My thesis is titled Spaces of positive scalar curvature metrics and parametrised Morse theory and has been written under the supervision of Johannes Ebert. I successfully defended it on 28 June 2023.

It is planned to turn the thesis into two papers, one on parametrised Morse theory and the other on the application to positive scalar curvature.

Abstract

In this thesis we study the space of positive scalar curvature metrics $\mathcal{R}^+(M)$ on a compact spin manifold $M^d$ of high dimension. As in previous results, we use Hitchin’s index difference to obtain a map into $K$-theory acting as a detector for non-trivial homotopy classes of $\mathcal{R}^+(M)$. Under favourable circumstances we prove this map to be (rationally) surjective.

Our results improve the state of the art in two ways:
firstly they hold in all dimensions $d\ge 5$, secondly they are valid for the higher index difference, which maps into the $K$-theory of the group $C^*$-algebra of $G= \pi_1(M)$. This is accomplished by rebuilding the machinery of previous results from scratch using “parametrised Morse theory” as the new driving force, which is an extension of the methods of Madsen and Weiss laid out by Perlmutter in the preprint arXiv:1703.01047.

We also give another application of our machinery in the form of a rigidity result for the action of the diffeomorphism group on the space $\mathcal{R}^+(M)$ via pullback. Here we show that the action map factors through the infinite loop space of the Madsen–Tillmann–Weiss spectrum associated to the tangential structure $\theta \colon B\mathrm{Spin}(d) \times B G \to B\mathrm{O}(d)$.

Sample .bib-file entry

@phdthesis{bantje-thesis,
    author = {Jannes Bantje},
    institution = {Universität Münster},
    title = {Spaces of positive scalar curvature metrics and parametrised Morse theory},
    year = {2023},
    url = {https://jbantje.gitlab.io/phd/},
    doi = {10.17879/59968634386}
}

(if you are using bibtex instead of biber you might need to change institution to school)