Topological Lower Bounds on the Sizes of Simplicial Complexes and Simplicial Sets

Sergey Avvakumov; Роман Карасев

doi:10.1007/s00454-026-00823-z

February 10, 2026Discrete & Computational GeometryOpen Access

Topological Lower Bounds on the Sizes of Simplicial Complexes and Simplicial Sets

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Key Points

The goal is to establish lower boundaries on the number of faces in simplicial complexes and sets associated with n-dimensional spaces under specific topological conditions.
Proves lower bounds based on topological conditions of n-dimensional space X.
Analyzes triangulations and simplicial set representations of these spaces.
Uses examples like the n-dimensional torus to illustrate findings.
Finds that any triangulation or simplicial set representation must have at least 2^n faces in dimension n.
Verifies that certain topological conditions are crucial for these lower bounds.

Implication

Demonstrates lower bounds on faces in simplicial complexes in n-dimensional spaces, indicating significant topological properties.

Abstract

We prove that if an n -dimensional space X satisfies certain topological conditions then any triangulation of X as well as any its representation as a simplicial set with contractible faces has at least 2ⁿ 2 n faces of dimension n . One example of such X is the n -dimensional torus (S¹)ⁿ ( S 1 ) n .

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Cite This Study

Avvakumov et al. (2026) studied this question.

synapsesocial.com/papers/698d6e4a5be6419ac0d53ecf https://doi.org/10.1007/s00454-026-00823-z

View Full Paper

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Topological Lower Bounds on the Sizes of Simplicial Complexes and Simplicial Sets

Key Points

Implication

Abstract

Discussion

Cite This Study

Topological Lower Bounds on the Sizes of Simplicial Complexes and Simplicial Sets

Key Points

Implication

Abstract

Discussion

Cite This Study

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