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Zhongzi Wang

Publications and source records attributed to Zhongzi Wang.

At least 19 recordsLinked to original sources
arXiv2026

Profinite rigidity of simple closed curves in surface groups

This paper establishes a new characterization of simple closed curves on a closed orientable surface. Let $Γ$ be the fundamental group of a closed orientable surface. We prove that if an element $g\inΓ$ has the same possible images as a given simple closed curve $γ\in Γ$ under epimorphisms from $Γ$ to every finite group, then $g$ belongs to the $\mathrm{Aut}(Γ)$-orbit of $γ$, i.e. $g$ is itself a simple closed curve with the same topological type as $γ$. Consequently, the set of simple closed curves in $Γ$ is closed in the profinite topology of $Γ$; and we obtain a new algorithm to decide whether a given element in $Γ$ can be represented by a simple closed curve. Proper powers of simple closed curves and the pro-$p$ cases are also discussed.

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arXiv2026

On the fiberedness of surgery 3-manifolds

Let $M$ be a closed orientable 3-manifold and $k$ be a knot in $M$. Then the Dehn surgery of $M$ along $k$ with slope $α$ is not surface fibered for all but a sparse set of slopes.

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arXiv2026

Flexible exponent of geometric 3-manifolds and Legendrian maps of Seifert spaces

A classical question in quantitative topology is to bound the mapping degree $\operatorname{deg}(f)$ in terms of its Lipchitz constant $\operatorname{Lip}(f)$. For a closed, oriented manifold $M$, the flexible exponent $α(M)$ is the infimum of $α\geq 0$ such that $|\operatorname{deg} f|\leq C(\operatorname{Lip} f)^α$ holds for all differentiable map $f:M\to M$. The flexible exponent measures how effectively a manifold can wrap itself through self-maps. For geometric 3-manifolds $M$ in the sense of Thurston, we give the complete result for $α(M)$: \[ α(M)= \begin{cases} 3 & M \text{ modeled on } \mathbb S^3,\mathbb E^3,\mathbb S^2\times\mathbb E^1,\\ \frac83 & M \text{ modeled on Nil},\\ 2 & M \text{ modeled on Sol},\\ 1 & M \text{ modeled on }\mathbb H^2\times\mathbb E^1,\\ 0 & M \text{ modeled on } \mathbb H^3,\widetilde{\rm SL_2}. \end{cases} \] To prove $α(M)=8/3$ for Nil 3-manifold $M$, we construct the so-called Legendrian map: a smooth self-map $f: M\to M$ such that $f$ is homotopic to the identity and $f$ maps all $S^1$-fibers into the orthogonal contact plane field simultaneously. Moreover, we prove that any Legendrian map must not be a diffeomorphism.

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arXiv2026

Flexible exponents of non-geometric 3-manifolds

A classical question in quantitative topology is to bound the mapping degree $\operatorname{deg}(f)$ in terms of its Lipchitz constant $\text{Lip}(f)$. For a closed, orientable, Riemannian manifold $M$, the flexible exponent $α(M)$ is the infimum of $α\geqslant 0$ such that $|\text{deg}(f)|\leqslant C\cdot (\text{Lip}(f))^α$ holds for any Lipschitz map $f:M\to M$. For a geometric 3-manifold $M$ in the sense of Thurston, $α(M)$ is determined in \cite{DLWWW}. In this paper, we determine $α(M)$ for non-geometric 3-manifolds.

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arXiv2026

Asymptotics of shortest filling closed multi-geodesics

In this paper, we investigate the asymptotics of shortest filling closed multi-geodesics of closed hyperbolic surfaces as systole $\to 0$ or as genus $\to \infty$. We first show that for a closed hyperbolic surface $X_g$ of genus $g$, the length of a shortest filling closed multi-geodesic of $X_g$ is uniformly comparable to $$\left(g+\sum\limits_{\textit{closed geodesic }γ\subset X_g, \ \ell(γ)<1}\log \left(\frac{1}{\ell(γ)}\right)\right).$$ As an application, we show that as $g\to \infty$, a Weil-Petersson random hyperbolic surface has a shortest closed multi-geodesic of length uniformly comparable to $g$. We also show that this is true for a random hyperbolic surface in the Brooks-Makover model.

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arXiv2025

Equivariant embeddings of Riemann surfaces in Euclidean spaces with minimal dimensions

Let $Σ_g$ be a closed Riemann surface of genus $g$. Let $G$ be a finite subgroup of the automorphism group of $Σ_g$. It is well known that there exists a smooth $G$-equivariant embedding from $Σ_g$ to some Euclidean space $\mathbb{R}^n$. Let $d_g(G)$ be the minimal possible $n$ for $(Σ_g,G)$. We compute the value of $d_g(G)$ in certain cases. Especially, we show that: for the automorphism group of the closed Riemann surface which comes from the principal congruence subgroup of level $p$, where $p\geq 7$ is prime, $d_g(G)=p+1$. As a corollary, the minimal $n$ for the Hurwitz action on the Klein quartic is equal to $8$. Three kinds of methods are used in the computation, which are related to the representations of groups, the equivariant triangulations, and the orbifold theory, respectively. The methods are also used to provide two kinds of upper bounds: $d_g(G)\leq |G|$ if $|G|\geq 5$; and $d_g(G)\leq 12(g-1)$ if $g\geq 2$.

