Talk:Lefschetz fixed-point theorem

Lefschetz number

is defined arbitrarily for maps $X\rightarrow X$ , then if we use the identity map we get $\Lambda _{id}=\#(\Delta ,\Delta ,M\times M)=\chi (M)$ is the intersection number of the diagonal with itself in the product manifold $M\times M$ , i.e., the Euler characteristic. On the algebraic topological level I'm sure this holds too, that $\chi (M)=\Lambda _{id}(M)$ . Anyone know more about this? MotherFunctor 05:55, 28 May 2006 (UTC)[reply]

The connection is now explained.24.58.63.18 (talk) 19:07, 4 June 2009 (UTC)[reply]

References for the statements about Frobenius

it would be nice to have a reference (to look up proofs) for the statements about the Frobenius. --79.83.77.245 (talk) 00:49, 19 January 2010 (UTC)[reply]

Brouwer's fixed point theorem

In the text it is stated that every continuous map on the unit ball gives non-zero mapping on the homology groups. This is of course false, just take the constant map. It has a fixed point, of course, but for reasons that have nothing to do with Lefschetz theorem. In general, Lefschetz theorem can't tell us anything about the null-homotopic maps. The Brouwer's theorem, on the other hand, does.