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

Abstract

We show that the manifold $({\mathbb{S}}^2\times {\mathbb{S}}^2)\#({\mathbb{S}}^2\times {\mathbb{S}}^2)$ 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 $H^{\ast }(M;\mathbb{R})$ of a conformally formal $n$-manifold $M$ admits an embedding of algebras $\Phi :H^{\ast }(M;\mathbb{R})\hookrightarrow {\wedge }^{\ast }{\mathbb{R}}^n$. 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 ${\wedge }^{\ast }{\mathbb{R}}^n$, which in turn is impossible for $M=({\mathbb{S}}^2\times {\mathbb{S}}^2)\#({\mathbb{S}}^2\times {\mathbb{S}}^2)$.

42 pages

References in corpus (3)

• Sharp cohomological bound for uniformly quasiregularly elliptic manifolds
• Uniform cohomological expansion of uniformly quasiregular mappings
• Branched coverings of $C{P}^2$ and other basic 4-manifolds