Model category structures on truncated multicomplexes for complex geometry

TL;DR

This paper develops a family of model category structures on N-multicomplexes, linking spectral sequence quasi-isomorphisms to complex geometry, aiding the study of homotopy types of complex manifolds.

Contribution

It introduces new model category structures on N-multicomplexes with weak equivalences based on spectral sequence pages, connecting algebraic topology with complex geometry.

Findings

Model structures on N-multicomplexes with spectral sequence-based weak equivalences.

Application to homotopy theory of complex and generalized complex manifolds.

Framework for analyzing complex geometric structures via homotopical methods.

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.