Friday, August 06, 2004

In memoriam Andrei Tyurin : Verdier Duality and Localised Euler classes 

Dedicated to Andrei Tyurin.

Andrei Tyurin had a profound effect on my mathematical career. While I was writing my thesis I ran into a review article on the differential classification of algebraic surfaces that was clearly written by some body who like me at the time had an algebraic geometry perpective (and from some one with a very deep insight and working experience too ! It was from this article that I learned the analogy between the Abel Jacobi and the Donaldson correspondence for example). It focussed on ideas, some of which were quite wild, and contained inventive new words like "topomodel". He is also one of the people in the mathematical community I really miss, so it was a shock to find that he died in oktober 2002.

I first met him (and Victor Pidstrigatch) on a conference in Goettingen organised by Stephan Bauer and Tom Dieck in 1992. I will never forget his enthousiasm trying to explain the spien poolynoomial invaariants, with his heavy Russian accent, waving his massive arms below his grey beard, and struggling with his English (which improved dramatically in later years). IIRC we also talked a little then. At the time those invariants seemed mostly a trick to get differentiable classification results for surfaces easier because the computation for spin polynomials involved only vector bundles with a section (how little did we see the value of the Spin structure back then, although later Victor told me that he toyed with the idea of doing something useful with that extra parameter). Later that year he came to see Oxford and he was very interested in my work on the pure Hodge type of the Donaldson Polynomials which fitted well with his own ideas on differentiably invariant lattices in the intersection form.

But at least as important is the fond personal memories I have. Despite being 20 years older than me, I got along with him very well. He was a very friendly man, with both a lot of humour, and erudite wise opinions. I remember a conference on the campus in Sophia Antolis. Andrei, a bunch of other vector bundloscenti (Steve Bradlow, Manfed Lehn, Oscar Garcia Prada ? ) and I collected coupons to drink very cheap wine and gossiped and talked math till late in the evening, enjoying the wonderful weather. I remember how much fun we had swimming in the mediterenean in that lovely place Cetraro (which with all the Italians and Fabrizio Catanese around was a merry place, and a very good summerschool) and that I discovered that he lost most of his toes while climbing in the caucasus. I also remember strolling through Bonn with him where he gave fatherly advise on our impending second child, or that he came to see us for dinner in Bielefeld, the house slightly smelly from the nappies of the children.
Having left mathematics without having said goodbye to him is so sad.

One idea that I can ultimately trace back to him and Victor Pidstrigatch is localised Euler classes of infinite dimensional bundles. In fact the first discussion I had on infinite dimensional Euler classes must have been with Victor on that that conference in Goettingen. I defined these classes with much more work on a much more hoc basis (and only for Banach manifolds ) in The canonical class and the $C^\infty$ properties of Kaehler surfaces. They were independently invented and refined by many people around the same time, mostly in the context of Gromov-Witten invariants, but the first account of them is in the context of Donaldson theory in the work of Pidstrigatch Tyurin (in I think [1]). I have been thinking about localised Euler classes in the context of Verdier Duality recently, so I think it is apt to write on some of those ideas here. If you the rest makes any sense to you then TeX should make sense as well so for the moment I will just leave it like that.

Let $\pi: E \to X$ be vectorbundle with a section $s$ and zero section $s_0$ over a locally compact space $X$. Then Verdier duality at end boils down to the fact that for a continuous map $f: X \to Y$, the pushforward with proper support $f_!$ functor has a adjoint $f^!$ on the full derived category $D(X)$ of (cohomologically constructible ?) sheaves. It is tradional to make some extra finiteness assumption and work with the bounded derived category, but with the approach to duality using Brown representability this assumption can be avoided.

let $\D_\pi$ be the relative dualising complex. It is well known that for a vector bundle of rank $r$, the complex is quasi isomorphic to $(or E)[r]$. We now define the Thom class
\theta \in [{s_0}_! \Z_X, \D_\pi] = [\Z_X , s_0^!\pi^!\Z_X] = [\Z_X,\Z_X]
as the class corresponding to $1$, which by the explicit form of the dualising complex is essentially the same thing as the "usual" Thom class of a vb (in fact we might take any fibration with a section e.g. the fibrewise cone of a spherical fibration and presumably get the Thom class and ultimately the euler class of the spherical fibration). Now pull the Thom class back with $s$, to get a localised Euler Class
e(E, s) = s^*\theta \in [s^*{s_o}_! \Z_X, s^*\D_\pi]
This is not a very nice group to work in, so we massage it into something more maneagable. The base change formula for the diagram
Z &\to ^i & X
\j \downto & &\downto s_0
\ X &\to^s & E
implies that $s^*{s_0}_! \Z_X = j_!i^* \Z_X$ and $i^*\Z_X$ is just $\Z_Z$. So we get
e(E,s) = s^*\theta \in [ j_!\Z_Z, s^*\D_\pi] = [ \Z_Z, j^!s^*\D_\pi]
This class already lives clearly on a group assoicated to $Z$, but let us rewrite it further.

Assume there is a complex $Ind \in D(X)$ such that $s^*\pi^! \Z_X = (Ind, \D_X)$ where $\D_X $ is the Verdier dualising complex of $X$ (i.e $D_X = p^! \Z_{pt}$ where $X \to^p pt$). For example if all complexes can be represented by bounded and finitely generated groups then $s^*\D_\pi = (s^!(\D_\pi,\D_E),\D_X)$. Also on a manifold of finte dimension $d$ we know that $\D_X = orX[d]$ and we get $Ind = or(E)^* \tensor or(X) [d-e]$ so $Ind$ for index is quite appropriate. Then since
j^!(Ind, \D_X) = (j^*Ind, j^!\D_X) = (j^*Ind, \D_Z)
(an adjoint of the projection formula) we finally find that
e(E,s) \in [ \Z_Z, (j^*Ind, \D_Z)] = [j^*Ind, \D_Z]
So the localised Euler class lives in ""twisted locally finite homology" and because this construction is functorial, it is certainly the "right" way and the right group to define the localised Euler class in.

Now infinite dimensional spaces are of course not locally compact, but using the full derived category, it may well be that this assumption is not really needed for the duality machinary and the computation above would work. The existence of $Ind$ in the bounded derived category $D^b(X)$ should be the proper definition of a Fredholm section, and the usual Fredholm condition should be enough to guarantee the existence.

[1] Invariants of the smooth structure of an algebraic surface defined by the Dirac operator, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 2, 279 English transl. in Russian Acad. Sci. Izv. Math. 40 (1993), no. 2, 267

© Copyright 2004 Rogier Brussee.
Sometimes I write about things related to my work, but the views expressed here are my personal and do not necessarily reflect the views of my employer.

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© Copyright 2004-2006 Rogier Brussee.
These are my personal views and do not necessarily reflect those of my employer.