On the topology of the moduli space of positive scalar curvature concordances
Boris Botvinnik, David J. Wraith
TL;DR
The paper analyzes the topology of the moduli space of positive scalar curvature concordances on $M\times I$, proving that for even-dimensional $M$ of sufficiently large dimension, the rational homotopy group $\pi_{4q}\mathscr{M}^{\mathsf{pos}}_{\sqcup}(M\times I)_g$ is nontrivial in a stable range, with an analogous result for $Sc_k>0$ with $k\ge2$. The authors develop two models for classifying spaces $BDiff_\partial$ and $BDiff_\sqcup$ (embedding and metric), establish their equivalence, and leverage Hatcher bundles equipped with fibrewise positively curved metrics to produce nonzero rational classes that lift into the moduli space of PSC concordances. A key step is showing that a nonzero map from $BDiff_\partial(D^n)$ to $BDiff_\partial(M\times I)$ persists under inclusion, and that these classes admit lifts to $\mathscr{M}^{\mathsf{pos}_*}_\partial(M\times I)_{\tilde{g}}$, leading to nontrivial elements in $\pi_{4q}\mathscr{M}^{\mathsf{pos}_*}_\sqcup(M\times I)_g$. The results extend prior unstable-range findings to a broad stable regime and highlight rich topological structure in PSC-concordance moduli, offering insights into pseudoisotopy phenomena and smoothing theory without reliance on spin or Dirac techniques.
Abstract
Let $M$ be a manifold which admits a metric with positive scalar curvature (or a positive intermediate curvature in a suitable sense). We study the moduli space ${\mathscr{M}}^{\mathsf{pos}_*}_{\sqcup}(M\times I)_g$ of concordances of such metrics (with appropriate boundary conditions) which restrict to a given metric $g$ on $M \times \{0\} \cup\partial M \times I$. We show that $π_{4*}{\mathscr{M}}^{\mathsf{pos}_*}_{\sqcup}(M \times I)_g \otimes {\mathbb Q} \neq 0$ in a stable range provided $\dim M$ is even. We obtain analogous results when positive scalar curvature is replaced by $k$-positive Ricci curvature for $k \ge 2$.