Spaces of diffeomorphisms and embeddings via algebraic K-theory

Abstract

This thesis comprises two papers that study the homotopy type of spaces of automorphisms and embeddings of high-dimensional manifolds via algebraic K-theory. In the first paper, presented in Chapter 1, we show that the mapping class group is not an h-cobordism invariant of high-dimensional manifolds by exhibiting h-cobordant manifolds whose mapping class groups have different cardinalities. To do so, we introduce a moduli space of “h-block” bundles and compare it to the moduli space of ordinary block bundles. In the second paper, spanning Chapters 2 and 3, we establish a pseudoisotopy result for embedding spaces. We describe, within a range of homotopical degrees, the difference between spaces of block and ordinary embeddings in terms of relative algebraic K-theory; this is analogous to a theorem of Weiss and Williams for spaces of automorphisms. We use our result to provide a full description of the homotopy type—localised away from 2 and in the aforementioned degree range—of the space of long knots of codimension at least 3. This analysis involves a detailed study of certain geometric involutions in algebraic K-theory spaces that can be of independent interest.