Homology, Homotopy and Applications

Graphs associated with simplicial complexes

A. Grigor'yan, Yu. V. Muranov, and Shing-Tung Yau

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Abstract

The cohomology of digraphs was introduced for the first time by Dimakis and Müller-Hoissen. Their algebraic definition is based on a differential calculus on an algebra of functions on the set of vertices with relations that follow naturally from the structure of the set of edges. A dual notion of homology of digraphs, based on the notion of path complex, was introduced by the authors, and the first methods for computing the (co)homology groups were developed. The interest in homology on digraphs is motivated by physical applications and relations between algebraic and geometrical properties of quivers. The digraph $G_B$ of the partially ordered set $B_{S}$ of simplexes of a simplicial complex $S$ has graph homology that is isomorphic to the simplicial homology of $S$. In this paper, we introduce the concept of cubical digraphs and describe their homology properties. In particular, we define a cubical subgraph $G_{S}$ of $G_B$, whose homologies are isomorphic to the simplicial homologies of $S$.

Article information

Source
Homology Homotopy Appl., Volume 16, Number 1 (2014), 295-311.

Dates
First available in Project Euclid: 3 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.hha/1401800084

Mathematical Reviews number (MathSciNet)
MR3211747

Zentralblatt MATH identifier
1297.05269

Subjects
Primary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65] 18G60: Other (co)homology theories [See also 19D55, 46L80, 58J20, 58J22] 52C99: None of the above, but in this section 55U10: Simplicial sets and complexes 57M15: Relations with graph theory [See also 05Cxx]

Keywords
(Co)homology of digraphs path complex of a digraph simplicial homology cubical digraph cubical complex simplicial complex

Citation

Grigor'yan, A.; Muranov, Yu. V.; Yau, Shing-Tung. Graphs associated with simplicial complexes. Homology Homotopy Appl. 16 (2014), no. 1, 295--311. https://projecteuclid.org/euclid.hha/1401800084


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