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arXiv2025

Thurston spine in a Teichmüller curve

To study the Thurston spine $\mathcal{P}_g \subseteq \mathcal{T}_g$, we construct a Teichmüller curve $V \subseteq \mathcal{T}_g$. Then we characterize $V \cap \mathcal{P}_g$. More specifically, we show it is a trivalent tree and is an equivariant deformation retract of $V$. Moreover, by our construction, a lot of essential loops in the Thurston spine, both reducible and pseudo-Anosov, are obtained.

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arXiv2025

Shortest filling geodesics on hyperbolic surfaces

In this paper, we obtain the minimal length of a filling (multi-)geodesic on a genus $g$ hyperbolic surface in the moduli space of hyperbolic surfaces and show that it is realized by the geodesic whose complement is a right-angled regular $(8g-4)$-gon. A single geodesic realizing this minimum is provided.

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arXiv2025

$π_1$-injective bounding and application to 3- and 4-manifolds

Suppose a closed oriented $n$-manifold $M$ bounds an oriented $(n+1)$-manifold. It is known that $M$ $π_1$-injectively bounds an oriented $(n+1)$-manifold $W$. We prove that $π_1(W)$ can be residually finite if $π_1(M)$ is, and $π_1(W)$ can be finite if $π_1(M)$ is. In particular, each closed 3-manifold $M$ $π_1$-injectively bounds a 4-manifold with residually finite $π_1$, and bounds a 4-manifold with finite $π_1$ if $π_1(M)$ is finite. Applications to 3- and 4-manifolds are given: (1) We study finite group actions on closed 4-manifolds and $π_1$-isomorphic cobordism of 3-dimensional lens spaces. Results including: (a) Two lens spaces are $π_1$-isomorphic cobordant if and only if there is a degree one map between them. (b) Each spherical 3-manifold $M\ne S^3$ can be realized as the unique non-free orbit type for a finite group action on a closed 4-manifold. (2) The minimal bounding index $O_b(M)$ for closed 3-manifolds $M$ are defined, %and bounding Euler charicteristic $χ_b(M)$. the relations between finiteness of $O_b(M)$ and virtual achirality of aspherical (hyperbolic) $M$ are addressed. We calculate $O_b(M)$ for some lens spaces $M$. Each prime is realized as a minimal bounding index. (3) We also discuss some concrete examples:Surface bundle often bound surface bundles, and prime 3-manifolds often virtually bound surface bundles, $W$ bounded by some lens spaces realizing $O_b$ is constructed.

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arXiv2025

Maps between circle bundles: Fiber-preserving, Finiteness and Realization of mapping degree sets

Let $E_i$ be an oriented circle bundle over a closed oriented aspherical $n$-manifold $M_i$ with Euler class $e_i\in H^2(M_i;\mathbb{Z})$, $i=1,2$. We prove the following: (i) If every finite-index subgroup of $π_1(M_2)$ has trivial center, then any non-zero degree map from $E_1$ to $E_2$ is homotopic to a fiber-preserving map. (ii) The mapping degree set of fiber-preserving maps from $E_1$ to $E_2$ is given by $$\{0\} \cup\{k\cdot \mathrm{deg}(f) \ | \, k\ne 0, \ f\colon M_1\to M_2 \, \text{with} \, \mathrm{deg}(f)\ne 0 \ \text{such that}\, f^\#(e_2)=ke_1\},$$ where $f^\# \colon H^2(M_2;\mathbb{Z})\to H^2(M_1;\mathbb{Z})$ is the induced homomorphism. As applications of (i) and (ii), we obtain the following results with respect to the finiteness and the realization problems for mapping degree sets: ($\mathcal F$) The mapping degree set $D(E_1, E_2)$ is finite if $M_2$ is hyperbolic and $e_2$ is not torsion. ($\mathcal R$) For any finite set $A$ of integers containing $0$ and each $n>2$, $A$ is the mapping degree set $D(M,N)$ for some closed oriented $n$-manifolds $M$ and $N$. Items (i) and ($\mathcal F$) extend in all dimensions $\geq 3$ the previously known $3$-dimensional case (i.e., for maps between circle bundles over hyperbolic surfaces). Item ($\mathcal R$) gives a complete answer to the realization problem for finite sets (containing $0$) in any dimension, establishing in particular the previously unknown cases in dimensions $n= 4, 5$.

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arXiv2025

On virtual chirality of 3-manifolds

We prove that if a prime 3-manifold M is not finitely covered by the 3-sphere or a product manifold, then M is virtually chiral, i.e. it has a finite cover that does not admit an orientation reversing self-homeomorphism. In general if a 3-manifold contains a virtually chiral prime summand, then it is virtually chiral.

