Results 1  10
of
11
Modern foundations of stable homotopy theory. Handbook of Algebraic Topology, edited by
, 1995
"... 2. Smash products and twisted halfsmash products 11 ..."
Abstract

Cited by 22 (7 self)
 Add to MetaCart
2. Smash products and twisted halfsmash products 11
Homotopy Coherent Category Theory
, 1996
"... this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on: ..."
Abstract

Cited by 22 (6 self)
 Add to MetaCart
this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on:
Elliptic cohomology
 In preparation
"... This paper is an expository account of the relationship between elliptic cohomology and the emerging subject of derived algebraic geometry. We begin in §1 with an overview of the classical theory of elliptic cohomology. In §2 we review the theory of E∞ring spectra and introduce the language of deri ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
This paper is an expository account of the relationship between elliptic cohomology and the emerging subject of derived algebraic geometry. We begin in §1 with an overview of the classical theory of elliptic cohomology. In §2 we review the theory of E∞ring spectra and introduce the language of derived algebraic geometry. We apply this theory in §3, where we introduce the notion of an oriented group scheme and describe connection between oriented group schemes and equivariant cohomology theories. In §4 we sketch a proof of our main result, which relates the classical theory of elliptic cohomology to the classification of oriented elliptic curves. In §5 we discuss various applications of these ideas, many of which rely upon a special feature of elliptic cohomology which we call 2equivariance. The theory that we are going to describe lies at the intersection of homotopy theory and algebraic geometry. We have tried to make our exposition accessible to those who are not specialists in algebraic topology; however, we do assume the reader is familiar with the language of algebraic geometry, particularly with the theory of elliptic curves. In order to keep our account readable, we will gloss over many details, particularly where the use of higher category theory is required. A more comprehensive account of the material described here, with complete definitions and proofs, will be given in [21]. In carrying out the work described in this paper, I have benefitted from the ideas of many people. I
THOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS
"... Abstract. This is very much an account of work in progress. We sketch the construction of an Atiyah dual (in the category of Tspaces) for the free loopspace of a manifold; the main technical tool is a kind of Tits building for loop groups, discussed in detail in an appendix. Together with a new loc ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
Abstract. This is very much an account of work in progress. We sketch the construction of an Atiyah dual (in the category of Tspaces) for the free loopspace of a manifold; the main technical tool is a kind of Tits building for loop groups, discussed in detail in an appendix. Together with a new localization theorem for Tequivariant Ktheory, this yields a construction of the elliptic genus in the string topology framework of ChasSullivan, CohenJones, Dwyer, Klein, and others. We also show how the Tits building can be used to construct the dualizing spectrum of the loop group. Using a tentative notion of equivariant Ktheory for loop groups, we relate the equivariant Ktheory of the dualizing spectrum to recent work of Freed, Hopkins and Teleman.
Brown Representability And Flat Covers
, 1999
"... this paper is devoted to proving the main result. To this end we need to recall our assumptions on the triangulated category T : ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
this paper is devoted to proving the main result. To this end we need to recall our assumptions on the triangulated category T :
The Incommunicability of Content
 Mind
, 1966
"... 1. Setting up the foundations 3 2. The EilenbergSteenrod axioms 4 3. Stable and unstable homotopy groups 5 4. Spectral sequences and calculations in homology and homotopy 6 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
1. Setting up the foundations 3 2. The EilenbergSteenrod axioms 4 3. Stable and unstable homotopy groups 5 4. Spectral sequences and calculations in homology and homotopy 6
Stable Algebraic Topology, 19451966
 Mind
, 1966
"... this paper appeared four years before Milnor's discovery of exotic dierential structures on spheres [Mil56a]. For an embedding f , he went further and showed that the homotopy type of a tubular neighborhood of f is independent of the dierentiable structure on the ambient manifold. He then introduced ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
this paper appeared four years before Milnor's discovery of exotic dierential structures on spheres [Mil56a]. For an embedding f , he went further and showed that the homotopy type of a tubular neighborhood of f is independent of the dierentiable structure on the ambient manifold. He then introduced the notion of ber homotopy equivalence and proved that the ber homotopy type of the tangent bundle of a manifold is independent of its dierentiable structure. He observed that the 10 J. P. MAY
IDEMPOTENTS AND LANDWEBER EXACTNESS IN BRAVE NEW ALGEBRA
, 2001
"... We explain how idempotents in homotopy groups give rise to splittings of homotopy categories of modules over commutative Salgebras, and we observe that there are naturally occurring equivariant examples involving idempotents in Burnside rings. We then give a version of the Landweber exact functor ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
We explain how idempotents in homotopy groups give rise to splittings of homotopy categories of modules over commutative Salgebras, and we observe that there are naturally occurring equivariant examples involving idempotents in Burnside rings. We then give a version of the Landweber exact functor theorem that applies to MUmodules.
Structured Vector Bundles Define Differential KTheory
, 810
"... A equivalence relation, preserving the ChernWeil form, is defined between connections on a complex vector bundle. Bundles equipped with such an equivalence class are called Structured Bundles, and their isomorphism classes form an abelian semiring. By applying the Grothedieck construction one obta ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
A equivalence relation, preserving the ChernWeil form, is defined between connections on a complex vector bundle. Bundles equipped with such an equivalence class are called Structured Bundles, and their isomorphism classes form an abelian semiring. By applying the Grothedieck construction one obtains the ring ˆ K, elements of which, modulo a complex torus of dimension the sum of the odd Betti numbers of the base, are uniquely determined by the corresponding element of ordinary K and the ChernWeil form. This construction provides a simple model of differential Ktheory, c.f. HopkinsSinger (2005), as well as a useful codification of vector bundles with connection.
IDEMPOTENTS AND LANDWEBER EXACTNESS IN BRAVE NEW ALGEBRA
"... Abstract. We explain how idempotents in homotopy groups give rise to splittings of homotopy categories of modules over commutative Salgebras, and we observe that there are naturally occurring equivariant examples involving idempotents in Burnside rings. We then give a version of the Landweber exact ..."
Abstract
 Add to MetaCart
Abstract. We explain how idempotents in homotopy groups give rise to splittings of homotopy categories of modules over commutative Salgebras, and we observe that there are naturally occurring equivariant examples involving idempotents in Burnside rings. We then give a version of the Landweber exact functor theorem that applies to MUmodules. In 1997, not long after [6] was written, I gave an April Fool’s talk on how to prove that BP is an E ∞ ring spectrum or equivalently, in the language of [6], a commutative Salgebra. Unfortunately, the problem of whether or not BP is an E ∞ ring spectrum remains open. However, two interesting remarks emerged and will be presented here. One concerns splittings along idempotents and the other concerns the Landweber exact functor theorem. One of the nicest things in [6] is its one line proof that KO and KU are commutative Salgebras. This is an application of the following theorem [6, VIII.2.2], or rather the special case that follows. Theorem 1. Let R be a cell commutative Salgebra, A be a cell commutative R