Vol. 25, No. 2, 1968

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 297: 1
Vol. 296: 1  2
Vol. 295: 1  2
Vol. 294: 1  2
Vol. 293: 1  2
Vol. 292: 1  2
Vol. 291: 1  2
Vol. 290: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
Fixed points for iterates

Benjamin Rigler Halpern

Vol. 25 (1968), No. 2, 255–275
Abstract

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
Milestones
Received: 15 February 1967
Published: 1 May 1968
Authors
Benjamin Rigler Halpern