Homotopy theory of bundles with fiber matrix algebra

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 $CW$-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 radio emission of the Geminga pulsar and RBS 1223 at the frequency of 111 MHz

I have searched for pulsed radio emission from the Geminga pulsar and for the nearby isolated neutron star 1RX J1308.6+2127 (RBS 1223) at the frequency of 111 MHz. No pulsed signals were detected from these sources. Upper limits for mean flux density are 0.4 - 4 mJy for the Geminga pulsar and 1.5 - 15 mJy for RBS 1223 depending on assumed duty cycle (.05 - .5) of the pulsars.Comment: 4 pages, 3 figure

On KK-theory automorphisms related to bundles of finite order

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

Formal groups over Hopf algebras

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 $H$-space $\hat{Gr}$. 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 ${\frak F}(x\otimes 1,1\otimes x) \in (H{\mathop{\hat{\otimes}}\limits_R}H)[[x\otimes 1,1\otimes x]]$ is a FGoHA over a Hopf algebra $(H,\mu,\nu, \Delta,\epsilon, S)$ over a ring $R$ without torsion, then there exists a logarithm, i.e. the formal series ${\frak g}(x)\in H_\mathbb{Q}[[x]]$ such that $(\Delta {\frak g})({\frak F}(x\otimes 1,1\otimes x))= {\frak c}+{\frak g}(x)\otimes 1+1\otimes {\frak g}(x),$ 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 ${\frak c}$ is a cocycle in the cobar complex of the Hopf algebra $H_\mathbb{\mathbb{Q}}$). 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

On scattering of Giant Pulses from the Crab Pulsar: a Scattering Function

Simultaneous dual-frequency observations of giant pulses from the Crab pulsar were performed at the frequencies of 61 and 111 MHz. It is shown that scattering of giant pulses from the Crab pulsar occurs at thick, and not at thin screen.Comment: 3 pages, 3 figure

Logarithms of formal groups over Hopf algebras

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

A generalization of the topological Brauer group

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 $K$-theory.Comment: 34 pages. v5: The part concerning the generalized Brauer group has been completely rewritten. An application to twisted $K$-theory is adde

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

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

A bordism theory related to matrix Grassmannians

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

Supplement to the paper "Floating bundles and their applications"

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 $F(x\otimes 1,1\otimes x)$ by the Hopf algebra $H=\Omega_U^*(Gr).$Comment: 4 pages, xypi