= Self-coincidence number of a map of nonzero degree
{title2=$L(f,f)=(\deg f)\chi(N)$}
For $f:M\to N$ between closed connected oriented manifolds of equal dimension, cyclic invariance of trace and $f^!f^*=(\deg f)\mathrm{id}$ 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 $2k+1$.
Back to article page
