Nielsen coincidence theory applied to Borsuk-Ulam geometric problems
The paper under review studies a Borsuk-Ulam type theorem for a triple \((X, \tau, Y)\) where \(X, Y\) are simplicial complexes and \(\tau\) is a free involution on \(X\). One of the main features of this work is that the authors begin to develop a sort of Nielsen-Borsuk-Ulam theory which has properties similar to classical Nielsen fixed point theory. More precisely, given a homotopy class of maps \([f]\in [X, Y]\) a number is defined which is expected to be a lower bound for the number of \(\tau\)-orbits which collapse for any map \(g\in [f]\). As expected, based in the experience with the usual Nielsen number, it is not an easy task to compute this new number. The main result of the paper is: {Theorem 4.7} Let \(X\) and \(Y\) be closed triangulable \(n\)-manifolds, \(\tau\) a free involution on \(X\) and \(f:X \to Y\) a continuous map. Suppose that \(BUCoin(f;\tau)\) is finite. If there exists a map \(f'\) homotopic to \(f\) such that \(BUCoin(f';\tau)=\emptyset\), then for each \(BU\)-coincidence class \(C=\{c_1, \tau(c_1),....,c_l, \tau(c_l)\}\) there exists a sequence of splits, joins and blends from \(\Sigma(C)\) that produces a null-signature. The paper leads naturally to many new questions related with the Borsuk-Ulam theorem, for example the ones which are modelled in the classical Nielsen theory.
scientific article; zbMATH DE number 6115439