Odd-dimensional lens-space Yamabe counterexamples

Abstract

Let $n\ge5$ be odd and let \(L_{3q}^n=S^n/C_{3q}\) be the standard lens space, where $q>3$ is prime. We prove that for all sufficiently large $q$, \[ σ(L_{3q}^n) > (3q)^{-2/n}Y(S^n)=Y(L_{3q}^n,[g_{\rm rd}]). \] Here \([g_{\rm rd}]\) denotes the quotient-round conformal class. Combining spin bordism with the surgery theorem of Ammann--Dahl--Humbert, we establish \[ σ(L_{3q}^n)\ge \min \{q^{-2/n}Y(S^n),Λ_n\}, \] where $Λ_n>0$ depends only on $n$. For sufficiently large $q$, this exceeds the quotient-round value, showing that the quotient-round conformal class need not maximize the smooth Yamabe invariant in any odd dimension $n\ge5$.