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Moduli Spaces of Commutative Ring Spectra
, 2003
"... Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E#E is flat over E# . We wish to address the following question: given a commutative E# algebra A in E#Ecomodules, is there an E# ring spectrum X with E#X = A as comodule algebras? We will formulate this as ..."
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Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E#E is flat over E# . We wish to address the following question: given a commutative E# algebra A in E#Ecomodules, is there an E# ring spectrum X with E#X = A as comodule algebras? We will formulate this as a moduli problem, and give a way  suggested by work of Dwyer, Kan, and Stover  of dissecting the resulting moduli space as a tower with layers governed by appropriate AndreQuillen cohomology groups. A special case is A = E#E itself. The final section applies this to discuss the LubinTate or Morava spectra En .
(Pre)sheaves of Ring Spectra over the Moduli Stack of Formal Group Laws
, 2004
"... In the first part of this article, I will state a realization problem for diagrams of structured ring spectra, and in the second, I will discuss the moduli space which parametrizes the problem. ..."
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In the first part of this article, I will state a realization problem for diagrams of structured ring spectra, and in the second, I will discuss the moduli space which parametrizes the problem.
Moduli problems for structured ring spectra
 DANIEL DUGGER AND BROOKE
, 2005
"... In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞ring spectra. In that paper, we discussed the the HopkinsMiller theore ..."
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In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞ring spectra. In that paper, we discussed the the HopkinsMiller theorem on the LubinTate or Morava spectra En; in particular, we showed how to prove that the moduli space of all E ∞ ring spectra X so that (En)∗X ∼ = (En)∗En as commutative (En) ∗ algebras had the homotopy type of BG, where G was an appropriate variant of the Morava stabilizer group. This is but one point of view on these results, and the reader should also consult [3], [38], and [41], among others. A point worth reiterating is that the moduli problems here begin with algebra: we have a homology theory E ∗ and a commutative ring A in E∗E comodules and we wish to discuss the homotopy type of the space T M(A) of all E∞ring spectra so that E∗X ∼ = A. We do not, a priori, assume that T M(A) is nonempty, or even that there is a spectrum X so that E∗X ∼ = A as comodules.
Surjectivity for Hamiltonian Gspaces in Ktheory
 T rans. Amer. Math. Soc
"... Abstract. Let G be a compact connected Lie group, and (M, ω) a Hamiltonian Gspace with proper moment map µ. We give a surjectivity result which expresses the Ktheory of the symplectic quotient M /G in terms of the equivariant Ktheory of the original manifold M, under certain technical conditions ..."
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Abstract. Let G be a compact connected Lie group, and (M, ω) a Hamiltonian Gspace with proper moment map µ. We give a surjectivity result which expresses the Ktheory of the symplectic quotient M /G in terms of the equivariant Ktheory of the original manifold M, under certain technical conditions on µ. This result is a natural Ktheoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. The main technical tool is the Ktheoretic AtiyahBott lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian Gspaces. We discuss this lemma in detail and highlight the differences between the Ktheory and rational cohomology versions of this lemma. We also introduce a Ktheoretic version of equivariant formality and prove that when the fundamental group of G is torsionfree, every compact Hamiltonian Gspace is equivariantly formal. Under these conditions, the forgetful map K ∗ G (M) → K ∗ (M) is surjective, and thus every complex vector bundle admits a stable equivariant structure. Furthermore, by considering complex line bundles, we show that every integral cohomology class in H 2 (M;�) admits an equivariant extension in H 2 G (M;�). 1
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 ..."
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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 ..."
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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
KTheory, DBranes and RamondRamond Fields
, 2008
"... This thesis is dedicated to the study of Ktheoretical properties of Dbranes and RamondRamond fields. We construct abelian groups which define a homology theory on the category of CWcomplexes, and prove that this homology theory is equivalent to the bordism representation of KOhomology, the dua ..."
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This thesis is dedicated to the study of Ktheoretical properties of Dbranes and RamondRamond fields. We construct abelian groups which define a homology theory on the category of CWcomplexes, and prove that this homology theory is equivalent to the bordism representation of KOhomology, the dual theory to KOtheory. We construct an isomorphism between our geometric representation and the analytic representation of KOhomology, which induces a natural equivalence of homology functors. We apply this framework to describe mathematical properties of Dbranes in type I String theory. We investigate the gauge theory of RamondRamond fields arising from type II String theory defined on global orbifolds. We use the machinery of Bredon cohomology and