Conformally formal manifolds and the uniformly quasiregular non-ellipticity of (S2 x S2)#(S2 x S2)

We show that the manifold (S-2 x S-2)#(S-2 x 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; R) of a conformally formal n-manifold M admits an embedding of algebras Phi: H* (M; R) hooked right arrow boolean AND*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 Phi is closed under the Euclidean Clifford product of boolean AND*R-n, which in turn is impossible for M = (S-2 x S-2)#(S-2 x S-2). (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

The study demonstrates that certain manifolds do not admit a non-constant, non-injective uniformly quasiregular self-map, providing the first example of a quasiregularly elliptic manifold not uniformly quasiregularly elliptic. Conformally formal manifolds are introduced, and it is shown that they have similarities to existing studies of geometrically formal manifolds.