Poincaré duality and module topological complexity
Abstract
Topological complexity, introduced by Farber, measures the number of continuous local motion-planning rules on a configuration space. For every simply connected space X of finite rational homotopy type whose rational cohomology is a finite-dimensional Poincaré duality algebra, we prove MTC(X)=TC0(X)=TC0M(X), where TC0(X)=TC(Xℚ) and M denotes the monoidal invariant. No formality or ellipticity assumption is required. The result extends to higher topological complexity and gives product additivity under duality. For every map f between simply connected spaces of finite rational homotopy type whose rationalization admits a homotopy retraction, we establish the sharp bound Msecat(fℚ)≤secat(fℚ)≤relcat(fℚ)≤Msecat(fℚ)+1. In particular, MTC(X)≤TC0(X)≤MTC(X)+1 holds without a duality assumption. The proofs use controlled Sullivan resolutions and an ideal-valued criterion for relative category. For sections of fibrations with Poincaré duality fibre, we prove that a derived self-intersection formula, vanishing of Thom multiplication on the augmentation ideal, and ideal-valued evaluation are equivalent. This criterion yields equality of the three rational sectional invariants for smooth sections of smooth bundles over simply connected smooth bases of finite rational homotopy type, with closed simply connected manifold fibre, including rationally hyperbolic fibres. Equality also holds when the fibre has finite-dimensional total rational homotopy, without a cohomological dimension bound on the base, and for suitable finite relative Frobenius models. Geometric applications include sphere, Grassmann, and projective bundles.
Related
BibTeX
@misc{parent2026poincaré,
title = {{Poincaré duality and module topological complexity}},
author = {Paul-Eugène Parent},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04153},
eprint = {2610.04153},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04153},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Poincaré duality and module topological complexity
AU - Paul-Eugène Parent
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.04153
UR - https://arxiv.org/abs/2610.04153
AB - Topological complexity, introduced by Farber, measures the number of continuous local motion-planning rules on a configuration space. For every simply connected space X of finite rational homotopy type whose rational cohomology is a finite-dimensional Poincaré duality algebra, we prove MTC(X)=TC_0(X)=TC_0^M(X), where TC_0(X)=TC(X_ℚ) and M denotes the monoidal invariant. No formality or ellipticity assumption is required. The result extends to higher topological complexity and gives product additivity under duality. For every map f between simply connected spaces of finite rational homotopy type whose rationalization admits a homotopy retraction, we establish the sharp bound Msecat(f_ℚ)≤secat(f_ℚ)≤relcat(f_ℚ)≤Msecat(f_ℚ)+1. In particular, MTC(X)≤TC_0(X)≤MTC(X)+1 holds without a duality assumption. The proofs use controlled Sullivan resolutions and an ideal-valued criterion for relative category. For sections of fibrations with Poincaré duality fibre, we prove that a derived self-intersection formula, vanishing of Thom multiplication on the augmentation ideal, and ideal-valued evaluation are equivalent. This criterion yields equality of the three rational sectional invariants for smooth sections of smooth bundles over simply connected smooth bases of finite rational homotopy type, with closed simply connected manifold fibre, including rationally hyperbolic fibres. Equality also holds when the fibre has finite-dimensional total rational homotopy, without a cohomological dimension bound on the base, and for suitable finite relative Frobenius models. Geometric applications include sphere, Grassmann, and projective bundles.
ER -