We give a new approach to intersection theory. Our "cycles" are closed manifolds mapping into compact manifolds and our "intersections" are elements of a homotopy group of a certain Thom space. The results are then applied in various contexts, including fixed point, linking and disjunction problems. Our main theorems resemble those of Hatcher and Quinn [17], but our proofs are fundamentally different. Errata Minor errors were corrected on page 967 (18 February 2008).

ABSTRACT. Manifold approximate fibrations arise in the geometric topology of manifolds and group actions on topological manifolds. The primary purpose of this paper is to classify manifold approximate fibrations in terms of the lifting problem for a certain bundle. Our classification meshes well with the classical classifications of fibrations and bundles and, hence, we are able to attack questions such as the following. When is a fibration controlled homotopy equivalent to a manifold approximate fibration? When is a manifold approximate fibration controlled homeomorphic to a bundle?Let Bi be a topological manifold. Recall that a manifold approximate fibration over B is a proper map q: M -+ B such that M is a manifold (topological or Hilbert cube) and such that q satisfies an approximate lifting condition (see [8] or § I.D). This "bundle" theory plays an important role in the study of topological manifolds. Consider the following examples. Let Bi be a topological manifold. Recall that a manifold approximate fibration over B is a proper map q: M -+ B such that M is a manifold (topological or Hilbert cube) and such that q satisfies an approximate lifting condition (see [8] or § I.D). This "bundle" theory plays an important role in the study of topological manifolds. Consider the following examples.

ABSTRACT. Manifold approximate fibrations arise in the geometric topology of manifolds and group actions on topological manifolds. The primary purpose of this paper is to classify manifold approximate fibrations in terms of the lifting problem for a certain bundle. Our classification meshes well with the classical classifications of fibrations and bundles and, hence, we are able to attack questions such as the following. When is a fibration controlled homotopy equivalent to a manifold approximate fibration? When is a manifold approximate fibration controlled homeomorphic to a bundle?Let Bi be a topological manifold. Recall that a manifold approximate fibration over B is a proper map q: M -+ B such that M is a manifold (topological or Hilbert cube) and such that q satisfies an approximate lifting condition (see [8] or § I.D). This "bundle" theory plays an important role in the study of topological manifolds. Consider the following examples. Let Bi be a topological manifold. Recall that a manifold approximate fibration over B is a proper map q: M -+ B such that M is a manifold (topological or Hilbert cube) and such that q satisfies an approximate lifting condition (see [8] or § I.D). This "bundle" theory plays an important role in the study of topological manifolds. Consider the following examples.

Abstract. For any finite group G, we define a bivariant functor from the Dress category of finite G-sets to the conjugation biset category, whose objects are subgroups of G, and whose morphisms are generated by certain bifree bisets. Any additive functor from the conjugation biset category to abelian groups yields a Mackey functor by composition. We characterize the Mackey functors which arise in this way.

Abstract. In this paper we identify the "nil-terms" for Waldhausen's algebraic K-theory of spaces functor as the reduced K-theory of a category of equivariant spaces equipped with a homotopically nilpotent endomorphism.