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arXiv2024

Embedding periodic maps of surfaces into those of spheres with minimal dimensions

It is known that any periodic map of order $n$ on a closed oriented surface of genus $g$ can be equivariantly embedded into $S^m$ for some $m$. In the orientable and smooth category, we determine the smallest possible $m$ when $n\geq 3g$. We show that for each integer $k>1$ there exist infinitely many periodic maps such that the smallest possible $m$ is equal to $k$.

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arXiv2024

Achirality of Sol 3-Manifolds, Stevenhagen Conjecture and Shimizu's L-series

A closed orientable manifold is {\em achiral} if it admits an orientation reversing homeomorphism. A commensurable class of closed manifolds is achiral if it contains an achiral element, or equivalently, each manifold in $\CM$ has an achiral finite cover. Each commensurable class containing non-orientable elements must be achiral. It is natural to wonder how many commensurable classes are achiral and how many achiral classes have non-orientable elements. We study this problem for Sol 3-manifolds. Each commensurable class $\CM$ of Sol 3-manifold has a complete topological invariant $D_{\CM}$, the discriminant of $\CM$. Our main result is: (1) Among all commensurable classes of Sol 3-manifolds, there are infinitely many achiral classes; however ordered by discriminants, the density of achiral commensurable classes is 0. (2) Among all achiral commensurable classes of Sol 3-manifolds, ordered by discriminants, the density of classes containing non-orientable elements is $1-ρ$, where $$ρ:=\prod_{j=1}^\infty \left(1+2^{-j}\right)^{-1} = 0.41942\cdots.$$

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arXiv2023

On the realisation problem for mapping degree sets

The set of degrees of maps $D(M,N)$, where $M,N$ are closed oriented $n$-manifolds, always contains $0$ and the set of degrees of self-maps $D(M)$ always contains $0$ and $1$. Also, if $a,b\in D(M)$, then $ab\in D(M)$; a set $A\subseteq\mathbb Z$ so that $ab\in A$ for each $a,b\in A$ is called multiplicative. On the one hand, not every infinite set of integers (containing $0$) is a mapping degree set [NWW] and, on the other hand, every finite set of integers (containing $0$) is the mapping degree set of some $3$-manifolds [CMV]. We show the following: (i) Not every multiplicative set $A$ containing $0,1$ is a self-mapping degree set. (ii) For each $n\in\mathbb N$ and $k\geq3$, every $D(M,N)$ for $n$-manifolds $M$ and $N$ is $D(P,Q)$ for some $(n+k)$-manifolds $P$ and $Q$. As a consequence of (ii) and [CMV], every finite set of integers (containing $0$) is the mapping degree set of some $n$-manifolds for all $n\neq 1,2,4,5$.

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arXiv2023

Length minima for an infinite family of filling closed curves on a one-holed torus

We explicitly find the minima as well as the minimum points of the geodesic length functions for the family of filling (hence non-simple) closed curves, $a^2b^n$ ($n\ge 3$), on a complete one-holed hyperbolic torus in its relative Teichmüller space, where $a, b$ are simple closed curves on the one-holed torus which intersect exactly once transversely. This provides concrete examples for the problem to minimize the geodesic length of a fixed filling closed curve on a complete hyperbolic surface of finite type in its relative Teichmüller space.

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arXiv2023

Extendability over the $4$-sphere and invariant spin structures of surface automorphisms

It is known that an automorphism of $F_g$, the oriented closed surface of genus $g$, is extendable over the 4-sphere $S^4$ if and only if it has a bounding invariant spin structure \cite{WsWz}. We show that each automorphism of $F_g$ has an invariant spin structure, and obtain a stably extendable result: Each automorphism of $F_g$ is extendable over $S^4$ after a connected sum with the identity map on the torus. Then each automorphism of an oriented once punctured surface is extendable over $S^4$. For each $g\neq 4$, we construct a periodic map on $F_g$ that is not extendable over $S^4$, and we prove that every periodic map on $F_4$ is extendable over $S^4$, which answer a question in \cite{WsWz}. We illustrate for an automorphism $f$ of $F_g$, how to find its invariant spin structures, bounding or not; and once $f$ has a bounding invariant spin structure, how to construct an embedding $F_g\hookrightarrow S^4$ so that $f$ is extendable with respect to this embedding.

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arXiv2023

Distributions of points on non-extensible closed curves in $\R^3$ realizing maximum energies

Let $G_n$ be a non-extensible, flexible closed curve of length $n$ in the 3-space $\R^3$ with $n$ particles $A_1$,...,$A_n$ evenly fixed (according to the arc length of $G_n$) on the curve. Let $f:(0, \infty)\to \R$ be an increasing and continuous function. Define an energy function $$E^f_n(G_n)= \sum_{p< q} f(|A_pA_q|),$$ where $|A_pA_q|$ is the distance between $A_p$ and $A_q$ in $\R^3$. We address a natural and interesting problem: {\it What is the shape of $G_n$ when $E^f_n(G_n)$ reaches the maximum? } In many natural cases, one such case being $f(t) = t^α$ with $0 < α\le 2$, the maximizers are regular $n$-gons and in all cases the maximizers are (possibly degenerate) convex $n$-gons with each edge of length 1.

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