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MODULI SPACES AND FORMAL OPERADS
, 2004
"... Abstract. Let Mg,l be the moduli space of stable algebraic curves of genus g with l marked points. With the operations which relate the different moduli spaces identifying marked points, the family (Mg,l)g,l is a modular operad of projective smooth DeligneMumford stacks, M. In this paper we prove t ..."
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Abstract. Let Mg,l be the moduli space of stable algebraic curves of genus g with l marked points. With the operations which relate the different moduli spaces identifying marked points, the family (Mg,l)g,l is a modular operad of projective smooth DeligneMumford stacks, M. In this paper we prove that the modular operad of singular chains C∗(M; Q) is formal; so it is weakly equivalent to the modular operad of its homology H∗(M; Q). As a consequence, the “up to homotopy ” algebras of these two operads are the same. To obtain this result we prove a formality theorem for operads analogous to DeligneGriffithsMorganSullivan formality theorem, the existence of minimal models of modular operads, and a characterization of formality for operads which shows that formality is independent of the ground field. 1.
Algebraic cycles and motivic generic iterated integrals. arXiv:math.NT/0506370
"... Abstract. Following [GGL], we will give a combinatorial framework for motivic study of iterated integrals on the affine line. We will show that under a certain genericity condition these combinatorial objects yield to elements in the motivic Hopf algebra constructed in [BK]. It will be shown that th ..."
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Abstract. Following [GGL], we will give a combinatorial framework for motivic study of iterated integrals on the affine line. We will show that under a certain genericity condition these combinatorial objects yield to elements in the motivic Hopf algebra constructed in [BK]. It will be shown that the Hodge realization of these elements coincides with the Hodge structure induced from the fundamental torsor of path of punctured affine line.
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