Nagoya Mathematical Journal

Averaging formula for Nielsen coincidence numbers

Seung Won Kim and Jong Bum Lee

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In this paper we study the averaging formula for Nielsen coincidence numbers of pairs of maps $(f, g) : M \to N$ between closed smooth manifolds of the same dimension. Suppose that $G$ is a normal subgroup of $\Pi = \pi_{1}(M)$ with finite index and $H$ is a normal subgroup of $\Delta = \pi_{1}(N)$ with finite index such that $f_{*}(G) \subset H$ and $g_{*}(G) \subset H$. Then we investigate the conditions for which the following averaging formula holds

N(f, g) = \frac{1}{[\Pi:G]} \sum_{\bar{\alpha} \in \Delta/H} N(\bar{\alpha} \bar{f}, \bar{g}),

where $(\bar{f}, \bar{g}) : M_{G} \to N_{H}$ is any pair of fixed liftings of $(f, g)$. We prove that the averaging formula holds when $M$ and $N$ are orientable infra-nilmanifolds of the same dimension, and when $M = N$ is a non-orientable infra-nilmanifold with holonomy group $\mathbb{Z}_{2}$ and $(f, g)$ admits a pair of liftings $(\bar{f}, \bar{g}) : \bar{M} \to \bar{M}$ on the nil-covering $\bar{M}$ of $M$.

Article information

Nagoya Math. J., Volume 186 (2007), 69-93.

First available in Project Euclid: 22 June 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55M20: Fixed points and coincidences [See also 54H25] 57S30: Discontinuous groups of transformations

Infra-nilmanifold Lefschetz coincidence number Nielsen coincidence number Semi-index


Kim, Seung Won; Lee, Jong Bum. Averaging formula for Nielsen coincidence numbers. Nagoya Math. J. 186 (2007), 69--93.

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  • L. Auslander and F. E. A. Johnson, On a conjecture of C. T. C. Wall, J. London Math. Soc. (2), 14 (1976), 331–332.
  • O. Baues, Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups, Topology, 43 (2004), 903–924.
  • G. E. Bredon, Topology and Geometry, Springer-Verlag, New York, 1993.
  • R. Brooks and P. Wong, On changing fixed points and coincidences to roots, Proc. Amer. Math. Soc., 115 (1992), 527–533.
  • R. Dobreńko and J. Jezierski, The coincidence Nielsen number on non-orientable manifolds, Rocky Mountain J. Math., 23 (1993), 67–85.
  • D. L. Gonçalves, The coincidence Reidemeister classes of maps on nilmanifolds, Topol. Meth. Nonlin. Anal., 12 (1998), 375–386.
  • D. L. Gonçalves and P. Wong, Nilmanifolds are Jiang-type spaces for coincidences, Forum Math., 13 (2001), 133–141.
  • V. Guillemin and A. Pollack, Differential topology, Prentice-Hall, Englewood Cliffs, New Jersey, 1974.
  • M. W. Hirsch, Differential topology, Graduate Text in Mathematics 33, Springer-Verlag, New York, 1976.
  • J. Jezierski, The Nielsen number product formula for coincidences, Fund. Math., 134 (1990), 183–212.
  • J. Jezierski, The semi-index product formula, Fund. Math., 140 (1992), 99–120.
  • S. W. Kim and J. B. Lee, Anosov theorem for coincidences on nilmanifolds, Fund. Math., 185 (2005), no. 3, 247–259.
  • S. W. Kim, J. B. Lee and K. B. Lee, Averaging formula for Nielsen numbers, Nagoya Math. J., 178 (2005), 37–53.
  • J. B. Lee and K. B. Lee, Lefschetz numbers for continuous maps, and periods for expanding maps on infra-nilmanifolds, J. Geom. Phys., 56 (2006), 2011–2023.
  • K. B. Lee and F. Raymond, Rigidity of almost crystallographic groups, Contemporary Math. Amer. Math. Soc., 44 (1985), 73–78.
  • C. K. McCord, Estimating Nielsen numbers on infrasolvmanifolds, Pacific J. Math., 154 (1992), 345–368.
  • H. Schirmer, Mindestzahlen von Koinzidenpunkten, J. Reine Angew. Math., 194 (1955), 21–39.
  • D. Vendrúscolo, Coincidence classes in nonorientable manifolds, Fixed Point Theory Appl. 2006, Special Issue, Art. ID 68513, 9 pp.
  • P. Wong, Reidemeister number, Hirsch rank, coincidence on polycyclic groups and solvmanifolds, J. reine angew. Math., 524 (2000), 185–204.