Immersions of manifolds and homotopy theory (Q2689562)

The paper under review is notes of a lecture the author gave at Harvard's Center of Mathematical Sciences and Applications as part of their Math-Science Literature Lecture Series. In these notes the author discusses topics at the interface between the study of differentiable manifolds and of algebraic topology. This interface has been a very rich area of study in topology since the 1950's. The author focuses on the following types of questions: Given two smooth manifolds \(M\) and \(N\), does \(M\) embed in \(N\). How can one tell when two embeddings are isotopic? What can be said about the topology of the space of embeddings, \(\mathrm{Emb}(M,N)\)? There are analogous questions concerning immersions. There has been much more success in the study of immersions than embeddings, primarily, because the questions about immersions can be translated into questions in homotopy theory. In particular, the author considers the problems of finding the smallest dimensional Euclidean space into which every closed \(n\)-manifold embeds or immerses. The embedding question remains unsolved. The immersion question was solved in the 1980's, the author discusses the homotopy theoretic techniques involved in the solution. He finishes the notes by discussing the Goodwillie-Weiss embedding calculus.