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The concept of acyclic models, developed by Eilenberg and MacLane [2], appears to be one of the most convenient tools of algebraic topology. It is the purpose of this paper to describe a generalization of the theory of Eilenberg and MacLane, and to apply the theory to proving de Rham's theorem. The generalized theory has other applications; in particular the proof of the spectral sequence theorem due to Gugenheim and Moore [4] could be slightly simplified by the results of this paper. The reader is assumed to know the theory of functors and categories; the general reference for this topic will be Eilenberg and Steenrod [3, Chapter IV, pp. 108-113]. The first two chapters are devoted to establishing the existence or uniqueness of chain and cochain maps; the third chapter is devoted to cup products. In the fourth chapter the singular Cr cubical cohomology theory is defined, and it is shown that for paracompact manifolds the result is independent of r. In the last chapter the de Rham cohomology is defined and shown to be isomorphic to the singular cohomology (de Rham's theorem) by means of the Stokes' map f* (integration over chains); it is also shown that the Stokes' map preserves products, i.e., that f*(ao3) = (f*a)U(f*). The relation of acyclic models to products will be treated at length in another paper.

The notion of prop models the operations with multiple inputs and multiple outpus, acting on some algebraic structures like the bialgebras or the Lie bialgebras. In this paper, we generalize the Koszul duality theory of associative algebras and operads to props., Last version before publication in Transactions of the A.M.S (statement about the properad of involutive Lie bialgebras removed)

We generalize the concept of "foliation" and define k-flat structures; these are smooth vector bundles with afflne connections whose characteristic forms vanish above a certain dimension. Using semisimplicial techniques we construct a classifying space for k-flat structures, and prove a classification theorem for these structures on smooth manifolds. Techniques from rational homotopy theory are used to relate the exotic characteristic classes of foliations to the rational homotopy groups and cohomology of the classifying space. Introduction. A familiar technique in algebraic topology is to reduce the probD lem of studying structures on a large collection of objects to the study of a universal example which "contains" all the others. Consider the problem of classifying vector bundles over CW complexes, in this case we need only look at spaces B0k and BUk, and their canonical k-plane bundles [9]. A nice feature of this example is that the algebraic invariants of k-plane bundles (i.e. Stiefel-Whitney classes, Chern classes, Euler class) are contained in the cohomology of the classifying spaces. In this paper, we look at a set of invariants which can be attached to foliated smooth manifolds. These are the so-called "exotic characteristic classes" of foliations which were constructed in the late 1960's independently by 13ott, Haefliger, Gelfand-Fuks, Kamber-Tondeur, and Malgrange; they can be defined using purely geometric techniques (i.e. see [1]). At about the same time, Haefliger [5] developed a topological classification of foliations using Br spaces, and has applied the theory to smooth manifolds via theorems of Thurston and Gromov-Phillips. One would like to demonstrate a relationship between the exotic characteristic classes and the cohomology of the classifying spaces as in the example above, although this has proven to be difficult. Nevertheless, a lot is known about the homotopy type of Brk, largely through the efforts of Hutsch, Hurder, Kamber-Tondeur, Rasmussen, Thurston, et al. In §1, we introduce k-flat structures, which generalize the concept of foliations. These are smooth vector bundles with affine connections whose characteristic forms vanish above dimension 2k; they constitute the largest classes of objects for which exotic characteristic classes can be defined. k-flat structures and exotic characteristic classes were studied by Lehmann in [17]. Using semisimplicial techniques, we will construct classifying spaces for k-flat structures and prove a classification theorem for these structures on smooth manifolds. §2 contains a brief review of semisimplicial methods, while §3 contains the constructions and proofs. Received by the editors March 29, 1984 and, in revised form, November 18, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 57R30; Secondary 57M40. (C)1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page

Introduction. In this paper I study covering spaces in which the base space is an arbitrary topological space. No use is made of arcs and no assumptions of a global or local nature are required. In consequence, the fundamental group 7r(X, xo) which we obtain frequently reflects local properties which are missed by the usual arcwise group 7ri(X, xo). We obtain a perfect Galois theory of covering spaces over an arbitrary topological space. One runs into difficulties in the non-locally connected case unless one introduces the concept of space [?1 ], a common generalization of topological and uniform spaces. The theory of covering spaces (as well as other branches of algebraic topology) is better done in the more general domain of spaces than in that of topological spaces. The Poincare (or deck translation) group plays the role in the theory of covering spaces analogous to the role of the Galois group in Galois theory. However, in order to obtain a perfect Galois theory in the non-locally connected case one must define the Poincare filtered group [??6, 7, 8] of a covering space. In ?10 we prove that every filtered group is isomorphic to the Poincare filtered group of some regular covering space of a connected topological space [Theorem 2]. We use this result to translate topological questions on covering spaces into purely algebraic questions on filtered groups. In consequence we construct examples and counterexamples answering various questions about covering spaces [?11]. In ?11 we also discuss other applications of the general theory of covering spaces-e.g., a theory of covering spaces with singularities [?2, Example 1; ?11, Example 1]; the defining of a "relative fundamental group" 7r(X, A) [?2, Example 2; ?11, Example 3]; the interpretation of the etale coverings in characteristic 0 of algebraic geometry as covering spaces [? 11, Example 2 ]; and the resolution of the classical problem of the classification of the covering spaces of the topological space: { (0, 0) } \JGraph [sin(1/x), x > 0] [?11, Example 4 ]. The following table summarizes some of the similarities between Galois theory and the theory of covering spaces

A central problem in topological dynamics is the classification of minimal sets. In full generality, this problem appears to be quite difficult. Therefore, it seems appropriate to single out certain minimal sets which are 'well behaved' in some sense, and therefore more amenable to classification. It is for this purpose that we introduce the regular minimal sets in this paper. They are the universal minimal sets for certain ('admissible') properties. Several characterizations are given (theorem 3); the class of regular minimal sets is shown to coincide with the minimal right ideals of enveloping semigroups of transformation groups. and also with those minimal sets having a maximal supply of endomorphisms. (Author)