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Model category structures on truncated multicomplexes for complex geometry

Joana Cirici, Muriel Livernet, Sarah Whitehouse

TL;DR

This work develops a symmetric homotopical framework for N-multicomplexes by introducing a two-parameter family of weak equivalences \\mathcal{E}_{r,s}$ defined via quasi-isomorphisms on the $E_r$-page of one spectral sequence and the $E_s$-page of the dual sequence. It proves a right proper, combinatorial model structure on $N$-multicomplexes for each pair $(r,s)$, with fibrations controlled by bidegreewise surjectivity on both spectral sequences up to the specified pages. The construction hinges on witness cycles and boundaries that model the pages of the spectral sequences, and it relates to prior asymmetric models while providing a natural setting for invariants in complex geometry and for studying almost complex manifolds. A parallel adjunction between bicomplexes and 4-multicomplexes is analyzed, offering a bridge to geometric contexts where the four-component differential arises, and illustrating how these homotopical tools might inform questions about integrability and refined biholomorphic invariants in complex geometry.

Abstract

To a bicomplex one can associate two natural filtrations, the column and row filtrations, and then two associated spectral sequences. This can be generalized to $N$-multicomplexes. We present a family of model category structures on the category of $N$-multicomplexes where the weak equivalences are the morphisms inducing a quasi-isomorphism at a fixed page $r$ of the first spectral sequence and at a fixed page $s$ of the second spectral sequence. Such weak equivalences arise naturally in complex geometry. In particular, the model structures presented here establish a basis for studying homotopy types of almost and generalized complex manifolds.

Model category structures on truncated multicomplexes for complex geometry

TL;DR

This work develops a symmetric homotopical framework for N-multicomplexes by introducing a two-parameter family of weak equivalences \\mathcal{E}_{r,s}E_rE_sN(r,s)$, with fibrations controlled by bidegreewise surjectivity on both spectral sequences up to the specified pages. The construction hinges on witness cycles and boundaries that model the pages of the spectral sequences, and it relates to prior asymmetric models while providing a natural setting for invariants in complex geometry and for studying almost complex manifolds. A parallel adjunction between bicomplexes and 4-multicomplexes is analyzed, offering a bridge to geometric contexts where the four-component differential arises, and illustrating how these homotopical tools might inform questions about integrability and refined biholomorphic invariants in complex geometry.

Abstract

To a bicomplex one can associate two natural filtrations, the column and row filtrations, and then two associated spectral sequences. This can be generalized to -multicomplexes. We present a family of model category structures on the category of -multicomplexes where the weak equivalences are the morphisms inducing a quasi-isomorphism at a fixed page of the first spectral sequence and at a fixed page of the second spectral sequence. Such weak equivalences arise naturally in complex geometry. In particular, the model structures presented here establish a basis for studying homotopy types of almost and generalized complex manifolds.
Paper Structure (12 sections, 9 theorems, 29 equations)

This paper contains 12 sections, 9 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Multicomplexes and involution
  4. Recollection from previous work
  5. The model category structures
  6. Main Theorem
  7. Spectral sequences of the witness cycles
  8. Proof of Theorem \ref{['T:model']}
  9. Comparison with previous results
  10. Comparing bicomplexes and $4$-multicomplexes
  11. Definition of the adjunction
  12. Geometric viewpoint

Key Result

Lemma 2.1.4

The category $N\text{-}\mathrm{mC}_R$ is isomorphic to the category of left $\mathcal{A}_N$-modules. For each $(p,q)\in\mathbb{Z}^2$, the $R$-module $\mathcal{A}_N^{p,q}$ is a free $R$-module of finite rank. A basis of $\mathcal{A}_N$ as a free bigraded $R$-module is given by the set

Theorems & Definitions (25)

  • Definition 2.1.1
  • Remark 2.1.3
  • Lemma 2.1.4
  • proof
  • Lemma 2.1.6
  • proof
  • Definition 2.2.1
  • Proposition 2.2.2: cf. fglw22
  • Remark 2.2.3
  • Definition 2.2.4
  • ...and 15 more