Algebraic objects via homotopy

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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.

Keywords

graphs / simplicial complexes / homotopy type / commutative rings / finite groups / cyclotomic polynomial

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