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The semi-index product formula. (English) Zbl 0811.55003

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Summary: We consider fibre bundle maps \[ \begin{tikzcd} E \ar[r,"{f,g}"] \ar[d,"p"'] & E^\prime \ar[d,"p^\prime"] \\ B \ar[r,"{\overline{f},\overline{g}}"'] & B^\prime \end{tikzcd} \] where all spaces involved are smooth closed manifolds (with no orientability assumptions). We find a necessary and sufficient condition for the formula \[ | \text{ind}| (f,g:A)= |\text{ind} |( \overline{f}, \overline{g}: p(A)) |\text{ind} | (f_ b,g_ b: p^{-1} (b)\cap A), \] where \(A\) stands for a Nielsen class of \((f,g)\), \(b\in p(A)\) and \(| \text{ind} |\) denotes the coincidence semi-index from R. Dobrenko and the author [Rocky Mt. J. Math. 23, No. 1, 67- 85 (1993; Zbl 0787.55003)]. This formula enables us to derive a relation between the Nielsen numbers \(N(f,g)\), \(N(\overline{f}, \overline{g})\) and \(N(f_ b, g_ b)\).

MSC:

55M20 Fixed points and coincidences in algebraic topology

Citations:

Zbl 0787.55003
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