Self-coincidence of fibre maps
Given two maps \(f_1, f_2\colon M\to N\), a basic problem in coincidence point theory is whether we can make them coincidence free under suitable homotopies, i.e. whether there are \(f'_1\simeq f_1\) and \(f'_2\simeq f_2\) such that \(f'_1(x) \neq f'_2(x)\) for all \(x\in M\). By a result of \textit{R. B. S. Brooks} [Pac. J. Math. 40, 45--52 (1972; Zbl 0235.55006)], if \(N\) is a manifold, deforming one rather than both of them is enough. The authors consider this problem in some special cases, where \(f_1\) and \(f_2\) are the same fibre maps \(p\) of \(S^1\)- or \(S^3\)-principal bundles. According to the classification of these bundles, they obtain necessary and sufficient conditions under which \(p\) is self-coincidence free. The main invariants to detect coincidence sets are the obstructions from \textit{E. Fadell} and \textit{S. Husseini} [Topology 20, 53--92 (1981; Zbl 0453.55002)]. Because the classical spaces and maps under consideration are well-studied in algebraic topology, the authors are successful in handling the non-trivialities of those obstructions. On the other hand, some methods to make maps self-coincidence free are also developed by using the homotopy lifting property and related techniques. Similar results for some non-bundle maps are also obtained.
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