CHAPTER 1 Introduction to LS-Category 1.1. Introduction Lusternik-Schnirelmann category is like a Picasso painting. Looking at category from different perspectives produces completely different impressions of category's beauty and applicability. In this chapter, we will introduce the notion of category and prove the basic results about it. This includes fundamental results from both the homotopical and analytical perspectives. The central motivating result of the chapter is the famous Lusternik-Schnirelmann theorem (Theorem 1.15) which esti- mates the number of critical points of smooth functions by the invariant category. In order to get to this result (and its proof), however, we first need some topological preliminaries which culminate in Lemma 1.13. Chapter 1 mixes together homotopy theory and critical point theory in essential ways and, therefore, provides a not-so- subtle hint of where we shall journey in the rest of this book. 1.2. The Definition and Basic Properties The classical reductionist paradigm of mathematics and science has been to decompose an object into simpler pieces and then understand the object by ana- lyzing the pieces and how they fit together to make up the object. For a homotopy theorist, the simplest possible pieces of a space are its contractible subsets. To re- late such subsets to the space in which they sit, it is better, however, to look at the subsets which are contractible in the space. So, for a space, we can simply ask, how many such (open) subsets are required to cover the space. This simple numerical invariant provides one measure of the complexity of a space and also provides the starting point in our study. DEFINITION 1.1. The (Lusternik-Schnirelmann or LS) category of a space X is the least integer n such that there exists an open covering J7i,..., Un+\ of X with each Ui contractible to a point in the space X. We denote this by cat(X) = n and we call such a covering {Ui} categorical. If no such integer exists, we write cat(X) = oo. Let AC. X. The subspace category of A in X x , denoted catx(^4) is the least integer n such that there exist open subsets, [/]_,..., Un+i of X which cover A and which are contractible in X. If no such integer exists, we write catx(^4) = oo. Note that catxPO = cat(X) and catx(^4) cat(X). REMARK 1.2. (1) Other authors choose to say that such a covering endows X with cat(X) = Subspace category is also known as relative category in the older literature. We prefer to reserve this term for a newer concept described in Section 7.2. 1
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