Quasiregular mappings and cohomology.(English)Zbl 1088.30011

In a series of deep investigations S. Rickman studied the question of existence of a nonconstant quasiregular mapping $$f: \mathbb R^n \to Y,$$ $$n\geq 2,$$ for target spaces of the form $$\mathbb R^n \setminus E$$ where $$E$$ is a finite set. In particular, he proved a counterpart of Picard’s theorem [J. Anal. Math. 37, 100–117 (1980; Zbl 0451.30012)] in this context and also proved that this result is qualitatively best possible for $$n=3$$ [Acta Math. 154, 195–242 (1985; Zbl 0617.30024)]. See also the survey [Quasiconformal space mappings, Collect. Surv. 1960-1990, Lect. Notes Math. 1508, 93–103 (1992; Zbl 0764.30017)] of Rickman.
The Rickman-Picard theorem was an important landmark for the development of quasiregular mappings and has also found many applications in the works of his colleagues and students, e.g. in P. Järvi and M. Vuorinen, J. Reine Angew. Math. 424, 31–45 (1992; Zbl 0733.30017), J. Jormakka, Ann. Acad. Sci. Fenn., Ser. A I. Math., Diss. 69. Helsinki: Suomalainen Tiedeakatemia; Univ. of Helsinki, Faculty of Science (1988; Zbl 0662.57007), J. Kankaanpää, Ann. Acad. Sci. Fenn., Ser. A I. Math., Diss. 110. Helsinki: Suomalainen Tiedeakatemia; Univ. of Helsinki, Faculty of Science (1997; Zbl 0909.30014), K. Peltonen, Ann. Acad. Sci. Fenn., Ser. A I. Math., Diss. 85. Helsinki: Suomalainen Tiedeakatemia; Univ. of Helsinki, Faculty of Science (1992; Zbl 0757.53024).
One of the chief themes is to study the case when the target space $$Y$$ is a suitable manifold. Rickman’s joint work with I. Holopainen [Andreian Cazacu, Cabiria (ed.) et al., Analysis and topology. A volume dedicated to the memory of S. Stoilow. Singapore: World Scientific, 315–326 (1998; Zbl 0945.30022)] has developped these ideas further.
The authors of the paper under review penetrate deeper into this difficult territory. Their main result is the following theorem.
Theorem Let $$N$$ be a closed, connected and oriented Riemannian $$n$$-manifold, $$n \geq 2 \, .$$ If there exists a nonconstant $$K$$-quasiregular mapping $$f: \mathbb R^n \to N,$$ then $$\dim H^*(N) \leq C(n,K)$$ where $$\dim H^*(N)$$ is the dimension of the de Rham cohomology ring $$H^n(N)$$ of $$N,$$ and $$C(n,K)$$ is a constant depending only on $$n$$ and $$K.$$
The tools the authors use include methods from Rickman’s value distribution theory and also some results of T. Iwaniec and his coauthors [e.g. Arch. Ration. Mech. Anal. 125, No. 1, 25–79 (1993; Zbl 0793.58002)] on differential forms.

MSC:

 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 58A12 de Rham theory in global analysis
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References:

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