We study a topological version of the T -duality relation between pairs consisting of a principal U (1)-bundle equipped with a degree-three integral cohomology class. We describe the homotopy type of a classifying space for such pairs and show that it admits a selfmap which implements a T -duality transformation.We give a simple derivation of a T -duality isomorphism for certain twisted cohomology theories. We conclude with some explicit computations of twisted K-theory groups and discuss an example of iterated T -duality for higher-dimensional torus bundles. 1.1.1In this paper, we describe a new approach to topological T -duality for U (1)-principal bundles E → B (E is the background space time) equipped with degree-three cohomology classes h ∈ H 3 (E, Z) (the H-flux in the language of the physical literature). 1.1.2We first define a T -duality relation between such pairs using a Thom class on an associated S 3 -bundle. Then we introduce the functor B → P(B) which associates to each space the set of isomorphism classes of pairs. We construct a classifying space R of P and characterize its homotopy type. It admits a homotopy class of selfmaps T : R → R which implements a natural T -duality transformation P → P of order two. This transformation maps a class of pairsWe conclude in particular that our definition of topological T -duality essentially coincides with previous definitions, based on integration of cohomology classes along the fibers. 1.1.3 We describe an axiomatic framework for a twisted generalized cohomology theory h. We further introduce the condition of T -admissibility. Examples of T -admissible theories are 1 INTRODUCTION 3 the usual twisted de Rham cohomology and twisted K-theory. For a T -admissible generalized twisted cohomology theory h we prove a T -duality isomorphism between h(E, c) and h(Ê,ĉ),where (E, c) and (Ê,ĉ) are T -dual pairs. 1.1.4We compute a number of examples. Iterating the construction of T -dual pairs, we can define duals of certain higher dimensional torus bundles. We show that with our definition of duality the isomorphism type of the dual of a torus bundle, even if it exists, is not always uniquely determined. 1.1.5We thank the referees for their useful comments, in particular with respect to the presentation and the physical interpretation of our results. Description of the results 1.2.1In this paper we try to explain our understanding of the results of the recent paper [2] and parts of [3] and [10] (Sec. 4.1) by means of elementary algebraic topology. The notion of T -duality originated in string theory. Instead of providing an elaborate historical account of T -duality here we refer to the two papers above and the literature cited therein. In fact, the first paper which studies T -duality is in some sense [12]. We will explain the relation with the present paper later in this introduction. However, a few motivating words what this paper is about, and more importantly whatit is not about, are in order.T -duality first came up in physics in the following situation. The spac...
We provide an axiomatic framework for the study of smooth extensions of generalized cohomology theories. Our main results are about the uniqeness of smooth extensions, and the identification of the flat theory with the associated cohomology theory with R/Z-coefficients.In particular, we show that there is a unique smooth extension of K-theory and of MU-cobordism with a unique multiplication, and that the flat theory in these cases is naturally isomorphic to the homotopy theorist's version of the cohomology theory with R/Z-coefficients. For this we only require a small set of natural compatibility conditions.
In string theory, the concept of T-duality between two principal Tn-bundles E and Ê over the same base space B, together with cohomology classes h ∈ H3(E,ℤ) and ĥ ∈ H3(Ê,ℤ), has been introduced. One of the main virtues of T-duality is that h-twisted K-theory of E is isomorphic to ĥ-twisted K-theory of Ê. In this paper, a new, very topological concept of T-duality is introduced. We construct a classifying space for pairs as above with additional "dualizing data", with a forgetful map to the classifying space for pairs (also constructed in the paper). On the first classifying space, we have an involution which corresponds to passage to the dual pair, i.e. to each pair with dualizing data exists a well defined dual pair (with dualizing data). We show that a pair (E, h) can be lifted to a pair with dualizing data if and only if h belongs to the second step of the Leray–Serre filtration of E (i.e. not always), and that in general many different lifts exist, with topologically different dual bundles. We establish several properties of the T-dual pairs. In particular, we prove a T-duality isomorphism of degree -n for twisted K-theory.
We study equivariant coarse homology theories through an axiomatic framework. To this end we introduce the category of equivariant bornological coarse spaces and construct the universal equivariant coarse homology theory with values in the category of equivariant coarse motivic spectra.As examples of equivariant coarse homology theories we discuss equivariant coarse ordinary homology and equivariant coarse algebraic K-homology.Moreover, we discuss the cone functor, its relation with equivariant homology theories in equivariant topology, and assembly and forget-control maps. This is a preparation for applications in subsequent papers aiming at split-injectivity results for the Farrell-Jones assembly map.
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