Smoothness filtration of the magnitude complex

Gomi, Kiyonori

doi:10.1515/forum-2019-0091

Cited by 5 publications

(2 citation statements)

References 15 publications

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“…The existence of a 4-cut is a source of the computational complexity in determining the magnitude homology groups. Indeed, Gomi [9] proved that the nonexistence of 4-cuts is equivalent to the E 2 -degeneration of the spectral sequence converging to MH ℓ * (X). If 0 < ℓ < m X , Kaneta and the second author gave a decomposition of the magnitude homology into framed ones.…”

Section: Frame Decompositionmentioning

confidence: 99%

Causal order complex and magnitude homotopy type of metric spaces

Yu¹,

Yoshinaga²

2023

Preprint

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In this paper, we construct a pointed CW complex called the magnitude homotopy type for a given metric space X and a real parameter ℓ ≥ 0. This space is roughly consisting of all paths of length ℓ and has the reduced homology group that is isomorphic to the magnitude homology group of X.To construct the magnitude homotopy type, we consider the poset structure on the spacetime X × R defined by causal (time-or light-like) relations. The magnitude homotopy type is defined as the quotient of the order complex of an intervals on X × R by a certain subcomplex.The magnitude homotopy type gives a covariant functor from the category of metric spaces with 1-Lipschitz maps to the category of pointed topological spaces. The magnitude homotopy type also has a "path integral" like expression for certain metric spaces.By applying discrete Morse theory to the magnitude homotopy type, we obtain a new proof of the Mayer-Vietoris type theorem and several new results including the invariance of the magnitude under sycamore twist of finite metric spaces. Contents

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Section: Frame Decompositionmentioning

confidence: 99%

Causal order complex and magnitude homotopy type of metric spaces

Yu¹,

Yoshinaga²

2023

Preprint

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“…As a categorification of the magnitude, Hepworth and Willerton defined magnitude homology for graphs [6], and later Leinster and Shulman generalized to metric spaces (furthermore, for enriched categories) [10]. Several techniques to compute the magnitude homology groups have been developed [6,3,7,4].…”

Section: Introductionmentioning

confidence: 99%

Magnitude homology of graphs and discrete Morse theory on Asao-Izumihara complexes

Tajima¹,

Yoshinaga²

2021

Preprint

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Recently, Asao and Izumihara introduced CW-complexes whose homology groups are isomorphic to direct summands of the graph magnitude homology group. In this paper, we study the homotopy type of the CWcomplexes in connection with the diagonality of magnitude homology groups. We prove that the Asao-Izumihara complex is homotopy equivalent to a wedge of spheres for pawful graphs introduced by Y. Gu. We also formulate a slight generalization of the notion of a pawful graph and find new non-pawful diagonal graphs of diameter 2.

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Causal Order Complex and Magnitude Homotopy Type of Metric Spaces

Tajima

Yoshinaga

2023

International Mathematics Research Notices

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In this paper, we construct a pointed CW complex called the magnitude homotopy type for a given metric space $X$ and a real parameter $\ell \geq 0$. This space is roughly consisting of all paths of length $\ell $ and has the reduced homology group that is isomorphic to the magnitude homology group of $X$. To construct the magnitude homotopy type, we consider the poset structure on the spacetime $X\times \mathbb{R}$ defined by causal (time- or light-like) relations. The magnitude homotopy type is defined as the quotient of the order complex of an intervals on $X\times \mathbb{R}$ by a certain subcomplex. The magnitude homotopy type gives a covariant functor from the category of metric spaces with $1$-Lipschitz maps to the category of pointed topological spaces. The magnitude homotopy type also has a “path integral” like expression for certain metric spaces. By applying discrete Morse theory to the magnitude homotopy type, we obtain a new proof of the Mayer–Vietoris-type theorem and several new results including the invariance of the magnitude under sycamore twist of finite metric spaces.

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Smoothness filtration of the magnitude complex

Abstract: We introduce an intrinsic filtration to the magnitude chain complex of a metric space, and study basic properties of the associated spectral sequence of the magnitude homology. As an application, the third magnitude homology of the circle is computed.

Cited by 5 publications

References 15 publications

Causal order complex and magnitude homotopy type of metric spaces

Causal order complex and magnitude homotopy type of metric spaces

Magnitude homology of graphs and discrete Morse theory on Asao-Izumihara complexes

Causal Order Complex and Magnitude Homotopy Type of Metric Spaces

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