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

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 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

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

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

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

Floating bundles and their applications

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

In the present paper we study bundles equipped with extra homotopy conditions, in particular so-called simplicial $n$-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