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This paper examines Euler characteristics and characteristic classes in the motivic setting. We establish a motivic version of the Becker–Gottlieb transfer, generalizing a construction of Hoyois. Making calculations of the Euler characteristic of the scheme of maximal tori in a reductive group, we prove a generalized splitting principle for the reduction from
to the normalizer of a maximal torus (in characteristic zero). Ananyevskiy’s splitting principle reduces questions about characteristic classes of vector bundles in
-invertible theories to the case of rank two bundles. We refine the torus-normalizer splitting principle for
to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and then generalize this to develop a general calculus of characteristic classes with values in Witt cohomology.
We study the equivariant oriented cohomology ring
of partial flag varieties using the moment map approach. We define the right Hecke action on this cohomology ring, and then prove that the respective Bott–Samelson classes in
can be obtained by applying this action to the fundamental class of the identity point, hence generalizing previously known results of Chow groups by Brion, Knutson, Peterson, Tymoczko and others. Our main result concerns the equivariant oriented cohomology theory
corresponding to the 2-parameter Todd genus. We give a new interpretation of Deodhar’s parabolic Kazhdan–Lusztig basis, i.e., we realize it as some cohomology classes (the parabolic Kazhdan–Lusztig (KL) Schubert classes) in
. We make a positivity conjecture, and a conjecture about the relationship of such classes with smoothness of Schubert varieties. We also prove the latter in several special cases.
We establish class field theory for three-dimensional manifolds and knots. For this purpose, we formulate analogues of the multiplicative group, the idèle class group, and ray class groups in a cocycle-theoretic way. Following the arguments in abstract class field theory, we construct reciprocity maps and verify the existence theorems.
We develop a theory of
-module Thom spectra for a commutative symmetric ring spectrum
and we analyze their multiplicative properties. As an interesting source of examples, we show that
-algebra Thom spectra associated to the special unitary groups can be described in terms of quotient constructions on
. We apply the general theory to obtain a description of the
-based topological Hochschild homology associated to an
-algebra Thom spectrum.
We prove that the
-completed Brown–Peterson spectrum is a retract of a product of Morava
-theory spectra. As a consequence, we generalize results of Kashiwabara and of Ravenel, Wilson and Yagita from spaces to spectra and deduce that the notion of a good group is determined by Brown–Peterson cohomology. Furthermore, we show that rational factorizations of the Morava
-theory of certain finite groups hold integrally up to bounded torsion with height-independent exponent, thereby lifting these factorizations to the rationalized Brown–Peterson cohomology of such groups.
The actions, anomalies and quantization conditions allow the M2-brane and the M5-brane to support, in a natural way, structures beyond spin on their world-volumes. The main examples are twisted string structures. This also extends to twisted stringc structures which we introduce and relate to twisted string structures. The relation of the C-field to Chern–Simons theory suggests the use of the string cobordism category to describe the M2-brane.
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