Magnitude homology, diagonality, and median spaces

Bottinelli, Rémi; Kaiser, Tom

doi:10.4310/hha.2021.v23.n2.a7

Cited by 5 publications

(7 citation statements)

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“…• The magnitude homotopy type satisfies a type of excision and Mayer-Vietoris formula as established in [14,5] (Proposition 5.13, Theorem 5.14, Corollary 5. 16).…”

Section: Introductionmentioning

confidence: 86%

“…equivalently, H * = H K, hence every element in int(H) is biased. This assumption is equivalent to "H projects to K" ( [14]) and "gated decomposition" ( [5]). Also, as in the previous sections, denote the union by X = G ∪ K H.…”

Section: Additivity (Mayer-vietoris Formula)mentioning

confidence: 99%

“…It should be noted that this space has already been dealt with in previous works by Hepworth-Willerton [14] (for graphs) and Bottinelli-Kaiser [5] (for metric spaces). Asao-Izumihara [3] has also constructed a closely related space for graphs.…”

Section: Introductionmentioning

confidence: 98%

“…The space M ℓ (X) was constructed by Hepworth-Willerton[14, Definition 8.1] for graphs and by Bottinelli-Kaiser[5, Definition 4.4] for metric spaces as the realization of certain simplicial set. Our Definition 4.4 realizes it as the quotient of a simplicial complex by a subcomplex.…”

mentioning

confidence: 99%

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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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“…• The magnitude homotopy type satisfies a type of excision and Mayer-Vietoris formula as established in [14,5] (Proposition 5.13, Theorem 5.14, Corollary 5. 16).…”

Section: Introductionmentioning

confidence: 86%

Section: Additivity (Mayer-vietoris Formula)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

mentioning

confidence: 99%

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Causal order complex and magnitude homotopy type of metric spaces

Yu¹,

Yoshinaga²

2023

Preprint

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“…By considering k = 2i + 5 or 2i + 6 in theorem 1.5, it turns out that the range 1 j i guaranteeing the vanishing of the magnitude homology groups in corollary 1. 4 is optimal (see table I).…”

Section: Introductionmentioning

confidence: 99%

Girth, magnitude homology and phase transition of diagonality

Asao¹,

Hiraoka²,

Kanazawa³

2023

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. The magnitude and magnitude homology are formulations of the Euler characteristic and the corresponding homology, respectively, for finite metric spaces, first introduced by Leinster and Hepworth–Willerton. Several authors study them restricting to graphs with path metric, and some properties which are similar to the ordinary homology theory have come to light. However, the whole picture of their behaviour is still unrevealed, and it is expected that they catch some geometric properties of graphs. In this article, we show that the girth of graphs partially determines the magnitude homology, that is, the larger girth a graph has, the more homologies near the diagonal part vanish. Furthermore, applying this result to a typical random graph, we investigate how the diagonality of graphs varies statistically as the edge density increases. In particular, we show that there exists a phase transition phenomenon for the diagonality.

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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.

show abstract

Magnitude homology, diagonality, and median spaces

Cited by 5 publications

References 0 publications

Causal order complex and magnitude homotopy type of metric spaces

Causal order complex and magnitude homotopy type of metric spaces

Girth, magnitude homology and phase transition of diagonality

Causal Order Complex and Magnitude Homotopy Type of Metric Spaces

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