Don Stanley

University of Regina · Department of Mathematics and Statistics

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

We define the bigraded persistent homology modules and the bigraded barcodes of a finite pseudo-metric space X using the ordinary and double homology of the moment-angle complex associated with the Vietoris-Rips filtration of X. We prove the stability theorem for the bigraded persistent double homology modules and barcodes.

A persistence module with $m$ discrete parameters is a diagram of vector spaces indexed by the poset $\mathbb{N}^m$. If we are only interested in the large scale behavior of such a diagram, then we can consider two diagrams equivalent if they agree outside of a ``negligeable'' region. In the $2$-dimensional case, we classify the indecomposable diag...

We use weights on objects in an abelian category to define what we call a path metric. We introduce three special classes of weight: those compatible with short exact sequences; those induced by their path metric; and those which bound their path metric. We prove that these conditions are in fact equivalent, and call such weights exact. As a specia...

We put a cochain complex structure ${CH}^*(\mathcal Z_K)$ on the cohomology of a moment-angle complex $\mathcal Z_K$ and call the resulting cohomology the double cohomology, ${HH}^*(\mathcal Z_K)$. We give three equivalent definitions for the differential, and compute ${HH}^*(\mathcal Z_K)$ for a family of simplicial complexes containing clique com...

Let $A$ be the quotient of a graded polynomial ring $\mathbb{Z}[x_1,\cdots,x_m]\otimes\Lambda[y_1,\cdots,y_n]$ by an ideal generated by monomials with leading coefficients 1. Then we constructed a space~$X_A$ such that $A$ is isomorphic to $H^*(X_A)$ modulo torsion elements.

We examine the rationality conjecture raised in [1] which states that (a) the formal power series ∑r≥1TCr+1(X)⋅xr represents a rational function of x with a single pole of order 2 at x=1 and (b) the leading coefficient of the pole equals cat(X). Here X is a finite CW-complex and for r≥2 the symbol TCr(X) denotes its r-th sequential topological comp...

We examine the rationality conjecture which states that (a) the formal power series $\sum_{r\ge 1} \tc_{r+1}(X)\cdot x^r$ represents a rational function of $x$ with a single pole of order 2 at $x=1$ and (b) the leading coefficient of the pole equals $\cat(X)$. Here $X$ is a finite CW-complex and for $r\ge 2$ the symbol $\tc_r(X)$ denotes its $r$-th...

We study aisles in the derived category of a hereditary abelian category. Given an aisle, we associate a sequence of subcategories of the abelian category by considering the different homologies of the aisle. We then obtain a sequence, called a narrow sequence. We then prove that a narrow sequence in a hereditary abelian category consists of a nond...

For a smooth manifold M, we define a topological space X(k,M), and show that polynomial functors O(M)--> C of degree <= k from the poset of open subsets of M to a simplicial model category can be classified be a version of linear functors from O(X(k,M)) to C.

In persistence theory and practice, measuring distances between modules is central. The Wasserstein distances are the standard family of L^p distances for persistence modules. They are defined in a combinatorial way for discrete invariants called persistence diagrams that are defined for certain persistence modules. We give an algebraic formulation...

For any object A in a simplicial model category C, we construct a topological space \^A which classifies linear functors whose value on an open ball is equivalent to A. More precisely for a manifold M, and O(M) its poset category of open sets, weak equivalence classes of such functors O(M) ---> C are shown to be in bijection with homotopy classes o...

Torus manifolds are topological generalization of smooth projective toric manifolds. We compute the rational cohomology ring of a class of smooth locally standard torus manifolds whose orbit space is a connected sum of simple polytopes.

A theorem of Lambrechts and Stanley is used to find the rational cohomology of the complement of an embedding S⁴ⁿ⁻¹ → S²ⁿ ×Sm as a module and demonstrate that it is not necessarily determined by the map induced on cohomology by the embedding, nor is it a trivial extension. This demonstrates that the theorem is an improvement on the classical Lefsch...

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)--> Top from O(M) to the category of spaces. Weiss showed that polynomial cofunctors of degree <= k are determined by their values on O...

Let M be a smooth manifold, and let O(M) be the poset of open subsets of M. Let C be a category that has a zero object and all small limits. A homogeneous functor (in the sense of manifold calculus) of degree k from O(M) to C is called very good if it sends isotopy equivalences to isomorphisms. In this paper we show that the category VGHF of such f...

