We define a relative coincidence Nielsen number $N_{rel}(f,g)$ for pairs of maps between manifolds, prove a Wecken type theorem for this invariant and give some formulae expressing $N_{rel}(f,g)$ by the ordinary Nielsen numbers.
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We prove that a self-map f: M → M of a compact PL-manifold of dimension ≥ 3 is homotopic to a map with no periodic points of period n iff the Nielsen numbers $N(f^{k})$ for k dividing n all vanish. This generalizes the result from [Je] to dimension 3.
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We generalize the coincidence semi-index introduced in [D-J] to pairs of maps between topological manifolds. This permits extending the Nielsen theory to this class of maps.
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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 $N(f_b,g_b)$.
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We generalize the Lefschetz coincidence theorem to non-oriented manifolds. We use (co-) homology groups with local coefficients. This generalization requires the assumption that one of the considered maps is orientation true.
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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 $D^m_r[f]$, 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 $D³_r[f]$ for all self-maps of S³.
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