Spanier-Whitehead Duality Research Articles

A geometric representative for the fundamental class in KK-duality of Smale spaces

A fundamental ingredient in the noncommutative geometry program is the notion of KK-duality, often called K-theoretic Poincaré duality, that generalises Spanier-Whitehead duality. In this paper we construct a θ-summable Fredholm module that represents the fundamental class in KK-duality between the stable and unstable Ruelle algebras of a Smale space. To find such a representative, we construct dynamical partitions of unity on the Smale space with highly controlled Lipschitz constants. This requires a generalisation of Bowen's Markov partitions. Along with an aperiodic point-sampling technique we produce a noncommutative analogue of Whitney's embedding theorem, leading to the Fredholm module.

The reciprocal Kirchberg algebras

For two unital Kirchberg algebras with finitely generated K-groups, we introduce a property, called reciprocality, which is proved to be closely related to the homotopy theory of Kirchberg algebras. We show the Spanier–Whitehead duality for bundles of separable nuclear UCT C*-algebras with finitely generated K-groups and conclude that two reciprocal Kirchberg algebras share the same structure of their bundles.

External Spanier–Whitehead duality and homology representation theorems for diagram spaces

We construct a Spanier-Whitehead type duality functor relating finite $\mathcal{C}$-spectra to finite $\mathcal{C}^{\mathrm{op}}$-spectra and prove that every $\mathcal{C}$-homology theory is given by taking the homotopy groups of a balanced smash product with a fixed $\mathcal{C}^{\mathrm{op}}$-spectrum. We use this to construct Chern characters for certain rational $\mathcal{C}$-homology theories.

Spanier–Whitehead duality for topological coHochschild homology

AbstractIn this work, we compute the topological coHochschild homology (coTHH) of interesting coalgebras such as the Steenrod algebra spectrum. For this, we start by extending the Hess–Shipley definition of coTHH to ‐categories, following the Nikolaus–Scholze approach to topological Hochschild homology (THH). Furthermore, we prove that coTHH of what we call quasi‐proper coalgebras can be obtained from THH via Spanier–Whitehead duality which provides further insight into coTHH and its relationship to THH.

Dualizing spheres for compact p-adic analytic groups and duality in chromatic homotopy

The primary goal of this paper is to study Spanier–Whitehead duality in the K(n)-local category. One of the key players in the K(n)-local category is the Lubin–Tate spectrum \(E_n\), whose homotopy groups classify deformations of a formal group law of height n, in the implicit characteristic p. It is known that \(E_n\) is self-dual up to a shift; however, that does not fully take into account the action of the automorphism group \({\mathbb {G}}_n\) of the formal group in question. In this paper we find that the \({\mathbb {G}}_n\)-equivariant dual of \(E_n\) is in fact \(E_n\) twisted by a sphere with a non-trivial (when \(n>1\)) action by \({\mathbb {G}}_n\). This sphere is a dualizing module for the group \({\mathbb {G}}_n\), and we construct and study such an object \(I_{{\mathcal {G}}}\) for any compact p-adic analytic group \({\mathcal {G}}\). If we restrict the action of \({\mathcal {G}}\) on \(I_{{\mathcal {G}}}\) to certain type of small subgroups, we identify \(I_{{\mathcal {G}}}\) with a specific representation sphere coming from the Lie algebra of \({\mathcal {G}}\). This is done by a classification of p-complete sphere spectra with an action by an elementary abelian p-group in terms of characteristic classes, and then a specific comparison of the characteristic classes in question. The setup makes the theory quite accessible for computations, as we demonstrate in the later sections of this paper, determining the K(n)-local Spanier–Whitehead duals of \(E_n^{hH}\) for select choices of p and n and finite subgroups H of \({\mathbb {G}}_n\).

Spanier–Whitehead duality in the $K(2)$-local category at $p=2$

The fixed point spectra of Morava E-theory $E_n$ under the action of finite subgroups of the Morava stabilizer group $\mathbb{G}_n$ and their K(n)-local Spanier--Whitehead duals can be used to approximate the K(n)-local sphere in certain cases. For any finite subgroup F of the height 2 Morava stabilizer group at p=2 we prove that the K(2)-local Spanier--Whitehead dual of the spectrum $E_2^{hF}$ is $\Sigma^{44}E_2^{hF}$. These results are analogous to the known results at height 2 and p=3. The main computational tool we use is the topological duality resolution spectral sequence for the spectrum $E_2^{h\mathbb{S}_2^1}$ at p=2.

