Nielsensche Realisierungssätze für Seifertfaserungen (Q1069515)

In an earlier paper [ibid. 30, 361-373 (1980; Zbl 0438.57005)], the author showed that for a closed aspherical manifold M and a finite subgroup A of the outer automorphism group Out \(\pi\) \({}_ 1M\) of \(\pi_ 1M\), one can always find a finite group B and a surjection \(B\to A\) with abelian kernel such that there exists an extension to the abstract kernel \((B,\pi_ 1M\), \(B\to A\hookrightarrow Out \pi_ 1M)\). The existence of such an extension to the abstract kernel \((A,\pi_ 1M\), \(A\hookrightarrow Out \pi_ 1M)\) is known to provide a necessary (and often sufficient) condition in the Nielsen realization problem for A. In the present paper, the author proves a cohomological characterization (for a given A) of all the groups B with the above property and which can be realized by the induced actions of isomorphic groups of homeomorphisms of M. This follows from a purely algebraic result which provides an obstruction \(h\in H^ 3(A,C)\), where C is the center of \(\pi_ 1M\), to the existence of an extension. It is then applied to a class of Seifert fiber spaces (corresponding to the codimension 2 case of the injective Seifert fiber spaces of \textit{P. E. Conner} and \textit{F. Raymond} [Bull. Am. Math. Soc. 83, 36-85 (1977; Zbl 0341.57003)]) and to the class of closed flat Riemannian manifolds. Some realization results are derived and a few examples of nonrealizability are included.