The Lefschetz-Hopf theorem and axioms for the Lefschetz number

• Main Authors: Brown Robert F, Arkowitz Martin
• Format: Article
• Language: English
• Published: SpringerOpen 2004-01-01
• Series: Fixed Point Theory and Applications
• Online Access: http://www.fixedpointtheoryandapplications.com/content/2004/465090

<p/> <p>The reduced Lefschetz number, that is, <inline-formula><graphic file="1687-1812-2004-465090-i1.gif"/></inline-formula> where <inline-formula><graphic file="1687-1812-2004-465090-i2.gif"/></inline-formula> denotes the Lefschetz number, is proved to be the unique integer-valued function <inline-formula><graphic file="1687-1812-2004-465090-i3.gif"/></inline-formula> on self-maps of compact polyhedra which is constant on homotopy classes such that (1) <inline-formula><graphic file="1687-1812-2004-465090-i4.gif"/></inline-formula> for <inline-formula><graphic file="1687-1812-2004-465090-i5.gif"/></inline-formula> and <inline-formula><graphic file="1687-1812-2004-465090-i6.gif"/></inline-formula>; (2) if <inline-formula><graphic file="1687-1812-2004-465090-i7.gif"/></inline-formula> is a map of a cofiber sequence into itself, then <inline-formula><graphic file="1687-1812-2004-465090-i8.gif"/></inline-formula>; (3) <inline-formula><graphic file="1687-1812-2004-465090-i9.gif"/></inline-formula>, where <inline-formula><graphic file="1687-1812-2004-465090-i10.gif"/></inline-formula> is a self-map of a wedge of <inline-formula><graphic file="1687-1812-2004-465090-i11.gif"/></inline-formula> circles, <inline-formula><graphic file="1687-1812-2004-465090-i12.gif"/></inline-formula> is the inclusion of a circle into the <inline-formula><graphic file="1687-1812-2004-465090-i13.gif"/></inline-formula>th summand, and <inline-formula><graphic file="1687-1812-2004-465090-i14.gif"/></inline-formula> is the projection onto the <inline-formula><graphic file="1687-1812-2004-465090-i15.gif"/></inline-formula>th summand. If <inline-formula><graphic file="1687-1812-2004-465090-i16.gif"/></inline-formula> is a self-map of a polyhedron and <inline-formula><graphic file="1687-1812-2004-465090-i17.gif"/></inline-formula> is the fixed point index of <inline-formula><graphic file="1687-1812-2004-465090-i18.gif"/></inline-formula> on all of <inline-formula><graphic file="1687-1812-2004-465090-i19.gif"/></inline-formula>, then we show that <inline-formula><graphic file="1687-1812-2004-465090-i20.gif"/></inline-formula> satisfies the above axioms. This gives a new proof of the normalization theorem: if <inline-formula><graphic file="1687-1812-2004-465090-i21.gif"/></inline-formula> is a self-map of a polyhedron, then <inline-formula><graphic file="1687-1812-2004-465090-i22.gif"/></inline-formula> equals the Lefschetz number <inline-formula><graphic file="1687-1812-2004-465090-i23.gif"/></inline-formula> of <inline-formula><graphic file="1687-1812-2004-465090-i24.gif"/></inline-formula>. This result is equivalent to the Lefschetz-Hopf theorem: if <inline-formula><graphic file="1687-1812-2004-465090-i25.gif"/></inline-formula> 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 <inline-formula><graphic file="1687-1812-2004-465090-i26.gif"/></inline-formula> is the sum of the indices of all the fixed points of <inline-formula><graphic file="1687-1812-2004-465090-i27.gif"/></inline-formula>.</p>