Families of diffeomorphisms and concordances detected by trivalent graphs

TL;DR

This paper investigates how certain non-trivial elements in the homotopy groups of diffeomorphism and moduli spaces, detected by trivalent graphs, can be lifted to higher structures including cobordisms and positive scalar curvature metrics.

Contribution

It demonstrates that specific elements in homotopy groups of diffeomorphism spaces are lifted to moduli spaces of cobordisms and positive scalar curvature metrics, revealing new geometric and topological connections.

Findings

Non-trivial homotopy elements are lifted to moduli spaces of h-cobordisms.

Elements are also lifted to moduli spaces of positive scalar curvature metrics.

Results connect diffeomorphism groups with geometric structures via trivalent graphs.

Abstract

We study families of diffeomorphisms detected by trivalent graphs via the Kontsevich classes. We specify some recent results and constructions of the second named author to show that those non-trivial elements in homotopy groups $\pi_*(B\mathrm{Diff}_{\partial}(D^d))\otimes \mathbb{Q}$ are lifted to homotopy groups of the moduli space of $h$-cobordisms $\pi_*(B\mathrm{Diff}_{\sqcup}(D^d\times I))\otimes \mathbb{Q}$. As a geometrical application, we show that those elements in $\pi_*(B\mathrm{Diff}_{\partial}(D^d))\otimes \mathbb{Q}$ for $d\geq 4$ are also lifted to the rational homotopy groups $\pi_*(\mathcal{M}^{\mathrm{psc}}_{\partial}(D^d)_{h_0})\otimes \mathbb{Q}$ of the moduli space of positive scalar curvature metrics. Moreover, we show that the same elements come from the homotopy groups $\pi_*(\mathcal{M}^{\mathrm{psc}}_{\sqcup} (D^d\times I; g_0)_{h_0})\otimes \mathbb{Q}$ of moduli space of concordances of positive scalar curvature metrics on $D^d$ with fixed round metric $h_0$ on the boundary $S^{d-1}$.