CHAPTER 1 Introduction One of the most important invariants of homotopy type of a topological space is the coproduct-structure on the chain-complex. Indeed, it determines the rational homo- topy type of a pointed simply-connected space (see [20]). Over the integers there are many additional invariants of homotopy type including Steenrod operations on the mod p cohomology rings for all primes p. In this paper we will present an algebraic theory that incorporates all of these invariants, and allows one to compute them for: the loop-space of a space (via the cobar construction) and the total space of a fibration A key element of computing the coproduct of the total space of a fibration is the determination of the coproduct on the chain complex of the loop space of the base. Since Adams showed that this chain-complex is given by the cobar construction (see [1]), we would like to know a geometrically induced coproduct on the cobar construction. Here, the term "geometrically induced coproduct" can be defined in several ways with varying degrees of strength. We essentially regard the Alexander-Whitney diagonal map on a simplicial chain-complex as being the canonical geometric one and any other diagonal homotopic to it as being geometric to some extent. In the rational case Quillen showed (in [20]) that the shuffle coproduct on the cobar construction is geometric, where this is a dual of the shuffle product defined by Eilenberg and MacLane for the bar construction in [5]. This result implied a number of other results that made it relatively easy to compute a geometric coproduct on the total space of a fibration and on chain-complexes of simply-connected pointed spaces in general. In the integral case the shuffle coproduct on the cobar construction remains well-defined, in some cases, but Quillen's proof of its geometricity is no longer valid. In fact, any attempt to find a geometrically valid coproduct on the cobar construction encounters the following two obstacles, discovered by Alain Proute: The shuffle coproduct on the integral cobar construction is demonstrably non-geometric see [19]. Here the term 'geometric' is defined in a very weak sense the shuffle coproduct is non-geometric to the extent that it even induces the wrong maps in homology. Received by the editor October 14,1992. 1
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