L-classes of symmetric products: a homological formula
Abstract
If M is a closed oriented manifold of even dimension, then its nth symmetric product M(n)=Mn/Sn is a rational homology manifold for any n ≥ 0 and hence has a canonically defined L-class in H*(M(n);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); ℚ) into the Hopf algebra H*(M(∞); 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.
Related
- On bordism classes of symmetric products of surfaces
- Sn-equivariant L-classes of Cartesian powers of Witt spaces and the Hodge index conjecture for symmetric products
- Splitting torsion classes in higher homological algebra
- Homological finiteness properties of wreath products
- Local Bousfield classes via homological support
BibTeX
@misc{semikina2026lclasses,
title = {{$L$-classes of symmetric products: a homological formula}},
author = {Julia Semikina and Don Zagier},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32793},
eprint = {2609.32793},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32793},
}
RIS
TY - UNPB
TI - L-classes of symmetric products: a homological formula
AU - Julia Semikina
AU - Don Zagier
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32793
UR - https://arxiv.org/abs/2609.32793
AB - If M is a closed oriented manifold of even dimension, then its nth symmetric product M(n)=M^n/S_n is a rational homology manifold for any n ≥ 0 and hence has a canonically defined L-class in H^*(M(n);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); ℚ) into the Hopf algebra H_*(M(∞); 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.
ER -