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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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Cited by 24 (2 self)
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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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Cited by 13 (4 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
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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Cited by 3 (2 self)
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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.
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).
UNITS OF RING SPECTRA, ORIENTATIONS, AND THOM SPECTRA VIA RIGID INFINITE LOOP SPACE THEORY
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MATH. SCAND. 103 (2008), 208–242 QUASISYMMETRIC FUNCTIONS FROM A TOPOLOGICAL POINT OF VIEW
"... 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 CP ∞ as a topological model for the ring of quasisymmetric functions QSymm. We exploit standard results from topology to shed l ..."
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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 CP ∞ 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 CP ∞ that gives rise to an algebraic structure which generalizes the Witt vector structure in the cohomology of BU. The canonical Thom spectrum over CP ∞ is highly noncommutative and we study some of its features, including the homology of its topological Hochschild homology spectrum.