arXiv:math/0607245v1 [math.AT] 10 Jul 2006 THICK SUBCATEGORIES IN STABLE HOMOTOPY THEORY (WORK OF DEVINATZ, HOPKINS, AND SMITH). SUNIL K. CHEBOLU In this series of lectures we give an exposition of the seminal work of Devinatz, Hopkins, and Smith which is surrounding the classification of the thick subcate- gories of finite spectra in stable homotopy theory. The lectures are expository and are aimed primarily at non-homotopy theorists. We begin with an introduction to the stable homotopy category of spectra, and then talk about the celebrated thick subcategory theorem and discuss a few applications to the structure of the Bous- field lattice. Most of the results that we discuss below were conjectured by Ravenel [Rav84] and were proved by Devinatz, Hopkins, and Smith [DHS88, HS98]. 1. The stable homotopy category of spectra Recall that in homotopy theory one is interested in studying the homotopy classes of maps between CW complexes (spaces that are built in a systematic way by attaching cells): If f and g are maps (continuous) between CW complexes X and Y , we say that they are homotopic if there is a map from the cylinder X × [0, 1] to Y whose restriction to the two ends (top and bottom) of the cylinder gives f and g respectively. The homotopy classes of maps between X and Y is denoted by [X, Y ]. In stable homotopy theory one studies a weaker notion of homotopy called stable homotopy – maps f and g as above are said to be stably homotopic if Σ n f and Σ n g are homotopic for some n. (Σ denotes the reduced suspension functor on the homotopy category of pointed CW complexes.) The notion of stable homotopy is much weaker than homotopy. For example, the obvious quotient map from the torus to the two sphere is not null homotopic but stably null homotopic. The importance of stable homotopy classes of maps comes from an old result due to Freudenthal which implies that if X and Y are finite CW complexes, then the sequence [X, Y ] → [ΣX, ΣY ] → [Σ 2 X, Σ 2 Y ] →··· eventually stabilises. The stable homotopy classes of maps from X to Y is precisely the above colimit. In particular when X is the n-sphere S n , we get the n-th stable homotopy group of Y , denoted π s n (Y ). Computing stable homotopy groups is, in general, a more manageable problem than that of homotopy groups. However, it became abundantly clear to homotopy theorists by 1960s that in order to do serious stable calculations efficiently it is absolutely essential to have a nice category in which the objects are stabilised analogue of spaces each of which represent a cohomology theory. The finite objects of such a category can be easily described. This is called the (finite) Spanier-Whitehead category which captures finite stable phenomena, and is defined as follows. The objects are ordered pairs (X, n) where X Date : September 10, 2018. 1