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arXiv · 2608.02292

The Asymptotic Development of Paths on Nilmanifolds

Mark T. Mac Lean
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Abstract

We prove a new Carnot-scale $n$-step nilpotent $\pi_1$-de Rham theorem: along every convergent scale sequence, the asymptotic homotopy class of a long trajectory is identified with the nilpotent development of its macroscopic horizontal path. The key analytic mechanism is an asymptotic-development theorem showing that uniform convergence of rescaled horizontal paths, together with a uniform bound on variation, forces uniform, layer-by-layer convergence of their full Carnot-rescaled nilpotent developments. The theorem recovers Schwartzman's theory of asymptotic cycles in step one and the 2-step asymptotic homotopy theory of Benardete and Mitchell in step two, while extending the correspondence to arbitrary nilpotent step.

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Mark T. Mac Lean. 2026-08-03. The Asymptotic Development of Paths on Nilmanifolds. https://arxiv.org/abs/2608.02292

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