On bordism classes of symmetric products of surfaces

Abstract

We explicitly determine the complex bordism class of the $n$-th symmetric product of a complex curve of genus $g$. In particular, we provide an explicit formula expressing this class as an integral linear combination of bordism classes of products of complex projective spaces. Similarly, we determine the unoriented bordism class of the $n$-th symmetric product of a non-orientable surface of genus $g$ (i.e., ${\#}_{i=1}^{g+1}\mathbb{R}{P}^2$). It turns out that these classes are also expressed as a ${\mathbb{Z}}_2$-linear combination of bordism classes of products of even-dimensional real projective spaces. We also give an affirmative answer to a non-orientable version of the Blagojević--Grujić--Živaljević conjecture stating that the topological type of any symmetric product of a noncompact surface of finite type determines the topological type of the surface itself.