Lefschetz coincidence number ($L(f,g)={\sum }_i(-1{)}^i\mathrm{tr}(g^{\ast }{f}^{!})$)

For maps $f,g:M\to N$ between closed connected oriented manifolds of equal dimension, the rational Lefschetz coincidence number is the displayed alternating trace, using the cohomological pushforward between closed oriented manifolds. Equivalently it evaluates the pullback by $(f,g)$ of the cohomology class of the diagonal on $[M]$. This formula fixes the signs with the product orientation on $N\times N$.

For $f:M\to N$ between closed connected oriented manifolds of equal dimension, cyclic invariance of trace and $f^{!}{f}^{\ast }=(\mathrm{deg}f)\mathrm{id}$ give the formula. Thus if both the degree and the target Euler characteristic are nonzero, $f$ coincides somewhere with every map homotopic to it, by the Lefschetz coincidence theorem. For target ${\mathbb{CP}}^{2k}$ the Euler characteristic is $2k+1$.

A nonzero Lefschetz coincidence number forces a coincidence of the two maps. If their paired map misses the diagonal, the diagonal Thom class restricts to zero on its image and its pullback vanishes, making the coincidence number zero. The argument works for continuous maps and requires neither isolated coincidences nor transversality.