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Čech complexes for finite sets
Abstract
Let ℱm={A⊂[m]}, endowed with the symmetric difference metric and the Cardinality Reverse-Lexicographic order. That way, we compute the homotopy type Čech(ℱ⪯ am; r/2=1) by relating it to consecutively adding vertices of 𝕀m, the unit hypercube of mth dimension, to point $0$. Further, for higher r, we propose a conjecture regarding contractibility of a subcomplex implying a formula for the exact number of wedges of 𝕊3's and 𝕊4's in the Čech complex.
Related
BibTeX
@misc{biswas2026čech,
title = {{Čech complexes for finite sets}},
author = {Anamitro Biswas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32786},
eprint = {2609.32786},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32786},
}
RIS
TY - UNPB
TI - Čech complexes for finite sets
AU - Anamitro Biswas
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32786
UR - https://arxiv.org/abs/2609.32786
AB - Let ℱ^m={A⊂[m]}, endowed with the symmetric difference metric and the Cardinality Reverse-Lexicographic order. That way, we compute the homotopy type Čech(ℱ_⪯ a^m; r/2=1) by relating it to consecutively adding vertices of 𝕀^m, the unit hypercube of mth dimension, to point $0$. Further, for higher r, we propose a conjecture regarding contractibility of a subcomplex implying a formula for the exact number of wedges of 𝕊^3's and 𝕊^4's in the Čech complex.
ER -