The Stolz' positive scalar curvature sequence for G-proper manifolds and depth-1 pseudomanifolds

TL;DR

This thesis investigates the Stolz positive scalar curvature sequence in the context of G-proper manifolds and depth-1 pseudomanifolds, establishing new isomorphisms, universal spaces, and connections to the Higson-Roe surgery sequence using advanced K-theory and index theorems.

Contribution

It introduces a generalized framework for Stolz's sequence on G-spaces and pseudomanifolds, including universal spaces and a delocalized APS-index theorem, advancing the understanding of positive scalar curvature in singular and equivariant settings.

Findings

Dependence of R-groups on the 2-skeleton.

Isomorphism between R-groups for spaces with isomorphic fundamental functors.

Lower bounds on the structure group rank of positive scalar curvature metrics.

Abstract

This thesis revolves around the Stolz' positive scalar curvature sequence: in particular adapted to the context of (G, F)-spaces, i.e. proper G-spaces with isotropy groups belonging to a family F of subgroups of G, and to that of manifolds with non-isolated singularities. In both cases, the sequence is studied for appropriate classes of metrics with positive scalar curvature, and it is shown how the Stolz R-groups have a strict dependence on the 2-skeleton. This latter result will then be used in the (G, F) framework to establish an isomorphism between the R-groups for spaces having isomorphic fundamental functors, a suitable generalization of the fundamental group. A universal space is also introduced, where universal means that each space with this characterization admit a map with values in it, inducing isomorphisms at the level of R-groups. Subsequently, the mapping of the Stolz sequence to the Higson-Roe surgery sequence for singular spaces with (L, G)-singularities will be studied. Specifically, a version of the delocalized APS-index theorem, which makes use of localization algebras, will be employed. This includes a description of K-theory for graded $C^{*}-algebras$, which will be introduced for Real $C^{*}-algebras$, with the advantage of simultaneously including both the real and complex case. Finally, as an application of these results, the wedge index difference homomorphism is studied in the singular context. This homomorphism, besides serving as an obstruction for two well-adapted wedge metrics to be concordant, is used to provide a lower bound on the rank of the structure group of such positive scalar curvature metrics, namely the bordism group $Pos_*^{spin,(L,G)}$.