New topological and index theoretical methods to study the geometry of manifolds

by Martin Nitsche
Doctoral thesis
Date of Examination:2018-02-06
Date of issue:2018-07-31
Advisor:Prof. Dr. Thomas Schick
Referee:Prof. Dr. Thomas Schick
Referee:Prof. Dr. Ralf Meyer
Referee:Prof. Dr. Dorothea Bahns
Referee:Prof. Dr. Viktor Pidstrygach
Referee:Prof. Dr. Karl-Henning Rehren
Referee:Prof. Dr. Max Wardetzky
Persistent Address: http://dx.doi.org/10.53846/goediss-6978

 

 

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Abstract

English

For a $\mathit{Spin}$ manifold $M$ the Rosenberg index $\alpha([M])$ is an obstruction against positive scalar curvature metrics. When $M$ is non-$\mathit{Spin}$ but $\mathit{Spin}^c$, Bolotov and Dranishnikov suggested to apply the Rosenberg index to a suitable $S^1$-bundle $L\to M$. We study this approach, in particular for the case $\pi_1(L)\neq\pi_1(M)$. We explain how the bundle construction can be turned into a non-trivial natural transformation of bordism groups $\Omega^{\mathit{Spin}^c}\to\Omega^\mathit{Spin}$. Then we show that $\alpha([L])\in\mathit{KO}(C^*(\pi_1(L)))$ always vanishes, but also give an example where $L$ nonetheless does not admit a positive scalar curvature metric. The second part of the thesis concerns the relation of $\alpha([N])$ and $\alpha([M])$ for certain codimension-2 submanifolds $N\subset M$. Following a construction of Engel we extend the Thom map $\mathit{KO}_*(M)\to\mathit{KO}_{*-2}(N)$ to $\mathit{KO}_*(\mathbf{B}\pi_1(M))\to\mathit{KO}_{*-2}(\mathbf{B}\pi_1(N))$, and then further to $\mathit{KO}_*^{\pi_1(M)}(\mathbf{\underline{E}}\pi_1(M))\to\mathit{KO}_{*-2}^{\pi_1(N)}(\mathbf{\underline{E}}\pi_1(N))$.
Keywords: positive scalar curvature; Rosenberg index; geometric K-homology; circle bundle; classifying space for proper actions; codimension-2 transfer; Spin^c
 
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