Algebraic objects via homotopy

Aleksandra Kostić Matijević; Nela Milošević; Zoran Z. Petrović; Aleksandra Kostić Matijević; Nela Milošević; Zoran Z. Petrović

doi:10.3934/era.2026351

Electronic Research Archive

2026, Volume 34, Issue 11: 8278-8294. doi: 10.3934/era.2026351

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Algebraic objects via homotopy

1.
Faculty of Mathematics, University of Belgrade, Studentski trg 16, Belgrade, Serbia
2.
Faculty of Applied Sciences, University of Donja Gorica, Oktoih 1, Podgorica, Montenegro

Received: 22 July 2026 Revised: 27 August 2026 Accepted: 07 September 2026 Published: 22 September 2026

Graphs associated to algebraic objects have been widely used to encode algebraic information through pairwise relations. Many algebraic relations, however, naturally involve finite collections rather than pairs. Simplicial complexes therefore provide a higher-dimensional framework for studying algebraic objects: after encoding algebraic data as a simplicial complex, one investigates the homology and homotopy type of its geometric realization. In this paper, we present several such constructions arising from commutative rings, finite groups, and cyclotomic polynomials. These examples illustrate how simplicial complexes can retain higher-order algebraic information and how the resulting topological invariants can reflect structural properties of the underlying algebraic object.
- graphs,
- simplicial complexes,
- homotopy type,
- commutative rings,
- finite groups,
- cyclotomic polynomial
Citation: Aleksandra Kostić Matijević, Nela Milošević, Zoran Z. Petrović. Algebraic objects via homotopy[J]. Electronic Research Archive, 2026, 34(11): 8278-8294. doi: 10.3934/era.2026351

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Abstract

Graphs associated to algebraic objects have been widely used to encode algebraic information through pairwise relations. Many algebraic relations, however, naturally involve finite collections rather than pairs. Simplicial complexes therefore provide a higher-dimensional framework for studying algebraic objects: after encoding algebraic data as a simplicial complex, one investigates the homology and homotopy type of its geometric realization. In this paper, we present several such constructions arising from commutative rings, finite groups, and cyclotomic polynomials. These examples illustrate how simplicial complexes can retain higher-order algebraic information and how the resulting topological invariants can reflect structural properties of the underlying algebraic object.

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