Concordances in Positive Scalar Curvature and Index Theory

by Thorsten Hertl
Doctoral thesis
Date of Examination:2022-09-02
Date of issue:2023-02-24
Advisor:Prof. Dr. Thomas Schick
Referee:Prof. Dr. Thomas Schick
Referee:Prof. Dr. Wolfgang Steimle
Persistent Address: http://dx.doi.org/10.53846/goediss-9746

 

 

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Abstract

English

We apply the strategy to study of diffeomorphisms via block diffeomorphisms to the world of positive scalar curvature (psc) metrics. For each closed psc manifold, we construct the cubical set of all psc block metrics, the so called concordance set, which only encodes concordance information of psc metrics within its homotopy type. We show that the concordance is a cubical Kan set, give a geometric description for the group structure of the combinatorial homotopy groups, and construct a comparison map from the cubical model of the space of psc metrics on the underlying manifold to the concordance set. Next, we build a concordance-themed model for real K-theory based on the notion of invertible block Dirac operators and use it to factor the index difference through the concordance set. In the final part of this thesis, we construct the psc Hatcher spectral sequence, which is a non-index-theoretic tool to get information about the difference of the space of psc metrics and the concordance set.
Keywords: Positive Scalar Curvature; Index Theory; Cubical Sets; Spectral Sequences; Classifying Spaces; K-theory; Dirac Operators
 
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