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HZalgebra spectra are differential graded algebras
 Amer. Jour. Math
, 2004
"... Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZalgebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZalgebra spectra. We also construct Qu ..."
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Cited by 32 (10 self)
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Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZalgebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZalgebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that bascially any rational stable model category is Quillen equivalent to modules over a differential graded Qalgebra (with many objects). 1.
Multiplicative properties of Atiyah duality
 Homology Homotopy Appl
"... Let M n be a closed, connected nmanifold. Let M −τ denote the Thom spectrum of its stable normal bundle. A well known theorem of Atiyah states that M −τ is homotopy equivalent to the SpanierWhitehead dual of M with a disjoint basepoint, M+. This dual can be viewed as the function spectrum, F(M, S) ..."
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Cited by 12 (1 self)
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Let M n be a closed, connected nmanifold. Let M −τ denote the Thom spectrum of its stable normal bundle. A well known theorem of Atiyah states that M −τ is homotopy equivalent to the SpanierWhitehead dual of M with a disjoint basepoint, M+. This dual can be viewed as the function spectrum, F(M, S), where S is the sphere spectrum. F(M, S) has the structure of a commutative, symmetric ring spectrum in the sense of [7], [12] [9]. In this paper we prove that M −τ also has a natural, geometrically defined, structure of a commutative, symmetric ring spectrum, in such a way that the classical duality maps of Alexander, SpanierWhitehead, and Atiyah define an equivalence of symmetric ring spectra, α: M −τ → F(M,S). We discuss applications of this to Hochschild cohomology representations of the ChasSullivan loop product in the homology of the free loop space of M.
Topological Hochschild homology of Thom spectra which are . . .
, 2008
"... We identify the topological Hochschild homology (THH) of the Thom spectrum associated to an E ∞ classifying map X → BG, for G an appropriate group or monoid (e.g. U, O, and F). We deduce the comparison from the observation of McClure, Schwanzl, and Vogt that THH of a cofibrant commutative Salgebra ..."
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Cited by 8 (2 self)
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We identify the topological Hochschild homology (THH) of the Thom spectrum associated to an E ∞ classifying map X → BG, for G an appropriate group or monoid (e.g. U, O, and F). We deduce the comparison from the observation of McClure, Schwanzl, and Vogt that THH of a cofibrant commutative Salgebra (E ∞ ring spectrum) R can be described as an indexed colimit together with a verification that the LewisMay operadic Thom spectrum functor preserves indexed colimits. We prove a splitting result THH(Mf) ≃ Mf ∧BX+ which yields a convenient description of THH(MU). This splitting holds even when the classifying map f: X → BG is only a homotopy commutative A ∞ map, provided that the induced multiplication on Mf extends to an E ∞ ring structure; this permits us to recover Bokstedt’s calculation of THH(HZ).
Units of ring spectra and Thom spectra
"... Abstract. We review and extend the theory of Thom spectra and the associated obstruction theory for orientations. Specifically, we show that for an E ∞ ring spectrum A, the classical construction of gl1A, the spectrum of units, is the right adjoint of the functor To a map of spectra Σ ∞ + Ω ∞ : ho(c ..."
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Cited by 4 (1 self)
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Abstract. We review and extend the theory of Thom spectra and the associated obstruction theory for orientations. Specifically, we show that for an E ∞ ring spectrum A, the classical construction of gl1A, the spectrum of units, is the right adjoint of the functor To a map of spectra Σ ∞ + Ω ∞ : ho(connective spectra) → ho(E ∞ ring spectra). f: b → bgl1A, we associate an E ∞ Aalgebra Thom spectrum Mf, which admits an E ∞ Aalgebra map to R if and only if the composition b → bgl1A → bgl1R is null; the classical case developed by [MQRT77] arises when A is the sphere spectrum. We develop the analogous theory for A ∞ ring spectra. If A is an A ∞ ring spectrum, then to a map of spaces f: B → BGL1A we associate an Amodule Thom spectrum Mf, which admits an Rorientation if and only if
DIAGRAM SPACES, DIAGRAM SPECTRA, AND SPECTRA OF UNITS
, 908
"... Abstract. We compare the infinite loop spaces associated to symmetric spectra, orthogonal spectra, and EKMM Smodules. Each of these categories of structured spectra has a corresponding category of structured spaces that receives the infinite loop space functor Ω ∞. We prove that these models for sp ..."
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Abstract. We compare the infinite loop spaces associated to symmetric spectra, orthogonal spectra, and EKMM Smodules. Each of these categories of structured spectra has a corresponding category of structured spaces that receives the infinite loop space functor Ω ∞. We prove that these models for spaces are Quillen equivalent and that the infinite loop space functors Ω ∞ agree. This comparison is then used to show that two different constructions of the spectrum of units gl1R of a structured ring spectrum R agree. Contents
What precisely are E∞ ring spaces . . .
, 2009
"... E∞ ring spectra were defined in 1972, but the term has since acquired several alternative meanings. The same is true of several related terms. The new formulations are not always known to be equivalent to the old ones and even when they are, the notion of “equivalence ” needs discussion: Quillen eq ..."
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E∞ ring spectra were defined in 1972, but the term has since acquired several alternative meanings. The same is true of several related terms. The new formulations are not always known to be equivalent to the old ones and even when they are, the notion of “equivalence ” needs discussion: Quillen equivalent categories can be quite seriously inequivalent. Part of the confusion stems from a gap in the modern resurgence of interest in E ∞ structures. E∞ ring spaces were also defined in 1972 and have never been redefined. They were central to the early applications and they tie in implicitly to modern applications. We summarize the relationships between the old notions and various new ones, explaining what is and is not known. We take the opportunity to rework and modernize many of the early results. New proofs and perspectives are sprinkled throughout.
INTERPRETING THE BÖKSTEDT SMASH PRODUCT AS THE NORM
"... Abstract. This note compares two models of the equivariant homotopy type of the smash powers of a spectrum, namely the “Bökstedt smash product ” and the HillHopkinsRavenel norm. ..."
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Abstract. This note compares two models of the equivariant homotopy type of the smash powers of a spectrum, namely the “Bökstedt smash product ” and the HillHopkinsRavenel norm.
Homology, Homotopy and Applications, vol.6(1), 2004, pp.269–281 MULTIPLICATIVE PROPERTIES OF ATIYAH DUALITY
"... Let Mn be a closed, connected nmanifold. Let M−τ denote the Thom spectrum of its stable normal bundle. A well known theorem of Atiyah states that M−τ is homotopy equivalent to the SpanierWhitehead dual ofM with a disjoint basepoint, M+. This dual can be viewed as the function spectrum, F (M,S), ..."
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Let Mn be a closed, connected nmanifold. Let M−τ denote the Thom spectrum of its stable normal bundle. A well known theorem of Atiyah states that M−τ is homotopy equivalent to the SpanierWhitehead dual ofM with a disjoint basepoint, M+. This dual can be viewed as the function spectrum, F (M,S), where S is the sphere spectrum. F (M,S) has the structure of a commutative, symmetric ring spectrum in the sense of [7], [12] [9]. In this paper we prove that M−τ also has a natural, geometrically defined, structure of a commutative, symmetric ring spectrum, in such a way that the classical duality maps of Alexander, SpanierWhitehead, and Atiyah define an equivalence of symmetric ring spectra, α:M−τ → F (M,S). We discuss applications of this to Hochschild cohomology representations of the ChasSullivan loop product in the homology of the free loop space of M.