Spaces of diffeomorphisms and embeddings via algebraic K-theory

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.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2025

Author dc:creator

• Muñoz Echániz, Samuel

Advisor dc:contributor.advisor

• Randal-Williams, Oscar

• h-cobordism
• algebraic K-theory
• diffeomorphism group
• embedding space
• pseudoisotopy theory
• mapping class group

Licence

• https://www.repository.cam.ac.uk/bitstreams/9c173734-3d20-4326-8b56-8e045f426678/download
• https://creativecommons.org/licenses/by/4.0/

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.118659

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/384761