I know that the Lefschetz number of the identity map is the Euler character. A map $f$ which is isotopic to $id$ will have the same number. And then we use the theorem to find fixed points. My question is whether we can request the orbits of these fixed points under the isotopy to be trivial in $\pi_1(M)$.
Denote the isotopy as $f_t$ and let $x$ be a fixed point of $f$, then the orbit of $x$ under this isotopy is the loop $f_t(x),t\in [0,1]$. Thanks to Qiaochu to point this out.