Geometric Kernels of Proper Maps Between Non-Compact Surfaces

Sumanta Das

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arXiv · 2508.21057

Geometric Kernels of Proper Maps Between Non-Compact Surfaces

Sumanta Das

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Abstract

A map between connected $2$-manifolds has a geometric kernel if it sends a non-contractible simple loop to a null-homotopic loop. While every non-$\pi_1$-injective map between compact surfaces admits a geometric kernel, this generally fails for compact bordered or non-compact surfaces. In this paper, we use Brown's proper fundamental group to give a sufficient condition under which a degree-one map between non-compact surfaces admits a geometric kernel. Furthermore, we characterize conjugacy classes in the proper fundamental group and use this characterization to establish sufficient conditions for the existence of geometric kernels.

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BibTeX RIS

Sumanta Das. 2025-08-28. Geometric Kernels of Proper Maps Between Non-Compact Surfaces. https://arxiv.org/abs/2508.21057

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