25,188 research outputs found
Twists of K-theory and TMF
We explore an approach to twisted generalized cohomology from the point of
view of stable homotopy theory and quasicategory theory provided by
arXiv:0810.4535. We explain the relationship to the twisted K-theory provided
by Fredholm bundles. We show how our approach allows us to twist elliptic
cohomology by degree four classes, and more generally by maps to the four-stage
Postnikov system BO. We also discuss Poincare duality and umkehr maps in
this setting
Orientability of Fredholm families and topological degree
We construct a degree theory for oriented Fredholm mappings of index zero between open subsets of Banach spaces and between Banach manifolds. Our approach is based on the orientation of Fredholm mappings: it does not use Fredholm structures on the domain and target spaces. We provide a computable formula for the change in degree through an admissible homotopy that is necessary for applications to global bifurcation. The notion of orientation enables us to establish rather precise relationships between our degree and many other degree theories for particular classes of Fredholm maps, including the Elworthy-Tromba degree, which have appeared in the literature in a seemingly unrelated manner
Controlled surgery and -homology
This paper presents an alternative approach to controlled surgery
obstructions. The obstruction for a degree one normal map with control map to complete controlled
surgery is an element , where are topological manifolds of dimension . Our proof uses
essentially the geometrically defined -spectrum as described by
Nicas (going back to Quinn) and some well known homotopy theory. We also
outline the construction of the algebraically defined obstruction, and we
explicitly describe the assembly map in terms of forms in the case . Finally, we
explicitly determine the canonical map
Topological Hochschild cohomology and generalized Morita equivalence
We explore two constructions in homotopy category with algebraic precursors
in the theory of noncommutative rings and homological algebra, namely the
Hochschild cohomology of ring spectra and Morita theory. The present paper
provides an extension of the algebraic theory to include the case when is
not necessarily a progenerator. Our approach is complementary to recent work of
Dwyer and Greenlees and of Schwede and Shipley. A central notion of
noncommutative ring theory related to Morita equivalence is that of central
separable or Azumaya algebras. For such an Azumaya algebra A, its Hochschild
cohomology HH^*(A,A) is concentrated in degree 0 and is equal to the center of
A. We introduce a notion of topological Azumaya algebra and show that in the
case when the ground S-algebra R is an Eilenberg-Mac Lane spectrum of a
commutative ring this notion specializes to classical Azumaya algebras. A
canonical example of a topological Azumaya R-algebra is the endomorphism
R-algebra F_R(M,M) of a finite cell R-module. We show that the spectrum of mod
2 topological K-theory KU/2 is a nontrivial topological Azumaya algebra over
the 2-adic completion of the K-theory spectrum widehat{KU}_2. This leads to the
determination of THH(KU/2,KU/2), the topological Hochschild cohomology of KU/2.
As far as we know this is the first calculation of THH(A,A) for a
noncommutative S-algebra A.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-29.abs.htm
Central extensions of mapping class groups from characteristic classes
Tangential structures on smooth manifolds, and the extension of mapping class groups they induce, admit a natural formulation in terms of higher (stacky) differential geometry. This is the literal translation of a classical construction in differential topology to a sophisticated language, but it has the advantage of emphasizing how the whole construction naturally emerges from the basic idea of working in slice categories. We characterize, for every higher smooth stack equipped with tangential structure, the induced higher group extension of the geometric realization of its higher automor- phism stack. We show that when restricted to smooth manifolds equipped with higher degree topological structures, this produces higher extensions of homotopy types of diffeomorphism groups. Passing to the groups of connected components, we obtain abelian extensions of mapping class groups and we derive sufficient conditions for these being central. We show as a special case that this provides an elegant re-construction of Segal’s approach to -extensions of mapping class groups of surfaces that provides the anomaly cancellation of the modular functor in Chern-Simons theory. Our construction generalizes Segal’s approach to higher central extensions of mapping class groups of higher dimensional manifolds with higher tangential structures, expected to provide the analogous anomaly cancellation for higher dimensional TQFTs
On the Adams Spectral Sequence for R-modules
We discuss the Adams Spectral Sequence for R-modules based on commutative
localized regular quotient ring spectra over a commutative S-algebra R in the
sense of Elmendorf, Kriz, Mandell, May and Strickland. The formulation of this
spectral sequence is similar to the classical case and the calculation of its
E_2-term involves the cohomology of certain `brave new Hopf algebroids' E^R_*E.
In working out the details we resurrect Adams' original approach to Universal
Coefficient Spectral Sequences for modules over an R ring spectrum.
We show that the Adams Spectral Sequence for S_R based on a commutative
localized regular quotient R ring spectrum E=R/I[X^{-1}] converges to the
homotopy of the E-nilpotent completion pi_*hat{L}^R_ES_R=R_*[X^{-1}]^hat_{I_*}.
We also show that when the generating regular sequence of I_* is finite,
hatL^R_ES_R is equivalent to L^R_ES_R, the Bousfield localization of S_R with
respect to E-theory. The spectral sequence here collapses at its E_2-term but
it does not have a vanishing line because of the presence of polynomial
generators of positive cohomological degree. Thus only one of Bousfield's two
standard convergence criteria applies here even though we have this
equivalence. The details involve the construction of an I-adic tower R/I <--
R/I^2 <-- ... <-- R/I^s <-- R/I^{s+1} <-- ... whose homotopy limit is
hatL^R_ES_R. We describe some examples for the motivating case R=MU.Comment: Published 7 April 2001 by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-9.abs.html . Erratum added 9
May 200
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