Conformally formal manifolds and the uniformly quasiregular non-ellipticity of $(\mathbb{S}^2 \times \mathbb{S}^2) \operatorname{\#} (\mathbb{S}^2 \times \mathbb{S}^2)$

Abstract

We show that the manifold $(\mathbb{S}^2 \times \mathbb{S}^2) \operatorname{\#} (\mathbb{S}^2 \times \mathbb{S}^2)$ does not admit a non-constant non-injective uniformly quasiregular self-map. This answers a question of Martin, Mayer, and Peltonen, and provides the first example of a quasiregularly elliptic manifold which is not uniformly quasiregularly elliptic. To obtain the result, we introduce conformally formal manifolds, which are closed smooth $n$-manifolds $M$ admitting a measurable conformal structure $[g]$ for which the $(n/k)$-harmonic $k$-forms of the structure $[g]$ form an algebra. This is a conformal counterpart to the existing study of geometrically formal manifolds. We show that, similarly as in the geometrically formal theory, the real cohomology ring $H^*(M; \mathbb{R})$ of a conformally formal $n$-manifold $M$ admits an embedding of algebras $\Phi \colon H^*(M; \mathbb{R}) \hookrightarrow \wedge^* \mathbb{R}^n$. We also show that uniformly quasiregularly elliptic manifolds $M$ are conformally formal in a stronger sense, in which the wedge product is replaced with a conformally scaled Clifford product. For this stronger version of conformal formality, the image of $\Phi$ is closed under the Euclidean Clifford product of $\wedge^* \mathbb{R}^n$, which in turn is impossible for $M = (\mathbb{S}^2 \times \mathbb{S}^2) \operatorname{\#} (\mathbb{S}^2 \times \mathbb{S}^2)$.