Tag Archives: exposition

A “well known interpolation formula”

In the Atiyah-Bott paper on their Lefschetz theorem for elliptic complexes, they give a very nice elementary example of the Lefschetz theorem for the Dolbeault complex, by considering the automorphism of   {\mathbb C}{\mathbb P}^n given by

\[ [z_0,\ldots, z_n] \mapsto [\gamma_0z_0, \ldots, \gamma_nz_n ] \]

in homogeneous coordinates, where the \(\gamma_i\) are distinct and nonzero complex numbers.  This has \( (n+1) \) simple fixed points and applying the holomorphic Lefschetz theorem gives

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

This is Example 2 on p. 460 of the second Atiyah-Bott paper.  They go on to describe this as a “well known interpolation formula”. Continue reading

Surgery for Amateurs

In 1996 I was the Ulam Visiting Professor at the University of Colorado, Boulder.  While I was there I gave a series of graduate lectures on high-dimensional manifold theory, which I whimsically titled Surgery for Amateurs.

The title was supposed to express that I was coming to the subject from outside – basically, trying to answer to my own satisfaction the question “What is this Novikov Conjecture you keep talking about?” Perhaps because of their amateurish nature, though, these lectures struck a chord, and I have received many requests for reprints of the lecture notes.  In 2004 I began a project of revising them with the help of Andrew Ranicki; but, alas, other parts of life intervened, and the proposed book never got finished.

Obviously some people still value the material, and my plan is to try and republish it in blog form, along with comments and discussion.  The Surgery for Amateurs blog is now live and your participation is welcomed!