On bordism classes of symmetric products of surfaces

Georgy Chernykh; Dmitry Gugnin

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arXiv · 2611.07813

On bordism classes of symmetric products of surfaces

Georgy Chernykh·Dmitry Gugnin

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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.

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BibTeX RIS

Georgy Chernykh, Dmitry Gugnin. 2026-11-05. On bordism classes of symmetric products of surfaces. https://arxiv.org/abs/2611.07813

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