On the locally finite chain algebra of a proper homotopy type

In the classical paper [A-H] Adams-Hilton constructed a free chain algebra which is an important algebraic model of a simply connected homotopy type. We show that this chain algebra (endowed with an additional structure given by a “height function”) yields actually an invariant of a proper homotopy type. For this we introduce the homotopy category of locally finite chain algebras without using the usual methods of pro-categories. As examples we consider the locally finite chain algebras of R, S×S−{point}, and CP2−{point}. 1 Proper homotopy types of locally finite polyhedra. Let Top be the category of topological spaces. A map f : X → Y is proper if both f is closed and the fibre f−1(y) is compact for each point y ∈ Y . Let Topp be the subcategory of Top consisting of topological spaces and proper maps. The unit interval I = [0, 1] ⊂ R yields the cylinder IX = X × I in Top and Topp such that these categories are I-categories in the sense of [BAH;I §3], compare [BP;I.3.9] or [ADQ1]. Hence the homotopy categories Top/' and Topp/' are defined, and isomorphism types in these categories are homotopy types and proper homotopy types respectively. We are interested in new algebraic invariants of the proper homotopy type of a locally finite polyhedron. A polyhedron X is a topological space homeomorphic to a simplicial complex; if every vertex belongs to only finitely many simplices the polyhedron is locally finite, this is the case if and only if the space X is Received by the editors May 1995. Communicated by Y. Félix. 1991 Mathematics Subject Classification : Primary: 55P15, 55P99. Secondary: 55P35, 55U15.