Abstract
AbstractWe prove practical formulas for the Reidemeister coincidence number, the Lefschetz coincidence number and the Nielsen coincidence number of continuous maps between oriented infra-nilmanifolds of equal dimension. In order to obtain these formulas, we use the averaging formulas for the Lefschetz coincidence number and for the Nielsen coincidence number and we develop an averaging formula for the Reidemeister coincidence number. We also give a simple proof of the averaging formula for the Lefschetz coincidence number.Mathematics Subject Classification 2000: 55M20; 57S30.
Highlights
In order to study the number of fixed points of a continuous selfmap f : M ® M on a closed, connected manifold M, three homotopy invariant numbers are associated to f: the Reidemeister number R(f), the Lefschetz number L(f) and the Nielsen number N(f)
Simple and practical formulas have only been obtained in specific cases and one often turns the attention to comparing the Nielsen number to other numbers that are relatively more easy to compute, such as the Lefschetz number and the Reidemeister number
For infra-nilmanifolds formulas for the Lefschetz coincidence number and the Nielsen coincidence number are still open for study. We address this problem and we prove in Theorem 6.11 explicit and practical formulas for the Reidemeister coincidence number, the Lefschetz coincidence number and the Nielsen coincidence number of a pair of continuous maps between oriented infra-nilmanifolds of equal dimension, generalizing [12] from fixed point theory to coincidence theory and generalizing [13] from nilmanifolds to infra-nilmanifolds
Summary
In order to study the number of fixed points of a continuous selfmap f : M ® M on a closed, connected manifold M, three homotopy invariant numbers are associated to f: the Reidemeister number R(f), the Lefschetz number L(f) and the Nielsen number N(f). We address this problem and we prove in Theorem 6.11 explicit and practical formulas for the Reidemeister coincidence number, the Lefschetz coincidence number and the Nielsen coincidence number of a pair of continuous maps between oriented infra-nilmanifolds of equal dimension, generalizing [12] from fixed point theory to coincidence theory and generalizing [13] from nilmanifolds to infra-nilmanifolds. To each isolated subset C of Coin(f, g), one associates an integer ind(f, g; C), called the coincidence index, which generalizes the well-known fixed point index to Nielsen coincidence theory in the setting of maps between oriented manifolds of the equal dimension by Schirmer [16] (see [17]). In other words: when the dimension is at least three, the Nielsen number coincides with the minimal number of coincidence points in the homotopy class of the map
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