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A uniqueness theorem for stable homotopy theory
 Math. Z
, 2002
"... Roughly speaking, the stable homotopy category of algebraic topology is obtained from the ..."
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Cited by 14 (9 self)
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Roughly speaking, the stable homotopy category of algebraic topology is obtained from the
Homotopy theory of comodules over a Hopf algebroid
, 2003
"... Given a good homology theory E and a topological space X, E∗X is not just an E∗module but also a comodule over the Hopf algebroid (E∗, E∗E). We establish a framework for studying the homological algebra of comodules over a wellbehaved Hopf algebroid (A, Γ). That is, we construct ..."
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Cited by 13 (3 self)
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Given a good homology theory E and a topological space X, E∗X is not just an E∗module but also a comodule over the Hopf algebroid (E∗, E∗E). We establish a framework for studying the homological algebra of comodules over a wellbehaved Hopf algebroid (A, Γ). That is, we construct
Monoidal uniqueness of stable homotopy theory
 Adv. in Math. 160
, 2001
"... Abstract. We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. As an application we show that with an added assumption about underlyi ..."
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Cited by 11 (7 self)
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Abstract. We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. As an application we show that with an added assumption about underlying model structures Margolis ’ axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence. Also, the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (Smodules, Wspaces, orthogonal spectra, simplicial functors) with symmetric spectra. The equivalences of modules, algebras and commutative algebras in these categories are also considered. 1.
Realizing coalgebras over the Steenrod algebra
 Topology
"... (co)algebra K ∗ over the mod p Steenrod algebra as the (co)homology of a topological space, and for distinguishing between the phomotopy types of different realizations. The theories are expressed in terms of the Quillen cohomology of K∗. 1. ..."
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Cited by 3 (2 self)
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(co)algebra K ∗ over the mod p Steenrod algebra as the (co)homology of a topological space, and for distinguishing between the phomotopy types of different realizations. The theories are expressed in terms of the Quillen cohomology of K∗. 1.
On Freyd’s generating hypothesis
"... Abstract. We revisit Freyd’s generating hypothesis in stable homotopy theory. We derive new equivalent forms of the generating hypothesis and some new consequences of it. A surprising one is that I, the BrownComenetz dual of the sphere and the source of many counterexamples in stable homotopy, is t ..."
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Cited by 1 (1 self)
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Abstract. We revisit Freyd’s generating hypothesis in stable homotopy theory. We derive new equivalent forms of the generating hypothesis and some new consequences of it. A surprising one is that I, the BrownComenetz dual of the sphere and the source of many counterexamples in stable homotopy, is the cofiber of a self map of a wedge of spheres. We also show that a consequence of the generating hypothesis, that the homotopy of a finite spectrum that is not a wedge of spheres can never be finitely generated as a module over π∗S, is in fact true for many finite torsion spectra.
TORSION INVARIANTS FOR TRIANGULATED CATEGORIES
"... The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasiisomorphisms. Such examples are called ‘algebraic ’ because they have underlying abelian (or at least additive) categories. Stable homotopy theory ..."
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The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasiisomorphisms. Such examples are called ‘algebraic ’ because they have underlying abelian (or at least additive) categories. Stable homotopy theory produces examples of triangulated categories by quite different means, and in this context the underlying categories are usually very ‘nonadditive ’ before passing to homotopy classes of morphisms. We call such triangulated categories topological, compare Definition 3.1; this class includes the algebraic triangulated categories. The purpose of this paper is to explain some systematic differences between these two kinds of triangulated categories. There are certain properties – defined entirely in terms of the triangulated structure – which hold in all algebraic examples, but which can fail in general. These differences are all torsion phenomena, and rationally every topological triangulated category is algebraic (at least under mild size restrictions). Our main tool is a new numerical invariant, the norder of an object in a triangulated category, for n a natural number (see Definition 1.1). The norder is a nonnegative integer (or infinity), and an object Y has positive norder if and only if n · Y = 0; the norder can be thought of
Contents
, 2006
"... Abstract. We show how methods from Ktheory of operator algebras can be applied in a completely algebraic setting to define a bivariant, M∞stable, homotopyinvariant, excisive Ktheory of algebras over a fixed unital ground ring H, (A, B) ↦ → kk∗(A, B), which is universal in the sense that it maps ..."
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Abstract. We show how methods from Ktheory of operator algebras can be applied in a completely algebraic setting to define a bivariant, M∞stable, homotopyinvariant, excisive Ktheory of algebras over a fixed unital ground ring H, (A, B) ↦ → kk∗(A, B), which is universal in the sense that it maps uniquely to any other such theory. It turns out kk is related to C. Weibel’s homotopy algebraic Ktheory, KH. We prove that, if H is commutative and A is central as an Hbimodule, then kk∗(H, A) = KH∗(A).
Equivariance of generalized Chern characters
, 904
"... In this note some generalization of the Chern character is discussed from the chromatic point of view. We construct a multiplicative Gn+1equivariant natural transformation Θ from some height n + 1 cohomology theory E ∗ (−) to the height n cohomology theory K ∗ (−)b⊗FL, where K ∗ (−) is essentially ..."
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In this note some generalization of the Chern character is discussed from the chromatic point of view. We construct a multiplicative Gn+1equivariant natural transformation Θ from some height n + 1 cohomology theory E ∗ (−) to the height n cohomology theory K ∗ (−)b⊗FL, where K ∗ (−) is essentially the nth Morava Ktheory. As a corollary, it is shown that the Gnmodule K ∗ (X) can be recovered from the Gn+1module E ∗ (X). We also construct a lift of Θ to a natural transformation between characteristic zero cohomology theories. 1
THE STABLE SHAPE OF COMPACT SPACES WITH COUNTABLE COHOMOLOGY GROUPS
"... Abstract. Let X be a compact Hausdorff space and q ∈ Z be an integer such that the integral cohomology groups H n (X; Z) are countable for n < q and the stable cohomotopy groups πn s (X) of X are countable for n ≥ q. Then there exists a compact metrizable compact space Y with the same stable shape a ..."
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Abstract. Let X be a compact Hausdorff space and q ∈ Z be an integer such that the integral cohomology groups H n (X; Z) are countable for n < q and the stable cohomotopy groups πn s (X) of X are countable for n ≥ q. Then there exists a compact metrizable compact space Y with the same stable shape as X.