Some problems in topology

Thesis

This thesis explores generalizations of category in the sense of Lusternik-Schnirelmann. The material falls naturally into two parts.

Chapter 1 is concerned with numerical invariants. Following Berstein, for a closed n-manifold M let N(M) (resp. n(M)) denote the minimum cardinality of a covering of M by open subsets, each of which embeds (resp. immerses) inIRn. In Chapter 1 the values of N(M) and n(M) are determined when M =112Pⁿ -- with the exception of N(1RPⁿ ) when n = 31 or 47. For a vector bundle C over a space X, James has defined Vecat(ξ) to be the minimum number of open subsets required to cover X with the restriction of C to each subset stably trivial. In Chapter 1 the categories of all vector bundles over all real projective spaces are also determined.

In Chapter 2 it is undertaken to define LusternikSchnirelmann category solely in terms of the category of spaces modulo weak equivalences. This is done by equipping the homotopy category with a notion of covering. The reason for doing this is to obtain a more conceptual definition of cocategory. It turns out that there are many possible definitions of cocategory. The relationship between two of them is discussed in Chapter 2.

The category of a connected space X is closely related to the Milnor filtration of X regarded as the classifying space of its loop space. This in turn is intimately related to the H-space structure on 2X. The framework of homotopy coverings is exploited in Chapter 3 to yield a filtration of a (k-1)-connected space analogously related to its k-fold loop space. There results a theory of higher associativity for k-fold loop spaces and a dual theory for k-fold suspensions.

Authors: Hopkins, MJ

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford