Quasiregular mappings and cohomology. (Q5953841)

In a series of deep investigations \textit{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. NEWLINENEWLINEThe 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 \textit{P. Järvi} and \textit{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), \textit{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), \textit{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). NEWLINENEWLINEOne of the chief themes is to study the case when the target space \(Y\) is a suitable manifold. Rickman's joint work with \textit{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. NEWLINENEWLINEThe authors of the paper under review penetrate deeper into this difficult territory. Their main result is the following theorem. NEWLINENEWLINETheorem 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 \(K.\)NEWLINENEWLINEThe tools the authors use include methods from Rickman's value distribution theory and also some results of \textit{T. Iwaniec} and his coauthors [e.g. Arch. Ration. Mech. Anal. 125, No. 1, 25--79 (1993; Zbl 0793.58002)] on differential forms.