generally, the present work is a continuation of the general work on homology isomorphism: How can one analyze and construct “all” homology circles? What are the possible homotopy groups and “k-invariants” of such spaces? What precise part of the (n + I)-type of C is determined by the n-type and what can be freely chosen? Such questions have relatively simple answers when one deals with e.g. acyclic spaces [3]. It is surprising to see that the case of homology circles turns out to be of the same general nature. But the details are more difficult since the fundamental group is not perfect. In the present paper we only deal with two types of homology circles: the (homotopy) finite case and the nilpotent case. We give a “classification” of these. It turns out that most of the important features of the general case arise already here and they assume here a concise and simple form. Roughly speaking the homotopy groups 7r” = TX appear naturally as extensions 0 + In, --, 7~, * nn IIT, + 0 where TV” is “a-perfect” and can be almost arbitrarily chosen and rr,,/I7r, depends on the (homologies of the) choices made in the lower dimension: In, for j <n. These considerations lead us to a general theorem which determines the structure of the second non-trivial homotopy group of a knot complement, in terms of the first non-trivial one (see 5.2). A word about the category in which we work: By “space” and “map” we mean objects and morphisms in the pointed category of CW complexes or Kan complexes. This small abuse of notation is harmless because we are interested in questions of homotopy type and the-geometric realizations provides an equivalence between the two categories [lo]. The paper is organized as follows. After formulating below the main results in general terms we recall some basic concepts and their properties in § I. $2 is devoted to the analysis of nilpotent homology circles. In 03 and 04 we explain and prove theorems relating to (homotopy) finite homology circles. Some applications and geometric remarks are given in $5.