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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 ) .

Minimal number of periodic points for smooth self-maps of S³

Grzegorz GraffJerzy Jezierski — 2009

Fundamenta Mathematicae

Let f be a continuous self-map of a smooth compact connected and simply-connected manifold of dimension m ≥ 3 and r a fixed natural number. A topological invariant D r m [ f ] , introduced by the authors [Forum Math. 21 (2009)], is equal to the minimal number of r-periodic points for all smooth maps homotopic to f. In this paper we calculate D ³ r [ f ] for all self-maps of S³.

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