Lefschetz numbers for homomorphisms of free groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T03:14:13Z http://mathoverflow.net/feeds/question/89688 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89688/lefschetz-numbers-for-homomorphisms-of-free-groups Lefschetz numbers for homomorphisms of free groups Mike Shulman 2012-02-27T18:33:43Z 2012-02-27T18:47:25Z <p>Let $G = F_X$ be the free group on a finite set $X$, and $\phi\colon G\to G$ a group homomorphism. Consider the number</p> <p>$$\sum_{x\in X} (\text{number of occurrences of the generator x in the word \phi(x)})$$</p> <p>where occurrences of $x^{-1}$ are counted negatively. Does this number bear any significance in group theory? Note that if $\phi$ is the identity map, the formula above reduces to the rank of $G$.</p> <p><em>Background</em>: a finitely generated free group $G$ has a classifying space $B G$ which is Spanier-Whitehead dualizable, hence the endomorphism $B \phi\colon B G \to B G$ has a Lefschetz number. The formula above computes $1 - L(B \phi)$.</p> http://mathoverflow.net/questions/89688/lefschetz-numbers-for-homomorphisms-of-free-groups/89691#89691 Answer by Benjamin Enriquez for Lefschetz numbers for homomorphisms of free groups Benjamin Enriquez 2012-02-27T18:47:25Z 2012-02-27T18:47:25Z <p>Isn't this the trace of the endomorphism induced by phi on the abelianization of G, which is the free abelian group over X? </p>