On bordism classes of symmetric products of surfaces

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₌₁ᵍ⁺¹ R P²). It turns out that these classes are also expressed as a Z₂-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.