TITLE: Periodic orbit property
QUESTION [24 upvotes]: A topological space $X$ satisfies the "Periodic orbit property", briefly POP, if for every continuous map $f:X \to X$, there exist  a natural number $n$ and  a point $x_{0}\in X$  such that $f^{n}(x_{0})=x_{0}$.
Obviously fixed point property (FPP) implies POP.
For a natural number $n$, a topological space $X$ is  called $n$-POP if for every continuous map $f$ on $X$, $f^{n}$ has a fixed point. (Ex: $\mathbb{S}^{2n}$ is a 2-POP manifold, because the degree of a fixed-point-free map on $\mathbb{S}^{2n}$ must be $-1$.)
The question:

Is there an example of a manifold $M$ which satisfies POP but for  every $n\in \mathbb{N}$, there is a continuous map $f$ on $M$ such that $f^{n}$ has no fixed point?

Namely: we search for a manifold for which every self-map has a periodic orbit, but there is no any control on periods.
Equivalently:

Is there a manifold $M$ which is POP but not $n$-POP for all $n\in \mathbb{N}$?

In particular, can we say:

"every compact POP manifold is necessarily a $n$-POP manifold, for some $n$"?

Motivated by the Lefschetz fixed-point theorem, we ask that:
What algebraic topological criterion, can be introduced for consideration of this property(POP)?
Edit: According to the very interesting answer of Qiaochu  Yuan, in the orientable case, the question is equivalent to the following:

Let M be a closed orientable  manifold. Is it true that $M$ is not  POP if and only if $\chi(M)=0$?

Note1 For a related question see this post and it is natural to ask that "Does $S^{2}\vee S^{2}$ satisfy the periodic orbit property?"
Note2 I think the continuation of the argument of Qiaochu Yuan  for his  first statement is  not easy, for arbitrary manifold. Because for the  simplest case $S^{3}$ we had the famous conjecture of "existence of  a vector field on $S^{3}$ without periodic orbit. In fact consideration of non vanishing vector fields is necessary but not sufficient. Periodic orbits of vector fields are important, too. Moreover, perhaps  an approach which is not based on "vector fields" could be useful, for example consideration of orientation-reversing diffeomorphisms.
Note 3: "pointwise periodic homeomorphism' is a concept which is indirectly similar to the subject of this post.

REPLY [5 votes]: The following is proved by
F. Brock Fuller in "The Existence of Periodic Points," Annals of Mathematics, Vol. 57, 1953, pp. 229-230:
Theorem. Let X be a  compact simplicial complex with  Betti numbers $B_i$, and nonzero Euler characteristic. If a continuous map
 $f: X\rightarrow X$ induces isomorphisms of homology groups, then $f$ has a point whose period is $\le \max_i \big (∑_i B_{2i},∑_i B_{2i-1} \big)$. Thus every homeomorphism of $X$ has a periodic orbit.