A combinatorial nerve theorem for effective homology computation
Abstract
The celebrated (homological) nerve theorem makes use of spectral sequences to determine the homology of a simplicial complex. However, this theorem cannot effectively compute the homology in every circumstance. In this paper, we develop an effective version of the nerve theorem, yielding a new and powerful tool for homology computation. The essence of our theorem can be formulated in the following manner. Suppose, X is a simplicial complex with covering subcomplexes A1, … ,Ak, that is, X= ∪i=1k Ai and 𝒩(X) is the nerve of X with respect to its covering. Let 𝒲α be a given gradient vector field on Aα(=∩i ∈ α Ai) for each α∈ 𝒩(X). Then, we use the mere information of the gradient trajectories in Aα for each α∈ 𝒩(X) to explicitly compute the homology groups of X. Furthermore, we point out here, that these gradient vector fields do not need to be coherent, that is, they do not need to coincide on the intersections, which gives us ample flexibility to apply our theorem. Moreover, we can further simplify the computation of the homology groups using a gradient vector field on the nerve of X. Our approach is purely combinatorial, in the sense that it does not involve any notions of geometric realisation, continuity or homotopy, which makes it more amenable to computation and coding.
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BibTeX
@misc{barik2026combinatorial,
title = {{A combinatorial nerve theorem for effective homology computation}},
author = {Sucharita Barik and Anupam Mondal and Sajal Mukherjee and Pritam Chandra Pramanik and Arundhati Rakshit},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2606.28047},
eprint = {2606.28047},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.28047},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A combinatorial nerve theorem for effective homology computation
AU - Sucharita Barik
AU - Anupam Mondal
AU - Sajal Mukherjee
AU - Pritam Chandra Pramanik
AU - Arundhati Rakshit
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2606.28047
UR - https://arxiv.org/abs/2606.28047
AB - The celebrated (homological) nerve theorem makes use of spectral sequences to determine the homology of a simplicial complex. However, this theorem cannot effectively compute the homology in every circumstance. In this paper, we develop an effective version of the nerve theorem, yielding a new and powerful tool for homology computation. The essence of our theorem can be formulated in the following manner. Suppose, X is a simplicial complex with covering subcomplexes A_1, … ,A_k, that is, X= ∪_i=1^k A_i and 𝒩(X) is the nerve of X with respect to its covering. Let 𝒲_α be a given gradient vector field on A_α(=∩_i ∈ α A_i) for each α∈ 𝒩(X). Then, we use the mere information of the gradient trajectories in A_α for each α∈ 𝒩(X) to explicitly compute the homology groups of X. Furthermore, we point out here, that these gradient vector fields do not need to be coherent, that is, they do not need to coincide on the intersections, which gives us ample flexibility to apply our theorem. Moreover, we can further simplify the computation of the homology groups using a gradient vector field on the nerve of X. Our approach is purely combinatorial, in the sense that it does not involve any notions of geometric realisation, continuity or homotopy, which makes it more amenable to computation and coding.
ER -