Pieper, Malte Mario: Assembly Maps and Pseudoisotopy Functors. - Bonn, 2019. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5n-53594
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5n-53594,
author = {{Malte Mario Pieper}},
title = {Assembly Maps and Pseudoisotopy Functors},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2019,
month = may,

note = {In this thesis we show the existence of a stable, smooth pseudoisotopyfunctor and construct in the topological, piecewise linear, and smooth category a zig-zag of natural weak equivalences between the stable pseudoisotopyfunctor and the corresponding functor of Whitehead spectra.
To achieve the former, we use the language of quasicategories to enhance the definition of a pseudoisotopy homotopy functor by Burghelea and Lashof to a functor of infinity categories. There are two main steps: First, we observe that most of the constructions by Burghelea and Lashof are unique in a homotopy coherent sense. Then, we give an explicit geometric construction to resolve coherence issues related to corners (of manifolds with corners) of arbitrary degree. This concludes the definition of a stable, smooth pseudoisotopyfunctor.
For the latter, we extend the definition of the Whitehead spectrum to a functor of infinity categories to reduce the problem to a natural transformation of infinity functors. At the center of the natural transformation lies yet again an explicit geometric construction. We define families of retraction maps curtailed to the category of choices, which is used in Enkelmann's PhD thesis to define the topological and piecewise linear pseudoisotopyfunctors, and show these families to be unique up to coherence via the Alexander trick.
The results of this thesis clarify the relation between the functor structures of pseudoisotopies, interesting due to their relations to automorphism spaces of manifolds, and the computationally accessible Whitehead spectrum. In conjunction with work on the Farrell-Jones conjecture for A-theory this resolves all questions concerning the original Farrell-Jones conjecture for pseudoisotopy. As an aside, we hope for eventual applications in the search of explicit nontrivial elements of homotopy groups of automorphism spaces of manifolds.},

url = {http://hdl.handle.net/20.500.11811/7872}

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