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arXiv · 2504.11136

Structure of some mapping spaces

Liangzhao Zhang·Xiangyu Zhou
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Abstract

We prove that the path space of a differentiable manifold is diffeomorphic to a Fr\'echet space, endowing the path space with a linear structure. Furthermore, the base point preserving mapping space consisting of maps from a cube to a differentiable manifold is also diffeomorphic to a Fr\'echet space. As a corollary of a more general theorem, we prove that the path fibration becomes a fibre bundle for manifolds M. Additionally, we discuss the mapping space from a compact topological space to a differentiable manifold, demonstrating that this space admits the structure of a smooth Banach manifold.

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Liangzhao Zhang, Xiangyu Zhou. 2025-04-15. Structure of some mapping spaces. https://arxiv.org/abs/2504.11136

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