ABSTRACT
This chapter describes homology manifolds and derive some of their local properties. The class of homology manifolds includes, among other things, all topological manifolds, so it is a broad and important class of spaces. The trian- gulable homology manifolds will be the basic objects involved in the various duality theorems. If one thinks of the universal coefficient theorem as expressing a kind of algebraic duality between cohomology and homology, then Poincare duality can be thought of as basically geometric in nature. This duality does not hold for an arbitrary space, but depends specifically on properties possessed by manifolds. There is a certain amount of interplay between these geometric and algebraic types of duality. There are further duality theorems; they bear the names of Lefschetz, Alexander, and Pontryagin. The chapter considers the cohomology ring as the natural object, and view the existence of the homology intersection ring for manifolds as a happy accident resulting from the Poincare duality isomorphism.
