Results 1  10
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136
Elliptic spectra, the Witten genus and the theorem of the cube
 Invent. Math
, 1997
"... 2. More detailed results 7 2.1. The algebraic geometry of even periodic ring spectra 7 ..."
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Cited by 63 (16 self)
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2. More detailed results 7 2.1. The algebraic geometry of even periodic ring spectra 7
Morava Ktheories and localisation
 Mem. Amer. Math. Soc
, 1999
"... Abstract. We study the structure of the categories of K(n)local and E(n)local spectra, using the axiomatic framework developed in earlier work of the authors with John Palmieri. We classify localising and colocalising subcategories, and give characterisations of small, dualisable, and K(n)nilpoten ..."
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Cited by 63 (18 self)
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Abstract. We study the structure of the categories of K(n)local and E(n)local spectra, using the axiomatic framework developed in earlier work of the authors with John Palmieri. We classify localising and colocalising subcategories, and give characterisations of small, dualisable, and K(n)nilpotent spectra. We give a number of useful extensions to the theory of vn self maps of finite spectra, and to the theory of Landweber exactness. We show that certain rings of cohomology operations are left Noetherian, and deduce some powerful finiteness results. We study the Picard group of invertible K(n)local spectra, and the problem of grading homotopy groups over it. We prove (as announced by Hopkins and Gross) that the BrownComenetz dual of MnS lies in the Picard group. We give a detailed analysis of some examples when n =1 or 2, and a list of open problems.
Quadratic functions in geometry, topology,and mtheory
"... 2. Determinants, differential cocycles and statement of results 5 ..."
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Cited by 51 (5 self)
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2. Determinants, differential cocycles and statement of results 5
Generalized group characters and complex oriented cohomology theories
 J. Amer. Math. Soc
, 2000
"... 2. Complex oriented descent and equivariant bundles 561 3. Rational equivariant stable homotopy and Artin’s Theorem 564 4. Complex oriented Euler characteristics 568 ..."
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Cited by 42 (3 self)
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2. Complex oriented descent and equivariant bundles 561 3. Rational equivariant stable homotopy and Artin’s Theorem 564 4. Complex oriented Euler characteristics 568
Ideals in triangulated categories: Phantoms, ghosts and skeleta
 Adv. in Math
, 1998
"... ABSTRACT. We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties, and that if I has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral seq ..."
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Cited by 41 (5 self)
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ABSTRACT. We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties, and that if I has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence. We use this to study various ideals. In the stable homotopy category we examine phantom maps, skeletal phantom maps, superphantom maps, and ghosts. (A ghost is a map which induces the zero map of homotopy groups.) We show that ghosts lead to a stable analogue of the Lusternik–Schnirelmann category of a space, and we calculate this stable analogue for lowdimensional real projective spaces. We also give a relation between ghosts and the Hopf and Kervaire invariant problems. In the case of A ∞ modules over an A ∞ ring spectrum, the ghost spectral sequence is a universal coefficient spectral sequence. From the phantom projective class we derive a generalized Milnor sequence for filtered diagrams of finite spectra, and from this it follows that the group of phantom maps from X to Y can always be described as a lim1 ←− group. The last two sections focus
The BaumConnes and the FarrellJones conjectures in K and Ltheory
 Preprintreihe SFB 478 — Geometrische Strukturen in der Mathematik, Heft 324
, 2004
"... Summary. We give a survey of the meaning, status and applications of the BaumConnes Conjecture about the topological Ktheory of the reduced group C ∗algebra and the FarrellJones Conjecture about the algebraic K and Ltheory of the group ring of a (discrete) group G. Key words: K and Lgroups o ..."
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Cited by 32 (24 self)
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Summary. We give a survey of the meaning, status and applications of the BaumConnes Conjecture about the topological Ktheory of the reduced group C ∗algebra and the FarrellJones Conjecture about the algebraic K and Ltheory of the group ring of a (discrete) group G. Key words: K and Lgroups of group rings and group C ∗algebras, BaumConnes
Invertible spectra in the E(n)local stable homotopy category
 J. London Math. Soc
"... Suppose C is a category with a symmetric monoidal structure, which we will refer to as the smash product. Then the Picard category is the full subcategory of objects which have an inverse under the smash product in C, and the Picard group Pic(C) is the collection of isomorphism classes of such inver ..."
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Cited by 28 (7 self)
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Suppose C is a category with a symmetric monoidal structure, which we will refer to as the smash product. Then the Picard category is the full subcategory of objects which have an inverse under the smash product in C, and the Picard group Pic(C) is the collection of isomorphism classes of such invertible objects. The
Algebraic topology and modular forms
 Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), Higher Ed
, 2002
"... The problem of describing the homotopy groups of spheres has been fundamental to algebraic topology for around 80 years. There were periods when specific computations were important and periods when the emphasis favored theory. Many ..."
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Cited by 24 (2 self)
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The problem of describing the homotopy groups of spheres has been fundamental to algebraic topology for around 80 years. There were periods when specific computations were important and periods when the emphasis favored theory. Many
Products on MUmodules
 Trans. Amer. Math. Soc
, 1999
"... modules over highly structured ring spectra to give new constructions of MUmodules such as BP, K(n) and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over MU [ 1] ∗ that are concentrated ..."
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Cited by 23 (5 self)
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modules over highly structured ring spectra to give new constructions of MUmodules such as BP, K(n) and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over MU [ 1] ∗ that are concentrated in 2 degrees divisible by 4; this guarantees that various obstruction groups are trivial. We extend these results to the cases where 2 = 0 or the homotopy groups are allowed to be nonzero in all even degrees; in this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity, and that the obstructions to commutativity are given by a certain power operation; this was inspired by parallel results of Mironov in BaasSullivan theory. We use formal group theory to derive various formulae for this power operation, and deduce a number of results about realising 2local MU∗modules as MUmodules. 1.