### A note on coalgebra gauge theory

A generalisation of quantum principal bundles in which a quantum structure group is replaced by a coalgebra is proposed.

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A generalisation of quantum principal bundles in which a quantum structure group is replaced by a coalgebra is proposed.

Given a smooth S¹-foliated bundle, A. Connes has shown the existence of an additive morphism ϕ from the K-theory group of the foliation C*-algebra to the scalar field, which factorizes, via the assembly map, the Godbillon-Vey class, which is the first secondary characteristic class, of the classifying space. We prove the invariance of this map under a bilipschitz homeomorphism, extending a previous result for maps of class C¹ by H. Natsume.

This survey of the work of the author with several collaborators presents the way groupoids appear and can be used in index theory. We define the general tools, and apply them to the case of manifolds with corners, ending with a topological index theorem.

The differential calculus on 'non-standard' h-Minkowski spaces is given. In particular it is shown that, for them, it is possible to introduce coordinates and derivatives which are simultaneously hermitian.

An extension of the category of local manifolds is considered. Instead of smooth mappings of neighbourhoods of linear spaces as morphisms we deal with formal operator power series (or formal maps). Analogues of the objects appearing on smooth manifolds and vector bundles (vector fields, sections of a bundle, exterior forms, the de Rham complex, connection, etc.) are considered in this way. All the examinations are carried out in algebraic language, for we do not care about the convergence of formal...

Building on the theory of elliptic operators, we give a unified treatment of the following topics: - the problem of homotopy invariance of Novikov’s higher signatures on closed manifolds, - the problem of cut-and-paste invariance of Novikov’s higher signatures on closed manifolds, - the problem of defining higher signatures on manifolds with boundary and proving their homotopy invariance.

The non-commutative torus ${C}^{*}({\mathbb{Z}}^{n},\omega )$ is realized as the ${C}^{*}$-algebra of sections of a locally trivial ${C}^{*}$-algebra bundle over $\widehat{{S}_{\omega}}$ with fibres isomorphic to ${C}^{*}({\mathbb{Z}}^{n}/{S}_{\omega},{\omega}_{1})$ for a totally skew multiplier ${\omega}_{1}$ on ${\mathbb{Z}}^{n}/{S}_{\omega}$. D. Poguntke [9] proved that ${A}_{\omega}$ is stably isomorphic to $C\left(\widehat{{S}_{\omega}}\right)\otimes {C}^{*}({\mathbb{Z}}^{n}/{S}_{\omega},{\omega}_{1})\cong C\left(\widehat{{S}_{\omega}}\right)\otimes {A}_{\varphi}\otimes {M}_{kl}\left(\u2102\right)$ for a simple non-commutative torus ${A}_{\varphi}$ and an integer $kl$. It is well-known that a stable isomorphism of two separable ${C}^{*}$-algebras is equivalent to the existence of equivalence bimodule between them. We construct an ${A}_{\omega}$-$C\left(\widehat{{S}_{\omega}}\right)\otimes {A}_{\varphi}$-equivalence bimodule.

We present the review of noncommutative symmetries applied to Connes' formulation of spectral triples. We introduce the notion of equivariant spectral triples with Hopf algebras as isometries of noncommutative manifolds, relate it to other elements of theory (equivariant K-theory, homology, equivariant differential algebras) and provide several examples of spectral triples with their isometries: isospectral (twisted) deformations (including noncommutative torus) and finite spectral triples.