The reduced Lefschetz number, that is, 
where 
denotes the Lefschetz number, is proved to be the unique integer-valued function
on self-maps of compact polyhedra which is constant on homotopy classes such that
(1) 
for 
and 
; (2) if 
is a map of a cofiber sequence into itself, then 
; (3) 
, where 
is a self-map of a wedge of 
circles, 
is the inclusion of a circle into the 
th summand, and 
is the projection onto the 
th summand. If 
is a self-map of a polyhedron and 
is the fixed point index of 
on all of 
, then we show that 
satisfies the above axioms. This gives a new proof of the normalization theorem:
if 
is a self-map of a polyhedron, then 
equals the Lefschetz number 
of 
. This result is equivalent to the Lefschetz-Hopf theorem: if 
is a self-map of a finite simplicial complex with a finite number of fixed points,
each lying in a maximal simplex, then the Lefschetz number of 
is the sum of the indices of all the fixed points of 
.
Research Article
The Lefschetz-Hopf theorem and axioms for the Lefschetz number
1 Department of Mathematics, Dartmouth College, Hanover, NH 03755-1890, USA
2 Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA
Fixed Point Theory and Applications 2004, 2004:465090 doi:10.1155/S1687182004308120
The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2004/1/465090
| Received: | 28 August 2003 |
| Published: | 3 March 2004 |
© 2004 Arkowitz and Brown
