Thin homotopy and the signature of piecewise linear surfaces
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909654061154304 |
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| author | Bischoff, Francis Lee, Darrick |
| author_facet | Bischoff, Francis Lee, Darrick |
| contents | We introduce a crossed module of piecewise linear surfaces and study the signature homomorphism, defined as the surface holonomy of a universal translation invariant $2$-connection. This provides a transform whereby surfaces are represented by formal series of tensors. Our main result is that the signature uniquely characterizes a surface up to translation and thin homotopy, also known as tree-like equivalence in the case of paths. This generalizes a result of Chen and positively answers a question of Kapranov in the setting of piecewise linear surfaces. As part of this work, we provide several equivalent definitions of thin homotopy, generalizing the plethora of definitions which exist in the case of paths. Furthermore, we develop methods for explicitly and efficiently computing the surface signature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_16657 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Thin homotopy and the signature of piecewise linear surfaces Bischoff, Francis Lee, Darrick Algebraic Topology Category Theory Differential Geometry Geometric Topology We introduce a crossed module of piecewise linear surfaces and study the signature homomorphism, defined as the surface holonomy of a universal translation invariant $2$-connection. This provides a transform whereby surfaces are represented by formal series of tensors. Our main result is that the signature uniquely characterizes a surface up to translation and thin homotopy, also known as tree-like equivalence in the case of paths. This generalizes a result of Chen and positively answers a question of Kapranov in the setting of piecewise linear surfaces. As part of this work, we provide several equivalent definitions of thin homotopy, generalizing the plethora of definitions which exist in the case of paths. Furthermore, we develop methods for explicitly and efficiently computing the surface signature. |
| title | Thin homotopy and the signature of piecewise linear surfaces |
| topic | Algebraic Topology Category Theory Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2506.16657 |