Donald Stanley

Algebraic models of Poincare embeddings

Let f : P curved right arrow W be an embedding of a compact polyhedron in a closed oriented manifold W, let T be a regular neighborhood of P in W and let C := (W) over bar be its complement. Then W is the homotopy push-out of a diagram C P. This homotopy push-out square is an example of what is called a Poincare embedding. We study how to construct algebraic models, in particular in the sense of Sullivan, of that homotopy push-out from a model of the map f. When the codimension is high enough this allows us to completely determine the rational homotopy type of the complement C similar or equal to Wf(P). Moreover we construct examples to show that our restriction on the codimension is sharp. Without restriction on the codimension we also give di ff erentiable modules models of Poincare embeddings and we deduce a refinement of the classical Lefschetz duality theorem, giving information on the algebra structure of the cohomology of the complement.

Fast searching games on graphs

Given a graph, suppose that intruders hide on vertices or along edges of the graph. The fast searching problem is to find the minimum number of searchers required to capture all the intruders satisfying the constraint that every edge is traversed exactly once and searchers are not allowed to jump. In this paper, we prove lower bounds on the fast search number. We present a linear time algorithm to compute the fast search number of Halin graphs and their extensions. We present a quadratic time algorithm to compute the fast search number of cubic graphs.

Lower Bounds on Fast Searching

Given a graph, suppose that intruders hide on vertices or along edges of the graph. The fast searching problem is to find the minimum number of searchers required to capture all intruders satisfying the constraint that every edge is traversed exactly once and searchers are not allowed to jump. In this paper, we prove lower bounds on the fast search number. We present a linear time algorithm to compute the fast search number of Halin graphs and their extensions. We present a quadratic time algorithm to compute the fast search number of cubic graphs.

Affine weakly regular tensor triangulated categories

We prove that the Balmer spectrum of a tensor triangulated category is homeomorphic to the Zariski spectrum of its graded central ring, provided the triangulated category is generated by its tensor unit and the graded central ring is noetherian and regular in a weak sense. There follows a classification of all thick subcategories, and the result extends to the compactly generated setting to yield a classification of all localizing subcategories as well as the analog of the telescope conjecture. This generalizes results of Shamir for commutative ring spectra.

Derived equivalences for hereditary Artin algebras

We study the role of the Serre functor in the theory of derived equivalences. Let A A be an abelian category and let (U,V) ( U , V ) be a t -structure on the bounded derived category D b A D b A with heart H H . We investigate when the natural embedding H→D b A H → D b A can be extended to a triangle equivalence D b H→D b A D b H → D b A . Our focus of study is the case where A A is the category of finite-dimensional modules over a finite-dimensional hereditary algebra. In this case, we prove that such an extension exists if and only if the t -structure is bounded and the aisle U U of the t -structure is closed under the Serre functor.

Polynomial functors in manifold calculus

Abstract Let M be a smooth manifold, and let O ( M ) be the poset of open subsets of M . Manifold calculus, due to Goodwillie and Weiss, is a calculus of functors suitable for studying contravariant functors (cofunctors) F : O ( M ) ⟶ Spaces from O ( M ) to the category of spaces. Weiss showed that polynomial cofunctors of degree ≤ k are determined by their values on O k ( M ) , where O k ( M ) is the full subposet of O ( M ) whose objects are open subsets diffeomorphic to the disjoint union of at most k balls. Afterwards Pryor showed that one can replace O k ( M ) by more general subposets and still recover the same notion of polynomial cofunctor. In this paper, we generalize these results to cofunctors from O ( M ) to any simplicial model category M . If F k ( M ) stands for the unordered configuration space of k points in M , we also show that the category of homogeneous cofunctors O ( M ) ⟶ M of degree k is weakly equivalent to the category of linear cofunctors O ( F k ( M ) ) ⟶ M provided that M has a zero object. Using a new approach, we also show that if M is a general model category and F : O k ( M ) ⟶ M is an isotopy cofunctor, then the homotopy right Kan extension of F along the inclusion O k ( M ) ↪ O ( M ) is also an isotopy cofunctor.

A classification of torsion classes in abelian categories

Abstract We give a classification of torsion classes (or nullity classes) in an abelian category by forming a spectrum of equivalence classes of premonoform objects. This is parallel to Kanda’s classification of Serre subcategories.

Implications of the Ganea condition

Suppose the spaces X and X × A have the same Lusternik- Schnirelmann category: cat(X × A) = cat(X). Then there is a strict inequality cat(X × (A ⋊ B)) < cat(X) + cat(A ⋊ B) for every space B, provided the connectivity of A is large enough (depending only on X). This is applied to give a partial verification of a conjecture of Iwase on the category of products of spaces with spheres. AMS Classification 55M30

The Cone Length of a Product of Co-H-Spaces and a Problem of Ganea

It is proved that the cone length or strong category of a product of two co-H-spaces is less than or equal to two. This yields the following positive solution to a problem of Ganea: Let α ∈ π2p(S) be an element of order p, p a prime ≥ 3, and let X(p) = S3 ∪α e2p+1. Then X(p) ×X(p) is the mapping cone of some map φ : Y → Z, where Z is a suspension.