Families of diffeomorphisms and concordances detected by 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 $S^{d-1}$.