Vol. 25, No. 2, 1968

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Fixed points for iterates

Benjamin Rigler Halpern

Vol. 25 (1968), No. 2, 255–275

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 i dimHi(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.

Mathematical Subject Classification
Primary: 55.25
Received: 15 February 1967
Published: 1 May 1968
Benjamin Rigler Halpern