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

• Published in: Advances in mathematics (New York. 1965) 2021-12, Vol.393, p.108103, Article 108103
• Main Author: Kangasniemi, Ilmari
• Format: Article
• Language: English
• Subjects: Conformally formal; Geometrically formal; Measurable conformal structure; p-Harmonic form; Uniformly quasiregular; UQR
• ISSN: 0001-8708; 1090-2082
• DOI: 10.1016/j.aim.2021.108103

We show that the manifold (S2×S2)#(S2×S2) 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 Φ:H⁎(M;R)↪∧⁎Rn. 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 Φ is closed under the Euclidean Clifford product of ∧⁎Rn, which in turn is impossible for M=(S2×S2)#(S2×S2).