ABSTRACT. This paper develops a new invariant of a C W-complex called the m-structure and uses it to perform homotopy-theoretic computations. The m-structure of a space encapsulates the coproduct structure, as well as higher-coproduct structures that de- termine Steenrod-operations. Algebraically, it amounts to an operad in the category of modules. In particular, given an m-structure on the chain complex of a reduced simplicial complex of a pointed simply-connected space, one can equip the cobar con- struction of this chain-complex with an natural m-structure. The m-structure of the cobar construction is shown to be geometrically meaningful, in the sense that it corre- sponds to the m-structure of the loop space of the original space under the map that carries the cobar construction to the loop space. This result allows one to form iterated cobar constructions that are shown to be homotopy equivalent to iterated loop-spaces. This homotopy equivalence is in the sense of chain-complexes equipped with m-structures. These results are applied to the computation of the cohomology algebra structure of total spaces of fibrations (actually, we compute m-structures, which determine the cohomology algebra). Key words and phrases, coproduct, cobar construction, twisted tensor products, co- homology operations.
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