On the spectral sequence associated to a multicomplex

Abstract

A multicomplex, also known as a twisted chain complex, has an associated spectral sequence via a filtration of its total complex. We give explicit formulas for all the differentials in this spectral sequence.Conventions. Throughout the paper k will be a commutative unital ground ring.

Morse Theory without Non-Degeneracy

“…In this section we will modify the usual definition of a multicomplex given in §4 (cf. for example [52]) to define multicomplexes supported on acyclic quivers (Definition 5.14 below). These will be used to generalise the Morse-Witten complex to the situation where ∶ → ℝ is any smooth function on a compact Riemannian manifold whose critical locus has finitely many connected components.…”

Dolbeault and J-Invariant Cohomologies on Almost Complex Manifolds

“…with respect to the differential d r , |d r | = (r , −r + 1). An explicit description up to isomorphism of the differential d r and of the pages of the spectral sequence is given in [14] for a general multicomplex, and was first described in [9] for the Frölicher spectral sequence of a complex manifold. For page 1, we have…”

Model category structures on multicomplexes

“…For the rest of this section, let r ≥ 0 be an integer. We consider the spectral sequence E * , * r (A) associated to the multicomplex A as described in [9,Proposition 2.8]. The following is a reformulation of the description in [9,Definition 2.6] to make the notation consistent with [2] in the case of bicomplexes.…”

“…A key ingredient of the model structures is the explicit description of the spectral sequence associated to a multicomplex in [9]. The main techniques imitate the work of [2], using representable versions of r-cycles and r-boundaries to provide generating (trivial) cofibrations for the model structures.…”