We study the role of the Serre functor in the theory of derived equivalences. Let be an abelian category and let be a t-structure on the bounded derived category with heart . We investigate when the natural embedding can be extended to a triangle equivalence . Our focus of study is the case where is the category of finite-dimensional modules over a...

For a commutative Noetherian ring $R$ with finite Krull dimension, we study the nullity classes in $D^c_{fg}(R)$, the full triangulated subcategory $D^c_{fg}(R)$ of the derived category $D(R)$ consisting of objects which can be represented by cofibrant objects with each degree finitely generated. In the light of perversity functions over the prime...

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, even, and regular in a weak sense. There follows a classification of all thick subcategories, and the re...

Let W be a compact simply connected triangulated manifold with boundary and $K \subset W$ be a subpolyhedron. We construct an algebraic model of the rational homotopy type of the complement $W \setminus K$ out of a model of the map of pairs $(K, K \cap \partial W) \to (W,\partial W)$ under some high codimension hypothesis. We deduce the rational ho...

We prove that a large class of Poincar\'e duality pairs admit rational models (in the sense of Sullivan) of a particularly nice form associated to some Poincar\'e duality CDGA. These models have applications in particular to the construction of rational models of configuration spaces in compact manifolds with boundary.

We study the role of the Serre functor in connection with derived equivalences. Let $\Lambda$ be a finite-dimensional hereditary algebra. Our main result is that the heart of a $t$-structure on $D^b \operatorname{mod} \Lambda$ is derived equivalent to $\operatorname{mod} \Lambda$ if and only if the $t$-structure is bounded and the aisle of the $t$-...

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 se...

For R a commutative Noetherian ring, wide and Serre subcategories of finitely generated R-modules have been classified by their support. This paper studies general torsion classes and introduces narrow subcategories. These are closed under fewer operations than wide and Serre subcategories, but still for finitely generated R-modules both narrow sub...

Let the circle act in a Hamiltonian fashion on a compact symplectic manifold $(M, \omega)$. Assume that the fixed point set $M^{S^1}$ has exactly two components, $X$ and $Y$, and that $\dim(X) + \dim(Y) +2 = \dim(M)$. We first show that $X$, $Y$ and $M$ are simply connected. Then we show that, under some minor dimension restrictions, up to $S^1$-eq...

A 7-dimensional CW-complex having Lusternik–Schnirelmann category equal to 2 is constructed. Using a divisibility phenomenon for Hopf invariants, it is proved that the Cartesian product of the constructed complex with a sphere of sufficiently large dimension also has category 2. This space hence constitutes the minimum dimensional known counterexam...

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...

Let M be a simply-connected closed manifold and consider the (ordered) configuration space of $k$ points in M, F(M,k). In this paper we construct a commutative differential graded algebra which is a potential candidate for a model of the rational homotopy type of F(M,k). We prove that our model it is at least a Sigma_k-equivariant differential grad...

We show that H-spaces with finitely generated cohomology, as an algebra or as an algebra over the Steenrod algebra, have homotopy exponents at all primes. This provides a positive answer to a question of Stanley.

Suppose that f:V->W is an embedding of closed oriented manifolds whose normal bundle has the structure of a complex vector bundle. It is well known in both complex and symplectic geometry that one can then construct a manifold W' which is the blow-up of W along V. Assume that dim(W)>2.dim(V)+2 and that H^1(f) is injective. We construct an algebraic...

We prove that every commutative differential graded algebra whose cohomology is a simply-connected Poincaré duality algebra is quasi-isomorphic to one whose underlying algebra is simply-connected and satisfies Poincaré duality in the same dimension. This has applications in rational homotopy, giving Poincaré duality at the cochain level, which is o...

We construct an invariant of t-structures on the derived category of a Noetherian ring. This invariant is complete when restricting to the category of quasi-coherent complexes, and also gives a classification of nullity classes with the same restriction. On the full derived category of $\mathbb Z$ we show that the class of distinct t-structures do...

We give an example of two homotopic embeddings j0, j 1: V → W of manifolds with isomorphic complex normal bundles but such that the blow-ups of W along j0 and along j1 have different rational homotopy types.

We give an example of two homotopic embeddings j 0 , j 1 : V ↪ W j_0,j_1\colon V\hookrightarrow W of manifolds with isomorphic complex normal bundles but such that the blow-ups of W W along j 0 j_0 and along j 1 j_1 have different rational homotopy types.

