arXiv subjects

Sebastian Chenery

Publications and source records attributed to Sebastian Chenery.

8 recordsLinked to original sources

arXiv2026

Local Inertness of Poincar\'{e} duality complexes

We prove that, under certain homological conditions, the attaching map of the top cell of a Poincar\'{e} duality complex is inert when localised away from a finite set of primes. This improves on a result of F\'elix and Tanr\'e in these cases. As an additional application of the methods, we give a loop space decomposition of simply-connected $6$-dimensional Poincar\'{e} duality complexes satisfying certain hypotheses. We also show that, under the hypotheses of the inertness theorem, the $(n-1)$-skeleton of an $n$-dimensional Poincar\'{e} duality complex satisfies the hyperbolic form of Moore's Conjecture after localising away from an explicit finite set of primes, and use this to obtain new examples of $p$-local maps between spheres that are not inert.

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arXiv2026

Gyration Stability for Products

A gyration is an operation on Poincar\'{e} Duality complexes that arises from a certain surgery on the product of a given complex $N$ and a sphere, parametrised by a chosen twisting. Of particular recent interest is the notion of gyration stability; that is, $N$ is gyration stable when all of its gyrations have the same homotopy type, regardless of the twisting used. We prove that a product $N\times M$ of two Poincar\'{e} Duality complexes is gyration stable when one of the product terms is itself gyration stable, and provide some examples of interest.

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arXiv2026

Pullbacks of Sphere Fibrations over Connected Sums

We prove conditions under which the total space of the pullback of a sphere fibration over a connected sum is homotopy equivalent to a connected sum with a gyration. Existing results of this type often depend on geometric methods. We develop new methods based only on homotopy theory, allowing for generalisations from manifolds to Poincar\'e Duality complexes and from integral settings to local ones. Several applications are given.

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arXiv2025

On Fico's Lemmata and the Homotopy Type of Certain Gyrations

We undertake to determine the homotopy type of gyrations of sphere products and of connected sums, thereby generalising results known in earlier literature as ''Fico's Lemmata'' which underpin gyrations in their original formulation from geometric topology. We provide applications arising from recasting these results into the modern homotopy theoretic setting.

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arXiv2024

Gyration Stability for Projective Planes

Gyrations are operations on manifolds that arise in geometric topology, where a manifold $M$ may exhibit distinct gyrations depending on the chosen twisting. For a given $M$, we ask a natural question: do all gyrations of $M$ share the same homotopy type regardless of the twisting? A manifold with this property is said to have gyration stability. Inspired by recent work by Duan, which demonstrated that the quaternionic projective plane is not gyration stable with respect to diffeomorphism, we explore this question for projective planes in general. We obtain a complete description of gyration stability for the complex, quaternionic, and octonionic projective planes up to homotopy.

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arXiv2023

Loop Space Decompositions of Connected Sums and Applications to the Vigué-Poirrier Conjecture

Recent work of Beben and Theriault on decomposing based loop spaces of highly connected Poincaré Duality complexes has yielded new methods for analysing the homotopy theory of manifolds. In this paper we will expand upon these methods, which we will then apply to give new examples supporting a long standing question of rational homotopy theory: the Vigué-Poirrier Conjecture.

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arXiv2023

A Homotopy Theoretic Analogue to a Theorem of Wall

It is a well-known result of C.T.C. Wall's that one may decompose a simply connected 6-manifold as a connected sum of two simpler manifolds. Recent work of Beben and Theriault on decomposing based loop spaces of highly connected Poincaré Duality complexes has yielded new methods for analysing the homotopy theory of manifolds. In this paper we will expand upon these methods, which we will then apply to prove a higher dimensional homotopy theoretic analogue to Wall's Theorem.

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arXiv2023

The Rational Homotopy Type of Homotopy Fibrations Over Connected Sums

We provide a simple condition on rational cohomology for the total space of a pullback fibration over a connected sum to have the rational homotopy type of a connected sum, after looping. This takes inspiration from recent work of Jeffrey and Selick, in which they study pullback fibrations of this type, but under stronger hypotheses compared to our result.

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