Groups of Transverse Stratifications and the Hierarchy of Configuration Spaces

Abstract

We introduce a broad family of groups G(Γ) associated with transverse stratifications of moduli spaces. For a wide class of stratifications satisfying a natural local condition (transverse niceness), we construct a homomorphism from the fundamental group of the moduli space to the corresponding group G(Γ). This general theorem, which is proved in the companion paper \cite{Manturov2026}, provides a far-reaching generalization of the G_n^k-theory. The main goal of the present paper is to discuss numerous examples and situations where the transverse niceness property actually occurs. We describe various moduli spaces and stratifications that are transversely nice. In particular, we study the hierarchy of configuration spaces of points in the plane defined by algebraic curves and establish the transversality property for conic configurations. The general hierarchy problem is formulated in the final section.

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BibTeX RIS

BibTeX

@misc{manturov2026groups,
  title = {{Groups of Transverse Stratifications and the Hierarchy of Configuration Spaces}},
  author = {Vassily Olegovich Manturov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02123},
  eprint = {2609.02123},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02123},
  note = {CC0},
}

RIS

TY  - UNPB
TI  - Groups of Transverse Stratifications and the Hierarchy of Configuration Spaces
AU  - Vassily Olegovich Manturov
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.02123
UR  - https://arxiv.org/abs/2609.02123
AB  - We introduce a broad family of groups G(Γ) associated with transverse stratifications of moduli spaces. For a wide class of stratifications satisfying a natural local condition (transverse niceness), we construct a homomorphism from the fundamental group of the moduli space to the corresponding group G(Γ). This general theorem, which is proved in the companion paper \cite{Manturov2026}, provides a far-reaching generalization of the G_n^k-theory. The main goal of the present paper is to discuss numerous examples and situations where the transverse niceness property actually occurs. We describe various moduli spaces and stratifications that are transversely nice. In particular, we study the hierarchy of configuration spaces of points in the plane defined by algebraic curves and establish the transversality property for conic configurations. The general hierarchy problem is formulated in the final section.
ER  -

Abstract

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