Computation of the Nielsen type numbers for maps on the Klein bottle

Hyun Jung Kim; Jong Bum Lee; Won Sok Yoo

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2008; 45(5): 1483-1503

Printed September 1, 2008

Computation of the Nielsen type numbers for maps on the Klein bottle

Hyun Jung Kim, Jong Bum Lee, and Won Sok Yoo

Sogang University, Sogang University, Kumoh National Institute of Technology

Abstract

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Abstract

Let $f:M\to M$ be a self-map on the Klein bottle $M$. We compute the Lefschetz number and the Nielsen number of $f$ by using the infra-nilmanifold structure of the Klein bottle and the averaging formulas for the Lefschetz numbers and the Nielsen numbers of maps on infra-nilmanifolds. For each positive integer $n$, we provide an explicit algorithm for a complete computation of the Nielsen type numbers $NP_n(f)$ and $N\Phi_n(f)$ of $f^n$.

Keywords: Klein bottle, Nielsen type numbers, weakly Jiang maps

MSC numbers: Primary 57S25; Secondary 37D20, 20F18