Combining Lefschetz numbers with Euler classes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T14:48:11Z http://mathoverflow.net/feeds/question/73379 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73379/combining-lefschetz-numbers-with-euler-classes Combining Lefschetz numbers with Euler classes Allen Knutson 2011-08-22T06:19:37Z 2011-08-22T06:19:37Z <p>Given an $n$-manifold $M$ (say), we can talk about its Euler characteristic $\chi(M)$.</p> <ol> <li><p>This can be generalized to the Euler number of any $n$-dimensional bundle ${\mathcal V}$. Or indeed, the Euler class of any $k$-dimensional oriented bundle.</p></li> <li><p>Or, given a map $f:M\to M$, we can talk about its Lefschetz number, where $\chi(M)$ is the Lefschetz number of the identity. </p></li> <li><p>Or, we can compute $M$'s Betti numbers, and see $\chi(M)$ as the alternating sum.</p></li> </ol> <p>I know how to combine 2 &amp; 3, replacing the Betti number by $Tr(f|_{H^i(M)})$.</p> <blockquote> <p>Are 1 &amp; 2 or 1 &amp; 3 combinable? Or 1, 2, and 3?</p> </blockquote> <p>I don't have an application in mind, just the usual mathematician's hankering to take LCMs wherever possible.</p>