Tag Archives: fixed point theorem

Comments for lecture 4

In this session we got around to stating the main theorem of the Atiyah-Bott Lefschetz theorem papers.  We imagine that we have an elliptic complex (as previously defined) and an endomorphism of that complex.  Since an elliptic complex is made up of two (or maybe three) components – manifold, bundles, operators – a geometric endomorphism similarly has several components: a smooth map  f\colon M\to M , a bunch of bundle maps from \(f^*E\to E\) for all the bundles \(E\) appearing in the complex, and the whole to give a (co)chain map and therefore an induced map on cohomology.  In these circumstances we can define the Lefschetz number to be the alternating sum of the traces of the induced cohomology maps.  The main theorem expresses this Lefschetz number as a sum over the fixed points of \(f\) (under the assumption that these fixed points are simple).

We reviewed what this theorem tells us in two cases: when we deal with the de Rham complex and an arbitrary smooth map \(M\to M\) (when we recover the classical Lefschetz fixed point theorem), and when we are considering the Dolbeault complex of a complex manifold \(M\), and \(f\) is a holomorphic map \(M\to M\) (this case supplied Atiyah and Bott’s original motivation).  There is a very pretty example on page 460 of the second paper in the series where they consider a self-map of projective space induced by multiplying each homogeneous coordinate by a different scalar factor.  Then the fixed point theorem gives an algebraic identity

\[1 = \sum_{i=0}^n \frac{\gamma_i^n}{\prod_{j\neq i} (\gamma_i-\gamma_j)}\]

for distinct nonzero complex numbers \(\gamma_0,\ldots,\gamma_n \). I wrote a post on my main blog explaining how this identity can be derived from the familiar Lagrange interpolation formula of numerical analysis.

In next class session we will return to the de Rham complex but we will also introduce a Riemannian metric, which makes it possible to get more subtle topological invariants from de Rham theory.  It would be appropriate to read section 6 of paper II in preparation for this session.