Automorphisms of Hopf manifolds

Fixed subgroups of automorphisms of hyperbolic 3-manifold groups

For fixed subgroups $Fix(ϕ)$ of automorphisms $ϕ$ on hyperbolic 3-manifold groups $π_{1}(M)$, we observed that $\text{rk}(Fix(ϕ))<2\text{rk}(π_{1}(M))$ and the constant 2 in the inequality is sharp; we also classify all possible groups $Fix(ϕ)$.

从霍普夫群代数出发的4维流形平坦连接不变量

Invariants of flat connections on 4-manifolds from Hopf group-algebras

本文提出通过霍普夫G-代数构造4维流形平坦连接的不变量，并在G为有限群时定义了基于所有平坦连接的不变量求和方法。

Corks and automorphisms of 3-manifolds

We investigate two specific contractible manifolds (one Stein, and the other non-Stein) whose boundaries have non-trivial mapping class groups. In both cases we show that every diffeomorphism of their boundary extends to a diffeomorphism of the full manifold. In particular, these manifolds cannot be corks. The methods are a mix of 3 and 4-manifold techniques.

Coperfectly Hopfian Groups and Shape Fibrator's Properties

This paper provides further investigation of the concept of shape m$_{\rm simpl}$-fibrators (previously introduced by the author). The main results identify shape m$_{\rm simpl}$-fibrators among direct products of Hopfian manifolds. First it is established that every closed orientable manifold homotopically determined by $π_1$ with coperfectly Hopfian group (a new class of Hopfian groups that are introduced here) is a shape m$_{\rm simpl}$o-fibrator if it is a codimension-2 fibrator (Theorem 5.4). The main result (Theorem 6.2) states that the direct product of two closed orientable manifolds (…

A Hennings type invariant of $3$-manifolds from a topological Hopf superalgebra

We prove the unrolled superalgebra $\mathcal{U}_ξ^{H}\mathfrak{sl}(2|1)$ has a completion which is a ribbon superalgebra in a topological sense where $ξ$ is a root of unity of odd order. Using this ribbon superalgebra we construct its universal invariant of links. We use it to construct an invariant of $3$-manifolds of Hennings type.

Geometric automorphism groups of symplectic 4-manifolds

Let $M$ be a closed, oriented, smooth $4-$manifold with intersection form $Γ$, $A(Γ)$ the automorphism group of $Γ$ and $D(M)$ the subgroup induced by orientation-preserving diffeomorphisms of $M$. In this note we study the question when $D(M)$ is of infinite index in $A(Γ)$ for a symplectic 4-manifold.

Generalized Kuperberg invariants of 3-manifolds

In the 90s, based on presentations of 3-manifolds by Heegaard diagrams, Kuperberg associated a scalar invariant of 3-manifolds to each finite dimensional involutory Hopf algebra over a field. We generalize this construction to the case of involutory Hopf algebras in arbitrary symmetric monoidal categories admitting certain pairs of morphisms called good pairs. We construct examples of such good pairs for involutory Hopf algebras whose distinguished grouplike elements are central. The generalized construction is illustrated by an example of an involutory super Hopf algebra.

对合Hopf群-余代数与4-流形上平坦丛的不变量

Involutory Hopf group-coalgebras and invariants of flat bundles over 4-manifolds

本文通过引入对合Hopf G-三元组结构，利用着色三剖分图构造了4-流形上平坦丛的不变量，推广了Chaidez-Cotler-Cui及Mochida的已有构造。

Higher dimensional topology and generalized Hopf bifurcations for discrete dynamical systems

In this paper we study generalized Poincaré-Andronov-Hopf bifurcations of discrete dynamical systems. We prove a general result for attractors in n-dimensional manifolds satisfying some suitable conditions. This result allows us to obtain sharper Hopf bifurcation theorems for fixed points in the general case and other attractors in low dimensional manifolds. Topological techniques based on the notion of concentricity of manifolds play a substantial role in the paper.

Reeb components of leafwise complex foliations and their symmetries III

The automorphisms group of the 3-dimensional Reeb component with complex leaves is computed in the case where the component is obtained by the Hopf construction and the holonomy of the boundary leaf is not tangent to the identity to the infinite order. Combined with a previous work, for 3-dimensional Reeb components obtained by the Hopf construction, we have an almost complete description of the groups of leafwise holomorphic smooth automorphisms.

Some finiteness results for groups of automorphisms of manifolds

We prove that in dimensions not equal to 4, 5, or 7, the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses homological stability, embedding calculus and the arithmeticity of mapping class groups. From this we deduce similar results for the homeomorphisms of $R^n$ and various types of automorphisms of 2-connected manifolds.

On Two Invariants of Three Manifolds from Hopf Algebras

We prove a 20-year-old conjecture concerning two quantum invariants of three manifolds that are constructed from finite dimensional Hopf algebras, namely, the Kuperberg invariant and the Hennings-Kauffman-Radford invariant. The two invariants can be viewed as a non-semisimple generalization of the Turaev-Viro-Barrett-Westbury $(\text{TVBW})$ invariant and the Witten-Reshetikhin-Turaev $(\text{WRT})$ invariant, respectively. By a classical result relating $\text{TVBW}$ and $\text{WRT}$, it follows that the Kuperberg invariant for a semisimple Hopf algebra is equal to the Hennings-Kauffman-Radfo…