Loop homology of moment-angle complexes in the flag case

Vylegzhanin, Fedor

doi:10.2140/agt.2025.25.5619

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Loop homology of moment-angle complexes in the flag case

Fedor Vylegzhanin

Algebraic & Geometric Topology 25 (2025) 5619–5663

DOI: 10.2140/agt.2025.25.5619

Abstract

We develop a general homological approach to presentations of connected graded associative algebras, and apply it to the loop homology of moment-angle complexes $Z_{K}$ that correspond to flag simplicial complexes $K$ . For an arbitrary coefficient ring, we describe generators of the Pontryagin algebra $H_{*} (Ω Z_{K})$ and defining relations between them. We prove that such moment-angle complexes are coformal over $ℚ$ , give a necessary condition for rational formality, and compute their homotopy groups in terms of homotopy groups of spheres.

Keywords

moment-angle complex, flag simplicial complex, loop homology, toric topology, extension of Hopf algebras, presentations of algebras

Mathematical Subject Classification

Primary: 16E30, 16W50, 57S12

Secondary: 55P35, 55Q52, 55U15, 57T05

References

Publication

Received: 12 April 2024

Revised: 4 October 2024

Accepted: 6 December 2024

Published: 18 December 2025

Authors

Fedor Vylegzhanin
Steklov Mathematical Insitute of Russian Academy of Sciences Moscow Russia

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Volume 25, issue 9 (2025)