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This chapter is meant to be a brief account of some of the recent developments and results that utilize some of the techniques developed in this book. As this is meant to be an overview, many details and proofs have been omitted, but ample references for further reading have been supplied. The central application is to introduce the calculus of functors, and we present two of its flavors – homotopy and manifold calculus – in Sections 10.1 and 10.2. Section 10.3 is an application of manifold calculus to spaces of embeddings, and Section 10.4 is an account of how manifold calculus, in combination with cosimplicial spaces and their spectral sequences, provides information about spaces of knots.
One important application that we did not have space to include is the Lusternik–Schnirelmann category. Some of the main results in that theory use Ganea's Fiber-Cofiber Construction (Proposition 4.2.14), Mather's Cube Theorems (Theorems 5.10.7 and 5.10.8) and other cubical techniques developed in this book. For more details, the reader should consult [CLOT03].
Homotopy calculus of functors
This and the next section are devoted to a brief outline of the calculus of functors, an organizing principle in topology which takes some inspiration from Taylor series in ordinary calculus. Our focus will be narrow, only briefly describing two flavors, known as “homotopy calculus” and “manifold calculus”, but we will pay special attention to how cubical diagrams play an important role in each of these theories.
We will not attempt to answer the very general question of what a calculus of functors is, but a few philosophical remarks are in order. Given a functor F : C → D, the general idea is to approximate F by a sequence of functors TkF : C → D which are “polynomial of degree k”, and with natural transformations F → TkF and TkF → Tk−1 F compatible in the obvious way. These functors and natural transformations form a “Taylor tower” for F, the analog of the Taylor series of a function f : R → R. We are typically interested in the homotopy type of the values of the functor F, so the category D should be one in which we have a reasonable notion of homotopy theory.