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Nielsen Fixed Point Theory
 Forum Math
, 1990
"... this article and are well written up in the textbooks [B 1 ] and [Ki]. This is the basic theory, and every topologist should know it. ..."
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Cited by 10 (2 self)
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this article and are well written up in the textbooks [B 1 ] and [Ki]. This is the basic theory, and every topologist should know it.
PERIODS, LEFSCHETZ NUMBERS AND ENTROPY FOR A CLASS OF MAPS ON A BOUQUET OF CIRCLES
, 2004
"... Abstract. We consider some smooth maps on a bouquet of circles. For these maps we can compute the number of fixed points, the existence of periodic points and an exact formula for topological entropy. We use Lefschetz fixed point theory and actions of our maps on both the fundamental group and the f ..."
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Abstract. We consider some smooth maps on a bouquet of circles. For these maps we can compute the number of fixed points, the existence of periodic points and an exact formula for topological entropy. We use Lefschetz fixed point theory and actions of our maps on both the fundamental group and the first homology group.
Reidemeister coincidence invariants of fiberwise maps
"... Given two fiberwise maps f1, f2 between smooth fiber bundles over a base manifold B, we develop techniques for calculating their Nielsen coincidence number. In certain settings we can describe the Reidemeister set of (f1, f2) as the orbit set of a group operation of pi1(B). The size and number of o ..."
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Given two fiberwise maps f1, f2 between smooth fiber bundles over a base manifold B, we develop techniques for calculating their Nielsen coincidence number. In certain settings we can describe the Reidemeister set of (f1, f2) as the orbit set of a group operation of pi1(B). The size and number of orbits captures crucial extra information. E.g. for torus bundles of arbitrary dimensions over the circle this determines the minimum coincidence numbers of the pair (f1, f2) completely. In particular we can decide when f1 and f2 can be deformed away from one another or when a fiberwise selfmap can be made fixed point free by a suitable homotopy. In two concrete examples we calculate the minimum and Nielsen numbers for all pairs of fiberwise maps explicitly. Odd order orbits turn out to play a special role.
Homotopy idempotents on manifolds and . . .
, 2007
"... The Bass trace conjectures are placed in the setting of homotopy idempotent selfmaps of manifolds. For the strong conjecture, this is achieved via a formulation of Geoghegan. The weaker form of the conjecture is reformulated as a comparison of ordinary and L²–Lefschetz numbers. ..."
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The Bass trace conjectures are placed in the setting of homotopy idempotent selfmaps of manifolds. For the strong conjecture, this is achieved via a formulation of Geoghegan. The weaker form of the conjecture is reformulated as a comparison of ordinary and L²–Lefschetz numbers.
Fixed Points of Maps of a Nonaspherical Wedge
, 2008
"... Let X be a finite polyhedron that is of the homotopy type of the wedge of the projective plane and the circle. With the aid of techniques from combinatorial group theory, we obtain formulas for the Nielsen numbers of the selfmaps of X. ..."
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Let X be a finite polyhedron that is of the homotopy type of the wedge of the projective plane and the circle. With the aid of techniques from combinatorial group theory, we obtain formulas for the Nielsen numbers of the selfmaps of X.