Title:

Some problems of algebraic topology

This thesis studies some aspects of the homotopy type of function spaces X^{Y} where X,Y are topological spaces. The thesis is in two parts. Part A (Chapters IIV) 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 lowdimensional homotopy type of (S^{n})^{X} when X = S^{r}u e^{r+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^{Y} by induction on the Postnikov system of X, are roughly of the type of determining k^{Y} : X^{Y}→A^{Y} when X,Y are spaces, A is a topological abelian group and k : X→A is a map. This problem we call here the "kinvariant 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 X^{Z&Y} and (X^{Y})^{Z} are homeomorphic. In fact it is usually the case that the only properties of the function space required are that as a set X^{Y} 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 wellknown weaktopological 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 csscomplexes (complete semisimplical complexes) with base point. In Chapter V the wellknown cssexponential law is extended to the category of cssMads, and the exponential law for complexes with base point obtained. The relation between the topological and cssfunction 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 A^{Y} where A is an FDcomplex, is related, by means of maps and functors, with mapping objects in the category of FDcomplexes 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 FDcomplex). The theories of cohomology operations and of EilenbergMaclane complexes are correspondingly generalised. Using these two sets of constructions, a solution of the kinvariant problem is given in terms of chain complexes (Chapter IX, S.2). The rest of Part B is concerned with obtaining the cohomological solution of the kinvariant problem, putting the results in a form suitable for computation, and obtaining applications.
