The Lefschetz coincidence class of \(p\) maps (Q2343366)

Let \(X\) be a topological space and \(Y\) a closed, connected, \(n\)-dimensional manifold and \(f_1,\dotsc,f_p\) maps from \(X\) to \(Y\). Let \(\mu\in H^n(Y\times Y,Y\times Y\setminus\Delta)\) denote the Thom class. For \(i=1,\dotsc,p-1\) let \(h_i:=(f_i,f_{i+1}):X\to Y\times Y\). Denote by \(j:Y\times Y\to(Y\times Y,Y\times Y\setminus\Delta)\) the inclusion and define the Lefschetz class of \(f_1,\dotsc,f_p\) by \(L(f_,\dotsc,f_p):= [(\undersetbrace p-1\to{(j\times\dotsb\times j)}\circ(h_1,\dotsc,h_{p-1})]^*(\undersetbrace p-1\to{(\mu\times\dotsb,\mu)}\in H^{n(p-1)}X\). The authors prove that \(L(f_1,\dotsc,f_p)\not=0\) implies that the coincidence set of \(f_1,\dotsc,f_p\) is nonempty. If \(Y\) is a homology \(n\)-sphere they derive several formulae to calculate \(L(f_1,\dotsc,f_p)\).