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arXiv · 2607.10956

Based maps to Lagrangian Grassmannians, Quivers, and Bott Periodicity

Jim Bryan·Ravi Vakil
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Abstract

We give a quiver description of the space of based algebraic maps from $\mathbb{P}^{1}$ to the Lagrangian Grassmannian (and its orthogonal counterpart). We show our descriptions lead to an algebro-geometric refinement of some of the homotopy equivalences in real Bott periodicity. In particular, we get an isomorphism in Larson and Vakil's ``naive algebro-geometric homotopy category'' whose topological realization (after specializing to $\mathbb{C}$) recovers the classical homotopy equivalences $\Omega^{2}(Sp/U) \simeq BO\times \mathbb{Z}$ and $\Omega^{2}(O/U) \simeq BSp\times \mathbb{Z}$.

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Jim Bryan, Ravi Vakil. 2026-07-12. Based maps to Lagrangian Grassmannians, Quivers, and Bott Periodicity. https://arxiv.org/abs/2607.10956

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