Model category structures on truncated multicomplexes for 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 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 complex manifolds.