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The minimizing of the Nielsen root classes

Daciberg Gonçalves, Claudemir Aniz (2004)

Open Mathematics

Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen...

The relative coincidence Nielsen number

Jerzy Jezierski (1996)

Fundamenta Mathematicae

We define a relative coincidence Nielsen number N r e l ( f , g ) for pairs of maps between manifolds, prove a Wecken type theorem for this invariant and give some formulae expressing N r e l ( f , g ) by the ordinary Nielsen numbers.

The semi-index product formula

Jerzy Jezierski (1992)

Fundamenta Mathematicae

We consider fibre bundle maps (...) where all spaces involved are smooth closed manifolds (with no orientability assumption). We find a necessary and sufficient condition for the formula    |ind|(f,g:A) = |ind| (f̅,g̅: p(A)) |ind| ( f b , g b : p - 1 ( b ) ∩ A ) to hold, where A stands for a Nielsen class of (f,g), b ∈ p(A) and |ind| denotes the coincidence semi-index from [DJ]. This formula enables us to derive a relation between the Nielsen numbers N(f,g), N(f̅,g̅) and N ( f b , g b ) .

The universal functorial Lefschetz invariant

Wolfgang Lück (1999)

Fundamenta Mathematicae

We introduce the universal functorial Lefschetz invariant for endomorphisms of finite CW-complexes in terms of Grothendieck groups of endomorphisms of finitely generated free modules. It encompasses invariants like Lefschetz number, its generalization to the Lefschetz invariant, Nielsen number and L 2 -torsion of mapping tori. We examine its behaviour under fibrations.

The Wecken property of the projective plane

Boju Jiang (1999)

Banach Center Publications

A proof is given of the fact that the real projective plane P 2 has the Wecken property, i.e. for every selfmap f : P 2 → P 2 , the minimum number of fixed points among all selfmaps homotopic to f is equal to the Nielsen number N(f) of f.

Theorem on signatures

Władysław Kulpa, Andrzej Szymański (2007)

Acta Universitatis Carolinae. Mathematica et Physica

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