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Quasisymmetric functions from a topological point of view arXiv: math.AT/0605743
"... Abstract. It is wellknown that the homology of the classifying space of the unitary group is isomorphic to the ring of symmetric functions Symm. We offer the cohomology of the space ΩΣCP ∞ as a topological model for the ring of quasisymmetric functions QSymm. We exploit standard results from topolo ..."
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Abstract. It is wellknown that the homology of the classifying space of the unitary group is isomorphic to the ring of symmetric functions Symm. We offer the cohomology of the space ΩΣCP ∞ as a topological model for the ring of quasisymmetric functions QSymm. We exploit standard results from topology to shed light on some of the algebraic properties of QSymm. In particular, we reprove the Ditters conjecture. We investigate a product on ΩΣCP ∞ that gives rise to an algebraic structure which generalizes the Witt vector structure in the cohomology of BU. The canonical Thom spectrum over ΩΣCP ∞ is highly noncommutative and we study some of its features, including the homology of its topological Hochschild homology spectrum.
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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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
WHAT ARE E∞
, 903
"... Abstract. Infinite loop space theory, both additive and multiplicative, arose largely from two basic motivations. One was to solve calculational questions in geometric topology. The other was to better understand algebraic Ktheory. The Adams conjecture is intrinsic to the first motivation, and Quil ..."
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Abstract. Infinite loop space theory, both additive and multiplicative, arose largely from two basic motivations. One was to solve calculational questions in geometric topology. The other was to better understand algebraic Ktheory. The Adams conjecture is intrinsic to the first motivation, and Quillen’s proof of that led directly to his original, calculationally accessible, definition of algebraic Ktheory. In turn, the infinite loop understanding of algebraic Ktheory feeds back into the calculational questions in geometric topology. For example, use of infinite loop space theory leads to a method for determining the characteristic classes for topological bundles (at odd primes) in terms of the cohomology of finite groups. We explain just a little about how all that works, focusing on the central role played by E ∞ ring spaces.
Thom Spectra that Are Symmetric Spectra
 DOCUMENTA MATH.
, 2009
"... We analyze the functorial and multiplicative properties of the Thom spectrum functor in the setting of symmetric spectra and we establish the relevant homotopy invariance. ..."
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We analyze the functorial and multiplicative properties of the Thom spectrum functor in the setting of symmetric spectra and we establish the relevant homotopy invariance.
HIGHER TOPOLOGICAL HOCHSCHILD HOMOLOGY OF THOM SPECTRA
, 811
"... Abstract. In this paper we analyze the higher topological Hochschild homology of commutative Thom Salgebras. This includes the case of the classical cobordism spectra MO, MSO, MU, etc. We consider the homotopy orbits of the torus action on iterated topological Hochschild homology and we describe th ..."
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Abstract. In this paper we analyze the higher topological Hochschild homology of commutative Thom Salgebras. This includes the case of the classical cobordism spectra MO, MSO, MU, etc. We consider the homotopy orbits of the torus action on iterated topological Hochschild homology and we describe the relationship to topological AndréQuillen homology. 1. introduction The simplicial model of the Hochschild homology complex associated to a commutative ring T and a Tmodule M was extended by Loday [18], [19], to a functor
Commutative Salgebras of prime characteristics and applications to unoriented bordism
, 2012
"... The notion of highly structured ring spectra of prime characteristic is made precise and is studied via the universal examples S//p for prime numbers p. These can be realised as Thom spectra, and therefore relate to other Thom spectra such as the unoriented bordism spectrum MO. We compute the Hochsc ..."
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The notion of highly structured ring spectra of prime characteristic is made precise and is studied via the universal examples S//p for prime numbers p. These can be realised as Thom spectra, and therefore relate to other Thom spectra such as the unoriented bordism spectrum MO. We compute the Hochschild and AndréQuillen invariants of the S//p. Among other applications, we show that S//p is not a commutative algebra over the EilenbergMac Lane spectrum HFp, although the converse is clearly true. 2010 MSC: 55P43 (primary), 13A35, 55P20, 55P42 (secondary).
A ∞ Structures on Thom Spectra A dissertation presented by
, 2009
"... Let R be an E ∞ ring spectrum. Given a map f: X → BGL1(R), we can construct a Thom spectrum X f. If f is a loop map, then there is an A ∞ ring structure on the Thom spectrum. I will consider various examples of these Thom spectra and construct A∞ structures on them. For these A ∞ structures, I will ..."
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Let R be an E ∞ ring spectrum. Given a map f: X → BGL1(R), we can construct a Thom spectrum X f. If f is a loop map, then there is an A ∞ ring structure on the Thom spectrum. I will consider various examples of these Thom spectra and construct A∞ structures on them. For these A ∞ structures, I will write down a convenient description for Topological Hochschild Homology and compute it in a few examples. Contents Abstract......................................... Acknowledgments.................................... iii