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The Nielsen Borsuk-Ulam number. (English) Zbl 1386.55003
The paper under review is an extension of previous work of the authors on maps between finite dimensionsal complexes [Topology Appl. 159, No. 18, 3738–3745 (2012; Zbl 1260.55003)].
The authors introduce a Nielsen-Borsuk-Ulam number \(\mathrm{NBU}(f,\tau)\) for a continuous map \(f:X \to Y\), where \(X\) and \(Y\) are closed orientable triangulable \(n\)-manifolds and \(\tau\) is a free involution on \(X\).
The number \(\mathrm{NBU}(f,\tau)\) shown to be a lower bound, in the homotopy class of \(f\), for the number of pairs of points in \(X\) satisfying \(f(x)=f(\tau(x))\). Further, it is proved that \(\mathrm{NBU}(f,\tau)\) can be realized (a Wecken type theorem) when \(n\geq3\).
55M20 Fixed points and coincidences in algebraic topology
Full Text: Euclid