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The Lefschetz-Hopf theorem and axioms for the Lefschetz number
Fixed Point Theory and Applications volume 2004, Article number: 465090 (2004)
Abstract
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
.
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Arkowitz, M., Brown, R.F. The Lefschetz-Hopf theorem and axioms for the Lefschetz number. Fixed Point Theory Appl 2004, 465090 (2004). https://doi.org/10.1155/S1687182004308120
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DOI: https://doi.org/10.1155/S1687182004308120