Details

Autor(en) / Beteiligte

• SHAW, EDMUND

Titel

EQUIVARIANT LUSTERNIK-SCHNIRELMANN CATEGORY (ORBIT TYPE, LIE GROUP)

Ort / Verlag

ProQuest Dissertations & Theses

Erscheinungsjahr

1991

Quelle

ProQuest Dissertations & Theses A&I

Beschreibungen/Notizen

• We consider the equivariant extension of Lusternik-Schnirelmann category, defining Gcat(X) to be the minimal number of open G-subsets of a G-space X required to cover X such that each G-subset is G-deformable in X to an orbit of X. We develop a general theory of this G-homotopy type invariant, establishing elementary relations between Gcat(X), the category of the fixed point set and the category of the orbit space. We find conditions for the existence of a G-categorical covering and conditions under which the covering can be replaced by one using closed sets. Some of this earlier work is equivalent to some independent work of Fadell. In our second chapter we consider actions by a compact Lie group of one orbit type: We obtain Gcat(X) when X is an n-dimensional cohomology n-sphere for any G except $G=N\sb{S{\sp1}}(SU\sb2)$, and lower bounds for the G-category in terms of the connectivity of the underlying space. For non-free actions on a space X we consider the relationship between G-category and the orbit structure, developing the concept of locally minimal orbit types to obtain lower bounds for Gcat(X). We generalise the product and dimension theorems of category, and extend Whitehead's definition of category to obtain upper bounds for the G-category of spaces whose G-homotopy groups vanish in low dimensions. In our fourth chapter, we obtain a formula for the Lusternik-Schnirelmann category of the total space E of a fibre bundle in terms of the category of the base, the category of the fibre and the action of the structure group on the fibre. We use this to obtain an improved upper bound for cat(E) when the bundle admits a section, and upper bounds in terms of the dimension of the fibre, and apply these results in various examples. We also obtain an upper bound for cat(E) in the case that the connectivity of the fibre is at least the dimension of the base. Finally, we calculate the index of nilpotency of the mod 2 cohomology of the real Stiefel manifolds, which gives a reasonably high lower bound for the category of these spaces. By way of an example, the last two results are used to show that $cat(SO\sb5$) = 9.