Some problems of algebraic topology

This thesis studies some aspects of the homotopy type of function spaces XY where X,Y are topological spaces.
The thesis is in two parts. Part A (Chapters I–IV) contains a discussion of some known facts on the homotopy type of function spaces under the heads of homology (Chapter II), homotopy groups (Chapter III) and Postnikov systems (Chapter IV). Also, in Chapter II, a theorem on duality is given which is useful in determining the low-dimensional homotopy type of (Sn)X when X = Sru er+q (r + q < n).
Chapter IV contains the statements of the problems whose solution is the motivation of the theory of Part B. These problems, which occur naturally in attempting to find the Postnikov system of X by induction on the Postnikov system of X, are roughly of the type of determining kY : XY→AY when X,Y are spaces, A is a topological abelian group and k : X→A is a map. This problem we call here the "k-invariant problem".
It is a commonplace that the most important property of function spaces is the "exponential law" which states that under certain restrictions the spaces XZ×Y and (XY)Z are homeomorphic. In fact it is usually the case that the only properties of the function space required are that as a set XY is the set of maps Y→X, and that the exponential law holds.
In Chapter I, as preparation for the work of Part B, a brief discussion of the exponential law in a general category is given. The rest of the chapter shows how the well-known weak-topological product may be used to obtain an exponential law for all (Hausdorff) spaces. The weak product is also shown to be convenient in the theory of the identification maps.
The theory of Part B is given in terns of css-complexes (complete semi-simplical complexes) with base point. In Chapter V the well-known css-exponential law is extended to the category of css-M-ads, and the exponential law for complexes with base point obtained. The relation between the topological and css- function spaces is discussed, and it is shown that the singular functor preserves the exponential law.
The further theoretical work of Part B is initially of two kinds* First, the function complex AY where A is an FD-complex, is related, by means of maps and functors, with mapping objects in the category of FD-complexes and chain complexes. This is done in such a way to preserve the exponential law. Second, a generalised cohomology of a complex is introduced, with the coefficient group replaced by an arbitrary chain complex (or FD-complex). The theories of cohomology operations and of Eilenberg–Maclane complexes are correspondingly generalised. Using these two sets of constructions, a solution of the k-invariant problem is given in terms of chain complexes (Chapter IX.§2).
The rest of Part B is concerned with obtaining the cohomological solution of the k-invariant problem, putting the results in a fora suitable for computation, and obtaining applications.
In detail Part B proceeds, after Chapter V, as follows. Chapter VI discusses chain complexes and the functor ⋔ of chain complexes, with particular attention to signs. The exponential map here is, for chain complexes A,B,C an isomorphism μ : (A⊗B)⋔C→A⋔(B⋔C).
In Chapter VII, on FD-complexes, a map product A⋔B of FD-complexes A,B is introduced. The exponential map here is an isomorphism (A×B)⋔C→A⋔(B⋔C), where A×B is the cartesian product of A and B. If Y is a css-complex, and C(Y) is essentially the free FD-complex generated by Y, it is shown that there is an isomorphism D : AY→C(Y)⋔A which preserves the exponential law, and by which these complexes may be identified. The well-known properties of the normalisation functor N and Dold-Kan functor R are given, and the generalised cohomology introduced. The fundamental classification theorem is proved, and the theory of operations derived.
Chapter VIII relates the Dold-Kan theory and generalised Eilenberg–Maclane complexes; the exactness properties of the latter are discussed.
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