A Characterization of Homology Manifolds

Abstract

A homology «-manifold is a space which has the same local homology at each point as Euclidean n-space. The principal result of this paper is the characterization of triangulable homology manifolds by a global property: The rc-circuit X is a homology manifold if and only if the diagonal cycle A in X x X is Poincare dual to a cocycle with support A (Theorem 1). If X is a smooth manifold, this cocycle represents the Thorn class of the tangent bundle of X. The homological properties of Thorn classes have been studied by Milnor [13; §11] and Spanier [14; Chapter 6]. The proof is based on their techniques. A corollary of this proof is that an w-circuit X satisfies Poincare duality if and only if there is a class U dual to the diagonal which has a certain symmetry with respect to the canonical involution T on XxX; namely U-^V = U^T*V for all V (Proposition 1). Furthermore, for any ^-circuit X, the diagonal cycle is dual to some cocycle U, if coefficients are in a field (Proposition 2). Thus U\(XxX—A) is the obstruction to X being a homology manifold. Propositions 1 and 2 have been obtained independently by P. Holm [8]. The ideas of Lefschetz about intersection theory and the topology of algebraic varieties have been my constant guide (cf. [15]). Theorem 1 can be interpreted in terms of the intersection pairing (Theorem 3). This paper is a revised version of part of my doctoral thesis at Brandeis University [11], written under the supervision of Professor Jerome Levine. I have also been helped by the questions and suggestions of P. Lynch, D. Stone, A. Landman, and especially D. Sullivan. My viewpoint has recently been influenced by the work of I. Fary [3]. Homology will be singular homology throughout, with integer coefficients in §§1 and 2, and field coefficients in §§3 and 4. Sign conventions for products are those of [14].