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Stephen Theriault

Publications and source records attributed to Stephen Theriault.

At least 19 recordsLinked to original sources
arXiv2026

Local Inertness of Poincaré duality complexes

We prove that, under certain homological conditions, the attaching map of the top cell of a Poincaré duality complex is inert when localised away from a finite set of primes. This improves on a result of Félix and Tanré in these cases. As an additional application of the methods, we give a loop space decomposition of simply-connected $6$-dimensional Poincaré duality complexes satisfying certain hypotheses. We also show that, under the hypotheses of the inertness theorem, the $(n-1)$-skeleton of an $n$-dimensional Poincaré 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

Iterated Whitehead products in the homotopy groups of polyhedral products

We study structure within the homotopy groups of the Davis-Januszkiewicz space DJ(K) associated with a simplicial complex K. The inclusion of each vertex in K induces a map from the two-sphere into DJ(K). These maps generate a quasi-Lie subalgebra QL(K) via the Whitehead product and a Pi-subalgebra S(K) via the Whitehead product and composition. We describe the quasi-Lie subalgebra QL(K), and show that the Pi-subalgebra S(K) coincides with the whole of the homotopy groups of DJ(K) if and only if K is a flag complex. Extensions to more general polyhedral products are also considered.

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arXiv2026

Realizing orders in rational sphere product algebras with three generators

The realization problem asks which algebras can be realized as the cohomology of spaces. We study this problem in the context of the orders in a graded rational exterior algebra on three generators. An order is a subring whose underlying additive group is a lattice. We give conditions for when such an order is realizable, and in particular show that in the simply-connected case any order is realizable if the generators of the exterior algebra are of odd degree.

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arXiv2026

Moment-angle manifolds associated to neighbourly triangulations of spheres

We show that a moment-angle manifold associated to a neighbourly triangulation of an odd dimensional sphere is homotopy equivalent to a connected sum of products of two spheres, resolving a problem of Buchstaber and Panov. The methods are entirely homotopy theoretic, allowing for an extension to a corresponding result in the case of generalized moment-angle manifolds.

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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 Poincaré Duality complexes and from integral settings to local ones. Several applications are given.

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arXiv2025

Loop spaces of $n$-dimensional Poincaré duality complexes whose $(n-1)$-skeleton is a co-$H$-space

Under certain hypotheses, we prove a loop space decomposition for simply-connected Poincaré Duality complexes of dimension $n$ whose $(n-1)$-skeleton is a co-$H$-space. This unifies many known decompositions obtained in different contexts and establishes many new families of examples. As consequences, we show that such a looped Poincaré Duality complex retracts off the loops of its $(n-1)$-skeleton and describe its homology as a one-relator algebra.

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arXiv2025

Homotopy of blow ups after looping

The homotopy theory of the blow up construction in algebraic and symplectic geometry is investigated via two approaches. The first approach introduces and develops fibrewise surgery theory, for which the fibrewise framing is characterized by the homotopy groups of a certain gauge group. This is used to obtain a homotopy decomposition of the based loop space on a blow up that holds $p$-locally for all but finitely many primes $p$ and holds rationally. The second approach is purely homotopy theoretic and obtains a homotopy decomposition of the based loop space on a blow up that holds integrally provided a certain condition is satisfied by an associated homotopy action. As applications, we obtain $p$-local homotopy decompositions of the based loop space of a focal genus $2$ manifold, improve an earlier result of the authors on the homotopy of manifolds stabilized by projective spaces, and obtain for blow-ups a refinement of the rational dichotomy.

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arXiv2025

Polyhedral products associated to pseudomanifolds

We study the homotopy theory of polyhedral products associated to a combinatorial generalisation of manifolds known as pseudomanifolds. As special cases, we show that loop spaces of moment-angle manifolds associated to triangulations of $S^2$ and $S^3$ decompose as a product of spheres and loops on spheres.

