L-classes of symmetric products: a homological formula (Iuliia Semikina)

Orateur : Iuliia Semikina

Lieu : salle des Séminaires M3

Résumé :

If M is a closed oriented manifold of even dimension, then its n-th symmetric product M(n)=M^n/S_n is a rational homology manifold and hence has a canonically defined L-class in H^∗(M(n); Q). A complicated formula for these classes was given many years ago by Don Zagier. We argue that the Poincaré dual classes l(M(n)) in H_*(M(n);Q) are the more natural objects to study. Embedding each H_∗(M(n);Q) into the Hopf algebra H_∗(M(\infty);Q)) we show that the generating function of the classes l(M(n)) is the product of an elementary factor depending only on the Euler characteristic of M and the exponential of a series consisting of odd Adams twists of l(M). We then use this result to obtain an expression for the cohomological L-classes of M(n).