Let f : X → X be a continuous
map of a compact polyhedron X into itself and H a homology theory with rational
coefficients. In the first section a variety of theorems are proved connecting the
existence or nonexistence of fixed points for certain iterates of f with a variety of
other information such as: conditions on the Betti numbers of X, f being a
homomorphism, certain induced homomorphisms fi∗: Hi(X) → Hi(X) being
isomorphisms, factors of the Lefschetz numbers Λ(fn), the gross behavior
of f with respect to the components of X, and certain other iterates of f
being fixed point free. One of the theorems proven is that if Hi(X) = 0
for odd i then there exists an x ∈ X and an n,1 ≦ n ≦∑idimHi(X),
such that fn(x) = x. Another theorem is that if fn is fixed point free for
1 ≦ n ≦ p∕2 and fp∗= identity then p divides the Euler characteristic of
X.
