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On removing coincidences of two maps when only one, rather than both, of them may be deformed by a homotopy. (English) Zbl 0235.55006

Summary: It is known that if \(f,g\colon X\to Y\) are maps of a topological space \(X\) into a topological manifold \(Y\), and that \(f\) and \(g\) can be deformed by homotopies to maps \(f'\) and \(g'\) which are coincidence-free, then \(f\) may be deformed by a homotopy to a map \(f''\) such that \(f''\) and \(g\) are coincidence-free. This result is generalized as follows: If \(f,g\colon X\to Y\) are maps of a topological space \(X\) into a topological manifold \(Y\) and \(f'\) and \(g'\) are homotopic to \(f\) and \(g\) respectively, then for any homotopy \(\{g_t\}\) from \(g\) to \(g'\), there is a homotopy \(\{f_t\}\) from \(f'\) such that the set of coincidences of \(f_t\) and \(g_{1-t}\) is the same for all \(t\in [0,1]\). Some applications of this result to fixed point theory and root theory are indicated.

MSC:

55M20 Fixed points and coincidences in algebraic topology
55P10 Homotopy equivalences in algebraic topology
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