Converse to Lefschetz fixed point theorem (counterexample) The celebrated Lefschetz fixed point theorem, in its simplest form, says (following Wikipedia) that if $f\colon X \to X$ is a continuous map of a compact triangulable space $X$ to itself, then $f$ has at least one fixed point, provided that the value $\Lambda_f$ given by:
$$
     \Lambda_f:=\sum_{k\geq 0}(-1)^k\mathrm{Tr}(f_*|H_k(X,\mathbb{Q})), 
$$
is non-zero.

What is an example when the converse fails to be true? I.e., what is an example of a space $X$ and a function $f$ such that $\Lambda_f$ is $0$, but nevertheless any function homotopic to $f$ needs to have a fixed point?
Recall that the Lefschetz number $\Lambda_f$ for $f:X \rightarrow X$ the identity map is just the Euler-Poincaré characteristic: so it is sufficient to take a compact CW complex with $0$ Euler characteristic, e.g. a circle, and you have an example of a map with $\Lambda_f=0$ but with fixed points.
Recall that the Lefschetz number $\Lambda_f$ for $f:X \rightarrow X$ the identity map is just the Euler-Poincaré characteristic: so it is sufficient to take a compact CW complex with $0$ Euler characteristic, e.g. a circle, and you have an example of a map with $\Lambda_f=0$ but with fixed points.