Groups with Spanier–Whitehead duality

Building on work by Kasparov, we study the notion of Spanier-Whitehead K-duality for a discrete group. It is defined as duality in the KK-category between two C*-algebras which are naturally attached to the group, namely the reduced group C*-algebra and the crossed product for the group action on the universal example for proper actions. We compare this notion to the Baum-Connes conjecture by constructing duality classes based on two methods: the standard "gamma element" technique, and the more recent approach via cycles with property gamma. As a result of our analysis, we prove Spanier-Whitehead duality for a large class of groups, including Bieberbach's space groups, groups acting on trees, and lattices in Lorentz groups.

Spanier–Whitehead K-duality for C∗-algebras

Classical Spanier–Whitehead duality was introduced for the stable homotopy category of finite CW complexes. Here we provide a comprehensive treatment of a noncommutative version, termed Spanier–Whitehead [Formula: see text]-duality, which is defined on the category of [Formula: see text]-algebras whose [Formula: see text]-theory is finitely generated and that satisfy the UCT, with morphisms the [Formula: see text]-groups. We explore what happens when these assumptions are relaxed in various ways. In particular, we consider the relationship between Paschke duality and Spanier–Whitehead [Formula: see text]-duality.

Duality and small functors

The homotopy theory of small functors is a useful tool for studying various questions in homotopy theory. In this paper, we develop the homotopy theory of small functors from spectra to spectra, and study its interplay with Spanier-Whitehead duality and enriched representability in the dual category of spectra. We note that the Spanier-Whitehead duality functor $D\colon \mathrm{Sp}\rightarrow \mathrm{Sp}^{\mathrm{op}}$ factors through the category of small functors from spectra to spectra and construct a new model structure on the category of small functors, which is Quillen equivalent to $\mathrm{Sp}^{\mathrm{op}}$. In this new framework for the Spanier-Whitehead duality, $\mathrm{Sp}$ and $\mathrm{Sp}^{\mathrm{op}}$ are full subcategories of the category of small functors and dualization becomes just a fibrant replacement in our new model structure.

On the Generating Hypothesis in Noncommutative Stable Homotopy

Freyd's Generating Hypothesis is an important problem in topology with deep structural consequences for finite stable homotopy. Due to its complexity some recent work has examined analogous questions in various other triangulated categories. In this short note we analyze the question in noncommutative stable homotopy, which is a canonical generalization of finite stable homotopy. Along the way we also discuss Spanier-Whitehead duality in this extended setup.

K-theoretic duality for hyperbolic dynamical systems

Abstract The K-theoretic analog of Spanier–Whitehead duality for noncommutative C * {C^{*}} -algebras is shown to hold for the Ruelle algebras associated to irreducible Smale spaces. This had previously been proved only for shifts of finite type. Implications of this result as well as relations to the Baum–Connes conjecture and other topics are also considered.

Gross-Hopkins duality and the Gorenstein condition

AbstractGross and Hopkins have proved that in chromatic stable homotopy, Spanier-Whitehead duality nearly coincides with Brown-Comenetz duality. We give a conceptual interpretation of this phenomenon in terms of a Gorenstein condition [8] for maps of ring spectra.

Modules over motivic cohomology

For fields of characteristic zero, we show that the homotopy category of modules over the motivic ring spectrum representing motivic cohomology is equivalent to Voevodsky's big category of motives. The proof makes use of some highly structured models for motivic stable homotopy theory, motivic Spanier–Whitehead duality, the homotopy theories of motivic functors and of motivic spaces with transfers as introduced from ground up in this paper. Working with rational coefficients, we extend the equivalence for fields of characteristic zero to all perfect fields by employing the techniques of alterations and homotopy purity in motivic homotopy theory.

