A model of the twisted $K$-theory related to bundles of finite order
In the present paper we propose a geometric model of the twisted K-theory corresponding to elements of finite order in $H^3(X, \mathbb{Z})\times [X, \BBSU_\otimes]$.
math.KT↗
arXiv subjects
A. V. Ershov
Publications and source records attributed to A. V. Ershov.
A bordism theory related to matrix Grassmannians
In the present paper we study a bordism theory related to pairs $(M,\, ξ),$ where $M$ is a closed smooth oriented manifold with a stably trivial normal bundle and $ξ$ 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.
math.AT↗
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.
math.AT↗
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.
math.KT↗
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.
math.KT↗
On bundles related to groupoids of matrix subalgebras
This paper has been withdrawn by the author.
math.KT↗
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.
math.KT↗
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.
math.AT↗
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 $(H,μ,ν, Δ,ε, S)$ 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 $(Δ{\frak g})({\frak F}(x\otimes 1,1\otimes x))= {\frak c}+{\frak g}(x)\otimes 1+1\otimes {\frak g}(x),$ where ${\frak c}\in H_\mathbb{Q}{\mathop{\hat{\otimes}}\limits_{R_ \mathbb{Q}}}H_\mathbb{Q}, (\id \otimes ε){\frak c}=0=(ε\otimes \id){\frak c}$ and $(\id \otimes Δ){\frak c}+1\otimes {\frak c}-(Δ\otimes \id){\frak c}-{\frak c}\otimes 1=0$ (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.
math.AT↗
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 $(\id \otimes ε){\frak c}=0= (ε\otimes \id){\frak c}.$
math.AT↗
Supplement 2 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 study some properties of category, connected with cobordism rings of FBSP. In particular, we shall show that it is the tensor category.
math.AT↗
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 $H=Ω_U^*(Gr).$
math.AT↗
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.
math.AT↗