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 ${\pi }_{\ast }\left(B{\mathrm{Diff}}_{\partial }\left(D^d\right)\right)\otimes Q$ are lifted to homotopy groups of the moduli space of $h$-cobordisms ${\pi }_{\ast }\left(B{\mathrm{Diff}}_{\bigsqcup }(D^d\times I)\right)\otimes Q$. As a geometrical application, we show that those elements in ${\pi }_{\ast }\left(B{\mathrm{Diff}}_{\partial }\left(D^d\right)\right)\otimes Q$ for $d⩾4$ are also lifted to the rational homotopy groups ${\pi }_{\ast }\left(M_{\partial }^{\mathrm{psc}}{\left(D^d\right)}_{h_0}\right)\otimes Q$ of the moduli space of positive scalar curvature metrics. Moreover, we show that the same elements come from the homotopy groups ${\pi }_{\ast }\left(M_{\bigsqcup }^{\mathrm{psc}}{(D^d\times I;g_0)}_{h_0}\right)\otimes Q$ of moduli space of concordances of positive scalar curvature metrics on $D^d$ with fixed-round metric $h_0$ on the boundary $S^{d-1}$.