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arXiv2025

Weighted polyhedral products and Steenrod's problem

We construct a weighted version of polyhedral products and compute its cohomology in special cases. This is applied to resolve Steenrod's cohomology realization problem in a case related to products of spheres.

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arXiv2025

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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arXiv2025

Stabilization of Poincaré duality complexes and homotopy gyrations

Stabilization of manifolds by a product of spheres or a projective space is important in geometry. There has been considerable recent work that studies the homotopy theory of stabilization for connected manifolds. This paper generalizes that work by developing new methods that allow for a generalization to stabilization of Poincaré Duality complexes. This includes the systematic study of a homotopy theoretic generalization of a gyration, obtained from a type of surgery in the manifold case. In particular, for a fixed Poincaré Duality complex, a criterion is given for the possible homotopy types of gyrations and shows there are only finitely many.

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arXiv2025

Calculating Higher Digraph Homotopy Groups

We give the first tractable and systematic examples of nontrivial higher digraph homotopy groups. To do this we define relative digraph homotopy groups and show these satisfy a long exact sequence analogous to the relative homotopy groups of spaces. We then define digraph suspension and Hurewicz homomorphisms and show they commute with each other. The existence of nontrivial digraph homotopy groups then reduces to the existence of corresponding groups in the degree 1 path homology of digraphs.

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arXiv2024

Homotopy theoretic properties of open books

We study the homotopy groups of open books in terms of those of their pages and bindings. Under homotopy theoretic conditions on the monodromy we prove an integral decomposition result for the based loop space on an open book, and under more relaxed conditions prove a rational loop space decomposition. The latter case allows for a rational dichotomy theorem for open books, as an extension of the classical dichotomy in rational homotopy theory. As a direct application, we show that for Milnor's open book decomposition of an odd sphere with monodromy of finite order the induced action of the monodromy on the homology groups of its page cannot be nilpotent.

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arXiv2024

Top cell attachment for a Poincare Duality complex

Let M be a simply-connected closed Poincare Duality complex of dimension n. Then M is obtained by attaching a cell of highest dimension to its (n-1)-skeleton M'. Conditions are given for when the skeletal inclusion i:M' --> M has the property that the based loops on i has a right homotopy inverse. This is an integral version of the rational statement that such a right homotopy inverse always exists provided the rational cohomology of M is not generated by a single element. New methods are developed in order to do the integral case. These lead to p-local versions and recover the full rational statement. Families for which the integral statement holds include moment-angle manifolds and quasi-toric manifolds.

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arXiv2023

Homotopy of manifolds stabilized by projective spaces

We study the homotopy of the connected sum of a manifold with a projective space, viewed as a typical way to stabilize manifolds. In particular, we show a loop homotopy decomposition of a manifold after stabilization by a projective space, and provide concrete examples. To do this, we trace the effect in homotopy theory of surgery on certain product manifolds by showing a loop homotopy decomposition after localization away from the order of the image of the classical $J$-homomorphism.

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arXiv2022

The suspension of a 4-manifold and its applications

Let $M$ be a smooth, orientable, closed, connected $4$-manifold and suppose that $H_1(M;\mathbb{Z})$ is finitely generated and has no $2$-torsion. We give a homotopy decomposition of the suspension of $M$ in terms of spheres, Moore spaces and $Σ\mathbb{C}P^{2}$. This is used to calculate any reduced generalized cohomology theory of $M$ as a group and to determine the homotopy types of certain current groups and gauge groups.

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arXiv2022

The Homotopy Types of $SU(4)$-Gauge Groups

Let $\mathcal{G}_k$ be the gauge group of the principal $SU(4)$-bundle over $S^4$ with second Chern class $k$ and let $p$ be a prime. We show that there is a rational or $p$-local homotopy equivalence $Ω\mathcal{G}_k\simeqΩ\mathcal{G}_{k'}$ if and only if $(60,k)=(60,k')$.

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