Smoothness filtration of the magnitude complex (Q2178794)

Let \((X, d)\) be a metric space. A sequence of points \(\langle x_0, x_1, \dots, x_n\rangle\) (\(x_i\in X\)) is said to be a proper \(n\)-chain of length \(\ell\) if \(x_{i-1}\neq x_i\) (\(i=1, \dots, n\)) and \(d(x_0, x_1)+d(x_1, x_2)+\dots +d(x_{n-1}, x_n)=\ell\). Let \(C_n^\ell(X)\) be the free abelian group generated by proper \(n\)-chains of length \(\ell\). One can naturally define the boundary map \(\partial : C_n^\ell(X)\longrightarrow C_{n-1}^\ell(X)\) whose homology group \(H_n^\ell(X):=H_n(C_*^\ell(X))\) is called the \textit{magnitude homology} of \((X, d)\). The notion of magnitude homology was first introduced by \textit{R. Hepworth} and \textit{S. Willerton} [Homology Homotopy Appl. 19, No. 2, 31--60 (2017; Zbl 1377.05088)] for a finite metric space defined by a graph. Later, it was generalized to a metric space (furthermore, enriched category) by \textit{T. Leinster} and \textit{M. Shulman} [``Magnitude homology of enriched categories and metric spaces'', Preprint, \url{arXiv:1711.00802}]. The computation of magnitude homology is, in general, difficult. In particular, if there exists a \textit{\(4\)-cut}, that is a chain \(\langle x_0, x_1, x_2, x_3\rangle\) satisfying \[ \begin{split} d(x_0, x_3) &< d(x_0, x_1) + d(x_1, x_2) + d(x_2, x_3)\\ &= d(x_0, x_2) + d(x_2, x_3) = d(x_0, x_1) + d(x_1, x_3), \end{split} \] then the computation becomes complicated. Indeed, a previous work by \textit{R. Kaneta} and \textit{M. Yoshinaga} [Bull. Lond. Math. Soc. 53, No. 3, 893--905 (2021; Zbl 1472.55007)] showed that if the metric space \((X, d)\) does not have \(4\)-cuts, then the computation of the magnitude homology is reduced to that of the order complexes for posets. The present paper extends and refines the previous works by using spectral sequences. In a proper chain \(\langle x_0, x_1, \dots, x_n\rangle\), a point \(x_i\) is said to be a \textit{smooth point} if \(d(x_{i-1}, x_{i})+d(x_{i}, x_{i+1})=d(x_{i-1}, x_{i+1})\) (otherwise, it is called a \textit{singular point}). Denote the number of smooth points in \(x\) by \(\sigma(x)\) and the submodule of \(C_n^\ell(X)\) generated by proper chains with \(\sigma(x)\leq p\) by \(F_pC_n^\ell(X)\). Then, \(F_pC_*^\ell(X)\) defines a filtration on the magnitude chain complex \(C_*^\ell(X)\). The associated spectral sequence is the main object of the present paper. The author completely describes at which page the spectral sequence degenerates. Precise results are as follows. \begin{itemize} \item[(a)] The spectral sequence always degenerates at \(E^4\). \item[(b)] The spectral sequence degenerates at \(E^2\) if and only if the metric space \((X, d)\) does not contain \(4\)-cuts. \item[(c)] The spectral sequence degenerates at \(E^3\) if and only if there does not exist a chain \(x=\langle x_0, x_1, x_2, x_3, x_4\rangle\) such that both \(\langle x_0, x_1, x_2, x_3\rangle\) and \(\langle x_1, x_2, x_3, x_4\rangle\) are \(4\)-cuts. \end{itemize} The author also applies the spectral sequence to a number of concrete examples.