6,486 research outputs found
Topological Hochschild homology of Thom spectra and the free loop space
We describe the topological Hochschild homology of ring spectra that arise as
Thom spectra for loop maps f: X->BF, where BF denotes the classifying space for
stable spherical fibrations. To do this, we consider symmetric monoidal models
of the category of spaces over BF and corresponding strong symmetric monoidal
Thom spectrum functors. Our main result identifies the topological Hochschild
homology as the Thom spectrum of a certain stable bundle over the free loop
space L(BX). This leads to explicit calculations of the topological Hochschild
homology for a large class of ring spectra, including all of the classical
cobordism spectra MO, MSO, MU, etc., and the Eilenberg-Mac Lane spectra HZ/p
and HZ.Comment: 58 page
Alcove path and Nichols-Woronowicz model of the equivariant -theory of generalized flag varieties
Fomin and Kirillov initiated a line of research into the realization of the
cohomology and -theory of generalized flag varieties as commutative
subalgebras of certain noncommutative algebras. This approach has several
advantages, which we discuss. This paper contains the most comprehensive result
in a series of papers related to the mentioned line of research. More
precisely, we give a model for the -equivariant -theory of a generalized
flag variety in terms of a certain braided Hopf algebra called the
Nichols-Woronowicz algebra. Our model is based on the Chevalley-type
multiplication formula for due to the first author and Postnikov;
this formula is stated using certain operators defined in terms of so-called
alcove paths (and the corresponding affine Weyl group). Our model is derived
using a type-independent and concise approach
Rational motivic path spaces and Kim's relative unipotent section conjecture
We initiate a study of path spaces in the nascent context of "motivic dga's",
under development in doctoral work by Gabriella Guzman. This enables us to
reconstruct the unipotent fundamental group of a pointed scheme from the
associated augmented motivic dga, and provides us with a factorization of Kim's
relative unipotent section conjecture into several smaller conjectures with a
homotopical flavor. Based on a conversation with Joseph Ayoub, we prove that
the path spaces of the punctured projective line over a number field are
concentrated in degree zero with respect to Levine's t-structure for mixed Tate
motives. This constitutes a step in the direction of Kim's conjecture.Comment: Minor corrections, details added, and major improvements to
exposition throughout. 52 page
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