The Stolz' positive scalar curvature sequence for G-proper manifolds and depth-1 pseudomanifolds
Massimiliano Puglisi
TL;DR
This work develops a comprehensive framework for Stolz-type obstructions to positive scalar curvature in two intertwined settings: (i) actions of a group $G$ with isotropy in a family $\mathcal{F}$ on CW-complexes, and (ii) smoothly stratified pseudomanifolds with depth-1 singularities modeled by depth-1 links $L=G/K$. It introduces $(G,\mathcal{F})$-Stolz groups and a universal space to compare $R$-groups, then extends these constructions to $L$-fibered and $(L,G)$-singular settings via well-adapted wedge metrics and stratified Morse theory. A central component is mapping the Stolz sequence to the Higson–Roe surgery sequence using graded Real $C^*$-algebras and localization algebras, enabling a KO-theoretic treatment of index data and secondary invariants $\rho^{\Gamma}$. The authors derive a Delocalized APS theorem in this stratified context and prove invariance results under 2-connected maps, yielding lower bounds for the structure group rank $Pos^{spin,(L,G)}$ in terms of cokernel data from fundamental maps, guided by rational Baum–Connes assumptions. Collectively, the results advance the interaction between geometric PSC problems, equivariant and stratified topology, and noncommutative geometry techniques, with concrete rank estimates for PSC-bordism groups in singular settings.
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 $Pos_*^{spin,(L,G)}$.