A duality between string topology and the fusion product in equivariant K-theory

Let $G$ be a compact Lie group. Let $G \to E \to M$ be a principal $G$-bundle over a closed manifold $M$, and $G \to Ad(E) \to M$ its adjoint bundle. In this paper we describe a new Frobenius algebra structure on $h_*(Ad(E)),$ where $h_*$ is an appropriate generalized homology theory. Recall that a Frobenius algebra has both a product and a coproduct. The product in this new Frobenius algebra is induced by the string topology product. In particular, the product can be defined when $G$ is any topological group and in the case that $E$ is contractible it is precisely the Chas-Sullivan string product on $H_*(LM).$ We will show that the coproduct is induced by completing the untwisted Freed-Hopkins-Teleman fusion product. Indeed, when $M$ is replaced by $BG$ and $h_*$ is $K$-theory the coproduct is the completion of the Freed-Hopkins-Teleman fusion structure. We will then show that this duality between the string and fusion products is realized by a Spanier-Whitehead duality between certain Thom spectra of virtual bundles over $Ad(E)$.

Dualité de Spanier–Whitehead en géométrie algébrique

In this Note, we study finitely presented objects in the stable A 1 -homotopy category introduced by Morel and Voevodsky and we use a theorem by Voevodsky and Ayoub on the four functors to get Spanier–Whitehead duality in A 1 -homotopy theory. To cite this article: J. Riou, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Duality and Pro-Spectra

Cofiltered diagrams of spectra, also called pro-spectra, have arisen in diverse areas, and to date have been treated in an ad hoc manner. The purpose of this paper is to systematically develop a homotopy theory of pro-spectra and to study its relation to the usual homotopy theory of spectra, as a foundation for future applications. The surprising result we find is that our homotopy theory of pro-spectra is Quillen equivalent to the opposite of the homotopy theory of spectra. This provides a convenient duality theory for all spectra, extending the classical notion of Spanier-Whitehead duality which works well only for finite spectra. Roughly speaking, the new duality functor takes a spectrum to the cofiltered diagram of the Spanier-Whitehead duals of its finite subcomplexes. In the other direction, the duality functor takes a cofiltered diagram of spectra to the filtered colimit of the Spanier-Whitehead duals of the spectra in the diagram. We prove the equivalence of homotopy theories by showing that both are equivalent to the category of ind-spectra (filtered diagrams of spectra). To construct our new homotopy theories, we prove a general existence theorem for colocalization model structures generalizing known results for cofibrantly generated model categories.

Generalized stable shape and duality

In this paper we formulate and prove a generalization of the Spanier–Whitehead duality theorem in the generalized stable shape category and discuss its consequences.

Gross–Hopkins duality

We give a new and simpler proof of a result of Hopkins and Gross relating Brown-Comenetz duality to Spanier-Whitehead duality in the K(n)-local stable homotopy category.

Duality in Waldhausen Categories

We develop a theory of Spanier-Whitehead duality in categories with cofibrations and weak equivalences (Waldhausen categories, for short). This includes L-theory, the involution on K-theory introduced by [Vogell, W.: The involution in the algebraic K-theory of spaces. Proc. of 1983 Rutgers Conf. on Alg. Topology. Springer Lect. Notes in Math. 1126, pp. 277–317] in a special case, and a map Ξ relating L-theory to the Tate spectrum of ℤ/2 acting on K-theory. The map Ξ is a distillation of the long exact Rothenberg sequences [Shaneson, J.: Wall's surgery obstruction groups for G × ℤ. Ann. of Math. 90 (1969), 296–334], [Ranicki, A.: Algebraic L-theory I. Foundations. Proc. Lond. Math. Soc. 27 (1973), 101–125], [Ranicki, A.: Exact sequences in the algebraic theory of surgery. Mathematical Notes, Princeton Univ. Press, Princeton, New Jersey 1981], including analogs involving higher K-groups. It goes back to [Weiss, M., Williams, B.: Automorphisms of manifolds and algebraic K-theory, Part II. J. Pure and Appl. Algebra 62 (1988), 47–107] in special cases. Among the examples covered here, but not in [Weiss, M., Williams, B.: Automorphisms of manifolds and algebraic K-theory, Part II. J. Pure and Appl. Algebra 62 (1988), 47–107], are categories of retractive spaces where the notion of weak equivalence involves control.

K-Theoretic Duality for Shifts of Finite Type

C*-algebras generalizing Cuntz-Krieger algebras can be associated to hyperbolic homeomorphisms of compact metric spaces. They satisfy a non-commutative form of Spanier-Whitehead duality with respect to K-theory. We prove this for the case of subshifts of finite type. The special feature of the present situation is that the constructions are all done on the full Fock space and are very explicit, while the general theorem requires much more abstract machinery.