New Developments in Topology
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In the last few years, a number of authors have developed competing theories of iterated loop spaces. Among the desirable properties of such a theory are:
(1) A recognition principle for n-fold loop spaces, 1 ≤ n ≤ ∞, which applies when n = ∞ to such spaces at Top, BF, PL/O, etc. and to the classifying spaces of categories with appropriate structure.
(2) An approximation theorem which describes the homotopy type of Ωn∑nX, 1 ≤ n ≤ ∞, in terms of iterated smash products of X and canonical spaces.
(3) A theory of homology operations on n-fold loop spaces, 1 ≤ n ≤ ∞, at least sufficient to describe H*Ωn ∑n X, with all structure in sight, as a functor of H*X.
(4) Computations and applications of the homology operations on interesting spaces to which (1) applies. In addition, rigor and aesthetics dictate (5) and suggest (6).
(5) Complete proofs of all non-trivial technical details are to be given.
(6) Only simple and easily visualized topological constructions are to be used.
Point (5) is particularly important since several quite plausible sketched proofs of recognition principles have foundered on seemingly minor technical details. At the moment, the author’s theory [17], which shall be referred to as [G], provides the only published solution to (1) and (2) which makes any claim to satisfy (5). For this reason, and because it includes the deeper cases 1 < n < ∞ of (1) and (2) as well as machinery designed for use in (3) and (4), [G] is quite lengthy.