Let f: P-->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:=closure(W-T) be its complement. Then W is the homotopy push-out of a diagram C<--dT-->P. This homotopy push-out square is an example of what is called a Poincare embedding. We study how to construct algebraic mo...

Suppose the spaces X and X cross A have the same Lusternik-Schnirelmann category: cat(X cross A)= cat(X). Then there is a strict inequality cat(X cross (A halfsmash B)) < cat (X) + cat(A halfsmash 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 conjectur...

The aim of this paper is twofold. On the one hand, we show that the kernel $\overline{C(A)}$ of the Bousfield periodization functor PA is cellularly generated by a space B, i.e., we construct a space B such that the smallest closed class C(B) containing B is exactly $\overline{C(A)}$ . On the other hand, we show that the partial order (Spaces, ≫) i...

For any collection of spaces A, we investigate two non-negative integer homotopy invariants of maps: LA(f), the A-cone length of f, and ℒA(f), the A-category of f. When A is the collection of all spaces, these are the cone length and category of f, respectively, both of which have been studied previously. The following results are obtained: (1) For...

For any collection of spaces A, we investigate two non-negative integer homotopy invariants of maps: l_A(f), the A-cone length of f, and L_A(f), the A-category of f. When A is the collection of all spaces, these are the cone length and category of f, respectively, both of which have been studied previously. The following results have been obtained:...

We prove that the rational homotopy type of the configuration space of two points in a 2-connected closed manifold depends only on the rational homotopy type of that manifold and we give a model in the sense of Sullivan of that configuration space. We also study the formality of configuration spaces.

We say that a nite CW-complex X embeds up to homotopy in a sphere Sn+1 if there exists a subpolyhedron K Sn+1 having the homotopy type of X. The main result of this paper is a sucient condition for the existence of such a homotopy embedding in a given codimension when X is a simply-connected two-cone (a two-cone is the homotopy cobre of a map betwe...

Let X be a finite CW-complex. We show that the image of the homotopy groups of X under suspension have an exponent at every prime. As a corollary we recover Long's result that finite H-spaces have exponents at all primes. We show that the stable homotopy groups of X have an exponent at p if and only if X is rationally equivalent to a point. Thi...

We say that a finite CW-complex X X embeds up to homotopy in a sphere S n + 1 S^{n+1} if there exists a subpolyhedron K ⊂ S n + 1 K\subset S^{n+1} having the homotopy type of X X . The main result of this paper is a sufficient condition for the existence of such a homotopy embedding in a given codimension when X X is a simply-connected two-cone (a...

We give conditions which determine if cat of a map go up when extending over a cofibre. We apply this to reprove a result of Roitberg giving an example of a CW complex Z such that cat(Z) = 2 but every skeleton of Z is of category 1. We also find conditions when cat(f × g) < cat(f) + cat(g). We apply our result to show that under suitable conditions...

In this paper we consider what happens when Adams self maps are modified by adding certain unstable maps. The unstable maps which are added are trivial after a single suspension. We can choose the modification so that the maps are still K-theory equivalences but the loops on the map are no longer K-theory equivalences. As a corollary we note that t...

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 ε π2p(S3) be an element of order p, p a prime ≥ 3, and let X(p) = S3∪αe2p+1. Then X(p) × X(p) is the mapping cone of some map ϕ : Y → Z where Z is a suspension. 2...

We give homological conditions which determine sectional cate- gory, secat, for rational spherical brations. In the odd dimensional case the secat is the least power of the Euler class which is trivial. In the even dimen- sional case secat is one when a certain homology class in twice the dimension of the sphere is 1 times a square. Otherwise secat...

We construct a series of spaces, X(n), for each n>0, such that cat(X(n))=n and cl(X(n))=n+1. We show that the Hopf invariants determine whether the category of a space goes up when attaching a cell of top dimension. We give a new proof of counterexamples to a conjecture of Ganea. Also we introduce some techniques for manipulating cone decomposition...

We give an example of a rational map, f , such that catf =c atf

Let the circle act in a Hamiltonian fashion on a compact symplectic manifold $(M, \omega)$. Assume that the fixed point set $M^{S^1}$ has exactly two components, $X$ and $Y$. We first show that, if $\dim(X) + \dim(Y) +2 = \dim(M)$, then $M$ is simply connected. Using this result and the results in \cite{LT} on the integral cohomology ring and the...