5,840 research outputs found

    Homotopy theory of bundles with fiber matrix algebra

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    In the present paper we consider a special class of locally trivial bundles with fiber a matrix algebra. On the set of such bundles over a finite CWCW-complex we define a relevant equivalence relation. The obtained stable theory gives us a geometric description of the H-space structure \BSU_\otimes on \BSU related to the tensor product of virtual \SU-bundles of virtual dimension 1.Comment: This is a version of the paper published as a preprint of Max Planck Institute for Mathematics. Several misprints are corrected. 24 page

    On KK-theory automorphisms related to bundles of finite order

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    In the present paper we describe the action of (not necessarily line) bundles of finite order on the KK-functor in terms of classifying spaces. This description might provide with an approach for more general twistings in KK-theory than ones related to the action of the Picard group.Comment: 18 page

    Formal groups over Hopf algebras

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    In this paper we study some generalization of the notion of a formal group over ring, which may be called a formal group over Hopf algebra (FGoHA). The first example of FGoHA was found under the study of cobordism's ring of some HH-space Gr^\hat{Gr}. The results, which are represented in this paper, show that some constructions of the theory of formal group may be generalized to FGoHA. For example, if F(xβŠ—1,1βŠ—x)∈(HβŠ—^RH)[[xβŠ—1,1βŠ—x]]{\frak F}(x\otimes 1,1\otimes x) \in (H{\mathop{\hat{\otimes}}\limits_R}H)[[x\otimes 1,1\otimes x]] is a FGoHA over a Hopf algebra (H,ΞΌ,Ξ½,Ξ”,Ο΅,S)(H,\mu,\nu, \Delta,\epsilon, S) over a ring RR without torsion, then there exists a logarithm, i.e. the formal series g(x)∈HQ[[x]]{\frak g}(x)\in H_\mathbb{Q}[[x]] such that (Ξ”g)(F(xβŠ—1,1βŠ—x))=c+g(x)βŠ—1+1βŠ—g(x),(\Delta {\frak g})({\frak F}(x\otimes 1,1\otimes x))= {\frak c}+{\frak g}(x)\otimes 1+1\otimes {\frak g}(x), where {\frak c}\in H_\mathbb{Q}{\mathop{\hat{\otimes}}\limits_{R_ \mathbb{Q}}}H_\mathbb{Q}, (\id \otimes \epsilon){\frak c}=0=(\epsilon \otimes \id){\frak c} and (\id \otimes \Delta){\frak c}+1\otimes {\frak c}-(\Delta \otimes \id){\frak c}-{\frak c}\otimes 1=0 (recall that the last condition means that c{\frak c} is a cocycle in the cobar complex of the Hopf algebra HQH_\mathbb{\mathbb{Q}}). On the other hand, FGoHA have series of new properties. For example, the convolution on a Hopf algebra allows us to get new FGoHA from given.Comment: 21page

    Logarithms of formal groups over Hopf algebras

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    The aim of this paper is to prove the following result. For any commutative formal group F(xβŠ—1,1βŠ—x),{\frak F}(x\otimes 1,1\otimes x), which is considered as a formal group over HQ,H_\mathbb{Q}, there exists a homomorphism to a formal group of the form c+xβŠ—1+1βŠ—x,{\frak c}+x\otimes 1+1\otimes x, where c∈HQβŠ—^RQHQ\frak c\in H_\mathbb{Q}{\mathop{\hat{\otimes}} \limits_{R_\mathbb{Q}}}H_\mathbb{Q} such that (\id \otimes \epsilon){\frak c}=0= (\epsilon \otimes \id){\frak c}.Comment: 5 page

    A generalization of the topological Brauer group

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    In the present paper we study some homotopy invariants which can be defined by means of bundles with fiber a matrix algebra. We also introduce some generalization of the Brauer group in the topological context and show that any its element can be represented as a locally trivial bundle with a group of invertible operators in a Hilbert space as the structure group. Finally, we discuss its possible applications in the twisted KK-theory.Comment: 34 pages. v5: The part concerning the generalized Brauer group has been completely rewritten. An application to twisted KK-theory is adde

    Topological obstructions to embedding of a matrix algebra bundle into a trivial one

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    In the present paper we describe topological obstructions to embedding of a (complex) matrix algebra bundle into a trivial one under some additional arithmetic condition on their dimensions. We explain a relation between this problem and some principal bundles with structure groupoid. Finally, we briefly discuss a relation of our results to the twisted K-theory.Comment: v.14: 29 pages, corrections and additions in Section

    Supplement to the paper "Floating bundles and their applications"

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    This paper is the supplement to the section 2 of the paper "Floating bundles and their applications" (math.AT/0102054). Below we construct the denumerable set of extensions of the formal group of geometric cobordisms F(xβŠ—1,1βŠ—x)F(x\otimes 1,1\otimes x) by the Hopf algebra H=Ξ©Uβˆ—(Gr).H=\Omega_U^*(Gr).Comment: 4 pages, xypi

    A bordism theory related to matrix Grassmannians

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    In the present paper we study a bordism theory related to pairs (M, ξ),(M,\, \xi), where MM is a closed smooth oriented manifold with a stably trivial normal bundle and ΞΎ\xi is a virtual \SU-bundle of virtual dimension 1 over MM. The main result is the calculation of the corresponding ring modulo torsion and the explicit description of its generators.Comment: 10 page

    Floating bundles and their applications

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    The aim of section 1 is to define the homotopic functor to category of Abelian groups, connected with the special classes of bundles with fiber matrix algebra or projective space. The aim of section 2 is to define some generalization of the notion of formal group. More precisely, we consider the analog of formal groups with coefficients belonging to a Hopf algebra. We also study some example of a formal group over a Hopf algebra, which generalizes the formal group of geometric cobordisms.Comment: 19 pages, xypi

    Theories of bundles with additional homotopy conditions

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    In the present paper we study bundles equipped with extra homotopy conditions, in particular so-called simplicial nn-bundles. It is shown that (under some condition) the classifying space of 1-bundles is the double coset space of some finite dimensional Lie group. We also establish some relation between our bundles and C*-algebras.Comment: 22 pages; v3: a new material in subsections 1.4. and 1.5. is added, minor changes and correction
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