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Spaces of diffeomorphisms and embeddings via algebraic K-theory https://doi.org/10.17863/CAM.118659 Muñoz Echániz, Samuel Randal-Williams, Oscar 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. University of Cambridge 2025-02-27 Thesis Doctoral Doctor of Philosophy (PhD) https://www.repository.cam.ac.uk/handle/1810/384761 https://www.repository.cam.ac.uk/bitstreams/90941cda-88d8-454b-87a4-ada4e99a870c/download fec4e277a4eb995c0eef401dac661a3e https://www.repository.cam.ac.uk/bitstreams/9c173734-3d20-4326-8b56-8e045f426678/download 87eda9de84448d1f82354d60eee3eb5f https://creativecommons.org/licenses/by/4.0/ h-cobordism algebraic K-theory diffeomorphism group embedding space pseudoisotopy theory mapping class group