Multicomplexes and spectral sequences

In this note we present some algebraic examples of multicomplexes whose differentials differ from those in the spectral sequences associated to the multicomplexes. The motivation for constructing examples showing the algebraic distinction between a multicomplex and its associated spectral sequence comes from the author’s work on Morse-Bott homology with A. Banyaga.

💡 Research Summary

The paper investigates the relationship between multicomplexes—a generalization of chain complexes equipped with a family of differentials (d_r) satisfying (\sum_{i+j=n} d_i d_j = 0)—and the spectral sequences that arise from filtering such structures. While it is common to assume that the differentials (d^r) appearing on the (E^r) pages of the spectral sequence coincide with the original multicomplex differentials (d_r), the author demonstrates that this is not generally true. Two explicit algebraic examples are constructed. The first example is a 2‑level multicomplex ((C^{,}, d_0, d_1)) where (d_0) acts horizontally and (d_1) vertically. After applying the natural filtration, the (E^0) page reflects (d_0) and the (E^1) page is built from the cohomology of (d_1). However, the induced differential (d^1) on (E^1) is not simply (d_1); it incorporates mixed terms arising from the interaction of (d_0) and (d_1). The second example extends to an infinite‑level multicomplex with higher differentials (d_2, d_3,\dots). In the early pages of the associated spectral sequence these higher differentials are invisible; they only emerge in later pages (e.g., (E^2) or (E^3)), illustrating that the spectral sequence can “hide” certain components of the multicomplex differential structure until convergence.

Motivated by the author’s work with A. Banyaga on Morse‑Bott homology, the paper explains how multicomplexes naturally appear when critical manifolds replace isolated critical points. In Morse‑Bott theory the differential on the chain complex is built from flow lines connecting critical submanifolds, leading to a multicomplex rather than a simple chain complex. The examples above show that relying solely on the spectral sequence to compute Morse‑Bott homology may miss essential higher‑order differential information. Consequently, a direct algebraic treatment of the multicomplex, rather than an indirect spectral‑sequence approach, is necessary for accurate homological calculations.

The conclusion emphasizes the need for new filtration techniques that preserve the full multicomplex differential data, systematic methods to trace higher differentials through the spectral sequence, and the development of computational frameworks that work directly with multicomplexes in geometric contexts such as Morse‑Bott theory. By exposing the subtle but crucial distinction between a multicomplex and its associated spectral sequence, the paper provides a foundation for future research aimed at refining algebraic tools used in topological and geometric applications.