Kazuo Habiro
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View article: The Johnson-Morita theory for the handlebody group
The Johnson-Morita theory for the handlebody group Open
The Johnson-Morita theory is an algebraic approach to the mapping class group of a surface, in which one considers its action on the successive nilpotent quotients of the fundamental group of the surface. In this paper, we develop an analo…
View article: On Borel's stable range of the twisted cohomology of $\mathrm{GL}(n,\mathbb{Z})$
On Borel's stable range of the twisted cohomology of $\mathrm{GL}(n,\mathbb{Z})$ Open
Borel's stability and vanishing theorem gives the stable cohomology of $\mathrm{GL}(n,\mathbb{Z})$ with coefficients in algebraic $\mathrm{GL}(n,\mathbb{Z})$-representations. We compute the improved stable range that Borel remarked about. …
View article: On the stable cohomology of the IA-automorphism groups of free groups
On the stable cohomology of the IA-automorphism groups of free groups Open
Borel's stability and vanishing theorem gives the stable cohomology of $\mathrm{GL}(n,\mathbb{Z})$ with coefficients in algebraic $\mathrm{GL}(n,\mathbb{Z})$-representations. By combining the Borel theorem with the Hochschild-Serre spectra…
View article: Ribbon Yetter--Drinfeld modules and tangle invariants
Ribbon Yetter--Drinfeld modules and tangle invariants Open
We define notions of pivotal and ribbon objects in a monoidal category. These constructions give pivotal or ribbon monoidal categories from a monoidal category which is not necessarily with duals. We apply this construction to the braided …
View article: The Kontsevich integral for bottom tangles in handlebodies
The Kontsevich integral for bottom tangles in handlebodies Open
Using an extension of the Kontsevich integral to tangles in handlebodies similar to a construction given by Andersen, Mattes and Reshetikhin, we construct a functor Z\colon \mathcal{B}\to \hat{A} , where \mathcal{B} is the category of bott…
View article: Double Johnson filtrations for mapping class groups
Double Johnson filtrations for mapping class groups Open
We first develop a general theory of Johnson filtrations and Johnson homomorphisms for a group $G$ acting on another group $K$ equipped with a filtration indexed by a "good" ordered commutative monoid. Then, specializing it to the case whe…
View article: Kirby calculus for null-homologous framed links in 3-manifolds
Kirby calculus for null-homologous framed links in 3-manifolds Open
A theorem of Kirby gives a necessary and sufficient condition for two framed\nlinks in S^3 to yield orientation-preserving diffeomorphic results of surgery.\nKirby's theorem is an important method for constructing invariants of\n3-manifold…
View article: Current algebras and categorified quantum groups
Current algebras and categorified quantum groups Open
We identify the trace, or 0th Hochschild homology, of type ADE categorified\nquantum groups with the corresponding current algebra of the same type. To\nprove this, we show that 2-representations defined using categories of modules\nover c…
View article: On the category of finitely generated free groups
On the category of finitely generated free groups Open
It is well known that the opposite F^{op} of the category F of finitely generated free groups is a Lawvere theory for groups, and also that F is a free symmetric monoidal category on a commutative Hopf monoid, or, in other words, a PROP fo…