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Lefschetz and Nielsen coincidence numbers on nilmanifolds and solvmanifolds. II. (English) Zbl 1001.55004

Summary: The author claimed in Part I [ibid. 43, No. 3, 249-261 (1992; Zbl 0748.55001)] that Nielsen coincidence numbers and Lefschetz coincidence numbers are related by the inequality \(N(f,g)\geqslant|L(f,g)|\) for all maps \(f,g: S_1\to S_2\) between compact orientable solvmanifolds of the same dimension. It was further claimed that \(N(f,g) = |L(f,g)|\) when \(S_2\) is a nilmanifold. A mistake in that paper has been discovered. In this paper, that mistake is partially repaired. A new proof of the equality \(N(f,g) = |L(f,g)|\) for nilmanifolds is given, and a variety of conditions for maps on orientable solvmanifolds are established which imply the inequality \(N(f,g)\geq |L(f,g)|\). However, it still remains open whether \(N(f,g)\geq|L(f,g)|\) for all maps between orientable solvmanifolds.

MSC:

55M20 Fixed points and coincidences in algebraic topology
57S25 Groups acting on specific manifolds

Citations:

Zbl 0748.55001
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Full Text: DOI

References:

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