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We introduce and study an algorithm that constructs a discrete gradient field on any simplicial complex. With a computational complexity similar to that of existing methods, our algorithmic gradient field is always maximal and in a number of cases even optimal. We make a thorough analysis of the resulting gradient field in the case of Munkres discrete model for Conf (Km,2), the configuration space of ordered pairs of noncolliding particles moving on the complete graph Km on m vertices. This allows us to describe in full the cohomology algebra H∗(Conf (Km,2);R) for any commutative unital ring R. As an application we prove that, although Conf (Km,2) is outside the “stable” regime, all its topological complexities are maximal when m ≥ 4.
González, Emilio J 1 ; González, Jesús 1
@article{10_2140_agt_2024_24_3719,
author = {Gonz\'alez, Emilio J and Gonz\'alez, Jes\'us},
title = {An algorithmic discrete gradient field and the cohomology algebra of configuration spaces of two points on complete graphs},
journal = {Algebraic and Geometric Topology},
pages = {3719--3758},
publisher = {mathdoc},
volume = {24},
number = {7},
year = {2024},
doi = {10.2140/agt.2024.24.3719},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.3719/}
}
TY - JOUR AU - González, Emilio J AU - González, Jesús TI - An algorithmic discrete gradient field and the cohomology algebra of configuration spaces of two points on complete graphs JO - Algebraic and Geometric Topology PY - 2024 SP - 3719 EP - 3758 VL - 24 IS - 7 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.3719/ DO - 10.2140/agt.2024.24.3719 ID - 10_2140_agt_2024_24_3719 ER -
%0 Journal Article %A González, Emilio J %A González, Jesús %T An algorithmic discrete gradient field and the cohomology algebra of configuration spaces of two points on complete graphs %J Algebraic and Geometric Topology %D 2024 %P 3719-3758 %V 24 %N 7 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.3719/ %R 10.2140/agt.2024.24.3719 %F 10_2140_agt_2024_24_3719
González, Emilio J; González, Jesús. An algorithmic discrete gradient field and the cohomology algebra of configuration spaces of two points on complete graphs. Algebraic and Geometric Topology, Tome 24 (2024) no. 7, pp. 3719-3758. doi: 10.2140/agt.2024.24.3719
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