$L$-classes of symmetric products: a homological formula

Abstract

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 for any $n \geq 0$ and hence has a canonically defined $L$-class in $H^*(M(n);\mathbb{Q})$. A complicated formula for these classes was given many years ago in the second author's thesis. We obtain a simpler result by studying the dual classes in homology, showing that their generating function (after embedding each $H_*(M(n); \mathbb{Q})$ into the Hopf algebra $H_*(M(\infty); \mathbb{Q})$) 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 the Poincaré dual of the $L$-class of $M$. We also use this to obtain both a simpler proof and a simpler expression for the cohomological $L$-classes than the previous one.