5,840 research outputs found

### 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 $K$-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

### 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

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