arXiv subjects

Paul-Eugène Parent

Publications and source records attributed to Paul-Eugène Parent.

5 recordsLinked to original sources

arXiv2026

Unbounded Nilpotency Gaps on Rationally Elliptic Manifolds

We construct simply connected, closed smooth manifolds whose rational homotopy types are elliptic and whose homology nilpotency exceeds rational category by arbitrarily large amounts. This separates short models from short quotients of minimal models. The construction starts from an explicit family of minimal Sullivan algebras with finite-dimensional cohomology. A finite quotient establishes Poincaré duality and gives an upper bound for homology nilpotency, while a closed orientation functional computes rational category. Two forced primitives yield the lower bound. An even regrading separates the relevant weights, extending this obstruction to every graded acyclic differential ideal. Rational surgery realizes the resulting models in their formal dimensions, with vanishing rational Pontryagin classes and arbitrarily large rational connectivity. The ratio of homology nilpotency to rational category tends to four thirds. Finally, under an external spherical stabilization statement, attaching a single cone can increase homology nilpotency by an arbitrarily large amount while leaving rational category unchanged.

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arXiv2026

Poincaré duality and module topological complexity

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 $\mathrm{MTC}(X)=\mathrm{TC}_0(X)=\mathrm{TC}_0^M(X)$, where $\mathrm{TC}_0(X)=\mathrm{TC}(X_{\mathbb Q})$ 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 $\mathrm{Msecat}(f_{\mathbb Q})\leq\mathrm{secat}(f_{\mathbb Q})\leq\mathrm{relcat}(f_{\mathbb Q})\leq\mathrm{Msecat}(f_{\mathbb Q})+1$. In particular, $\mathrm{MTC}(X)\leq\mathrm{TC}_0(X)\leq\mathrm{MTC}(X)+1$ 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.

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arXiv2026

Homology Nilpotency in Rational Homotopy Theory: Cell Attachments, Retractive Towers, Divergence, and Stabilization

Homology nilpotency of a simply connected minimal Sullivan algebra $M$ is the least integer $n$ for which $(M^+)^{n+1}$ is contained in an acyclic differential ideal. We develop lower- and upper-bound criteria, including methods for rational cell attachments, and construct an explicit finite-type minimal Sullivan algebra $M$ satisfying $nil_h(M)=3<Hnil(M)=4$. Complete strictly coordinated retractive towers identify the obstruction: they characterize homotopical nil-length while exposing the additional quotient defect measured by homology nilpotency. We then show that this rigid defect disappears after stabilization by degreewise finite wedges of simply connected rational spheres: $$ Hnil_s(X)=cat_o(X) $$ for every simply connected rational space $X$ of finite type.

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arXiv2026

Cone length and Lusternik-Schnirelmann category in rational homotopy

Lusternik-Schnirelmann category (LS-category) of a topological space is the least integer $n$ such that there is a covering of $X$ by $n+1$ open sets, each of them being contractible in $X$. The cone length is the minimum number of cofibations necessary to get a space in the homotopy type of $X$, starting from a suspension and attaching suspensions. The LS-category of a space is always less than or equal to its cone length. Moreover, these two invariants differ by at most one. In 1981, J.-M. Lemaire and F. Sigrist conjectured that they are always equal for rational spaces. This conjecture is clearly true for spaces of LS-category 1 and, in 1986, Y. Félix and J-C. Thomas verify it for spaces of LS-category 2. But, in 1999, the general conjecture is invalidated by N. Dupont who built a rational space of cone-length 4 and LS-category 3. In this work, we provide examples of rational spaces of cone-length $(k+1)$ and LS-category $k$ for any $k>2$.

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arXiv2005

A canonical enriched Adams-Hilton model for simplicial sets

For any 1-reduced simplicial set $K$ we define a canonical, coassociative coproduct on $\Om C(K)$, the cobar construction applied to the normalized, integral chains on $K$, such that any canonical quasi-isomorphism of chain algebras from $\Om C(K)$ to the normalized, integral chains on $GK$, the loop group of $K$, is a coalgebra map up to strong homotopy. Our proof relies on the operadic description of the category of chain coalgebras and of strongly homotopy coalgebra maps given in math.AT/0505559.

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