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Coincidence points of fiber maps on \(S^n\)-bundles. (English) Zbl 1196.55001
The authors study coincidences for fiber-preserving maps between \(S^n\)-bundles. If \(E\to B\) is an \(S^n\)-bundle they first show the existence of a fiber-preserving map \(\tau:E\to E\) which is fixed point free and such that the degree of \(\tau|\,S^n\) is 1 if \(n\) is odd and \(-1\) if \(n\) is even. They then obtain the following result: Let \(E_1\to B\) and \(E_2\to B\) be \(S^n\)-bundles and \(f:E_1\to E_2\) a fiber-preserving map. Then there is a unique homotopy class of maps given by \(\mathcal{C}=[\tau\circ f]\) such that for \(g\in\mathcal{C}(f)\) the pair \((f,g)\) can be deformed over \(B\) to a coincidence free pair. The authors then study the case of \(S^n\)-bundles over \(S^1\).

MSC:
55M20 Fixed points and coincidences in algebraic topology
55R10 Fiber bundles in algebraic topology
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