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Simple closed curves, non-kernel homology and Magnus embedding

Adam Klukowski

TL;DR

The paper addresses whether lifts of simple closed curves generate the whole first homology in covers of surfaces by proving the existence of unbranched covers where the simple-curve–generated subspace is proper, for genus $g\ge2$, puncture count $n\ge0$, and odd primes $r$. It develops a two-stage strategy: first relate different cover-homology notions and reduce to $d$-primitive homology, then construct polynomial substitutes via Magnus embedding, enabling embeddings of surface groups into units of graded associative algebras with controlled fixed vectors. This approach unifies and extends prior branched-cover results and furnishes a streamlined, algebraic method that also yields embeddings with independent algebraic interest. The work thereby advances understanding of how curve lifts interact with homology in covers and provides new tools (polynomial substitutes, graded-algebra embeddings) for studying surface groups and mapping-class-group-related phenomena with potential broader applications in geometric group theory.

Abstract

We consider the subspace of the homology of a covering space spanned by lifts of simple closed curves. Our main result is the existence of unbranched covers of surfaces where this is a proper subspace. More generally, for a fixed finite solvable quotient of the fundamental group we exhibit a cover whose homology is not generated by the lifts of curves in the complement of its kernel. We explain how the existing approach of Malestein and Putman (for branched covers) relates to the Magnus embedding, and by doing so we simplify their construction. We then generalise it to unbranched covers by producing embeddings of surface groups into units of certain graded associative algebras, which may be of independent interest.

Simple closed curves, non-kernel homology and Magnus embedding

TL;DR

The paper addresses whether lifts of simple closed curves generate the whole first homology in covers of surfaces by proving the existence of unbranched covers where the simple-curve–generated subspace is proper, for genus , puncture count , and odd primes . It develops a two-stage strategy: first relate different cover-homology notions and reduce to -primitive homology, then construct polynomial substitutes via Magnus embedding, enabling embeddings of surface groups into units of graded associative algebras with controlled fixed vectors. This approach unifies and extends prior branched-cover results and furnishes a streamlined, algebraic method that also yields embeddings with independent algebraic interest. The work thereby advances understanding of how curve lifts interact with homology in covers and provides new tools (polynomial substitutes, graded-algebra embeddings) for studying surface groups and mapping-class-group-related phenomena with potential broader applications in geometric group theory.

Abstract

We consider the subspace of the homology of a covering space spanned by lifts of simple closed curves. Our main result is the existence of unbranched covers of surfaces where this is a proper subspace. More generally, for a fixed finite solvable quotient of the fundamental group we exhibit a cover whose homology is not generated by the lifts of curves in the complement of its kernel. We explain how the existing approach of Malestein and Putman (for branched covers) relates to the Magnus embedding, and by doing so we simplify their construction. We then generalise it to unbranched covers by producing embeddings of surface groups into units of certain graded associative algebras, which may be of independent interest.
Paper Structure (24 sections, 16 theorems, 48 equations)

This paper contains 24 sections, 16 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Homologies of covers and subgroups
  3. Algebraic interpretation.
  4. Other results
  5. Outline and organisation
  6. Relationships between homologies of covers
  7. Stability of curve characteristics under taking covers
  8. Reduction to $d$-primitive homology
  9. Properness of $d$-primitive homology
  10. Existence of polynomial substitutes
  11. Free groups and proof of Proposition \ref{['prop:polynomial_substitute_free']}
  12. Magnus embedding.
  13. Construction of a polynomial substitute for a free group.
  14. Verification of the polynomial substitute property.
  15. Surface groups and proof of Proposition \ref{['prop:polynomial_substitute_surface']}
  16. ...and 9 more sections

Key Result

Theorem 1

For any $g \geq 2$, $n \geq 0$, and odd prime $r$ there is a cover $\widetilde{\Sigma} \rightarrow \Sigma_{g, n}$, whose degree is a power of $r$, and such that $\textnormal{H}_1^{\textnormal{scc} | \Sigma_{g, n}} (\widetilde{\Sigma} ; \mathbb{Q}) \neq \textnormal{H}_1 (\widetilde{\Sigma} ; \mathbb{

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Proposition 4
  • Remark 5
  • Remark 6
  • Theorem 7
  • Proposition : Lemma \ref{['lmm:hom_to_quat']}
  • Lemma 8
  • proof
  • ...and 22 more