arXiv subjects
Publications and source records attributed to Weizhe Niu.
We survey computations and tools concerning the mapping class group of a compact, oriented, connected 3-manifold $M$. We provide a guide to the literature and sketch proofs for various families of irreducible and geometric 3-manifolds. We also consider JSJ and prime decompositions of 3-manifolds, and consequences for their mapping class groups.
Let $Σ_g^0\subset S^4$, $g\ge3$, be the standard unknotted closed oriented surface, and let $a\subsetΣ_g^0$ be an oriented nonseparating curve. For every nontrivial knot $J\subset S^3$, let $Σ_{g,a,J}\subset S^4$ be the surface obtained from $Σ_g^0$ by ordinary untwisted rim surgery along $a$. We compute its extendable mapping-class subgroup exactly: $$ E(Σ_{g,a,J}) = \operatorname{Stab}_{\operatorname{Mod}(Σ_g)}(q_0) \cap \operatorname{Stab}_{\operatorname{Mod}(Σ_g)} (Γ_μ(J)\cdot[a]). $$ Here $q_0$ is the Rokhlin quadratic form of the standard embedding, $[a]\in H_1(Σ_g;\mathbb{Z})$ is the oriented rim homology class, and $Γ_μ(J)\subset\{\pm1\}$ records whether a meridian-preserving diffeomorphism of the knot exterior can preserve or reverse the preferred longitude. Thus ordinary rim surgery cuts Hirose's unknotted extendable subgroup by the stabilizer of the rim homology class, with the only additional ambiguity coming from this peripheral symmetry of $J$. We also prove a prescribed-mapping-class classification for such ambient pairs $(S^4,Σ_{g,a,J})$. More precisely, given two such pairs and $f\in\operatorname{Mod}(Σ_g)$, we characterize when $f$ is induced by an orientation-preserving pair diffeomorphism in terms of the Rokhlin quadratic form, the rim homology classes, and the meridian--longitude symmetries of the knot exteriors.
For every $g\geq 3$, every closed, connected, oriented, simply connected smooth $4$-manifold $X$, and every knot $K\subset S^3$, we construct infinitely many pairwise topologically inequivalent smoothly embedded oriented genus-$g$ surfaces $F\subset X$ whose orientation-preserving extendable mapping class subgroups are trivial in both the topological and smooth categories and whose first Alexander modules are isomorphic to the Alexander module of $K$. In particular, there are infinitely many such surfaces with vanishing first Alexander module. The construction is supported in a $4$-ball. Although their Alexander data are prescribed independently, the surfaces are distinguished, and their mapping-class rigidity is detected, by the nonabelian centralizer structure of their exterior groups.
We construct, for every $g\geq 2$, infinite families of homotopic smooth embeddings of a closed genus-$g$ surface whose images are pairwise not smoothly image-concordant, while each surface is $π_1$-injective. The main closed examples lie in one-fold stabilizations of closed aspherical mapping tori with torsion-free fundamental group: after stabilization by $S^2\times S^2$, the surfaces have a common framed dual sphere and the inclusion of each complement induces a $π_1$-isomorphism. The image-nonconcordance already occurs before stabilization, in the underlying closed aspherical mapping torus, and persists after every finite number of $S^2\times S^2$-stabilizations. The obstruction is a computable mod-two coordinate of Freedman--Quinn/Dax-type self-intersection data for concordance tracks, indexed by self-dual double-cosets of a possibly non-normal surface subgroup $H\leqπ_1X$. The geometric source of the relevant labels is a M"obius-band square-root relation: elements $t\notin H$ with $t^2\in H$ produce self-dual labels in torsion-free ambient groups. These square roots are realized naturally in Klein-bottle $I$-bundle pieces and persist in closed graph-manifold mapping-torus examples.
Freedman and Krushkal introduced a triple torsion linking form for rational homology $3$-spheres and used it to obstruct locally flat embeddings in $S^4$. For every odd prime $p$, we identify their triple torsion form, computed with parameter $t=p$ on rational homology $3$-spheres whose first homology has exponent $p$, with the mod-$p$ triple cup product under torsion-linking duality. For algebraically split $\pm p$-framed surgery links, this gives a signed formula in terms of Milnor's integral length-three invariants $\barμ_{ijk}$, with the framing-sign factor dictated by torsion-linking duality. We then use Borromean band-sums to realize arbitrary mod-$p$ triple cup tensors on rational homology $3$-spheres with $H_1\cong(\mathbb Z/p)^6$ and fixed hyperbolic ordinary torsion linking form. Finally, using the classical spinor/Klein model for the split six-dimensional quadratic space, we classify the tensors with no dual null Hantzsche pair. This produces, for every odd prime $p$, a rational homology $3$-sphere with hyperbolic ordinary torsion linking form but with no locally flat embedding in $S^4$, and indeed no locally flat embedding in any integer homology $4$-sphere.
We show that the $2$-component unlink in $S^4$ admits infinitely many isotopy classes of spanning $3$-disks that are Brunnian.
We define an invariant $(W_3)_m$ for $π_0\mathrm{Diff}(\natural_m S^1\times D^3,\partial)$ for $m\geq 1$ that generalizes Budney--Gabai's $W_3$ invariant. We give a computational framework inspired by Budney--Gabai and use it to calculate the invariant for all unknotted barbell difeomorphisms of $\natural_m S^1\times D^3$ for $m=1,2$. This allows us to detect more linearly independent elements in $π_0\mathrm{Diff}(S^1\times D^3,\partial)$, and to prove that $π_0\mathrm{Diff}( \natural_2 S^1\times D^3,\partial)/ \left( π_0 \mathrm{Diff}(S^1\times D^3,\partial)\right)^2$ admits infinitely generated subgroups generated by unknotted barbell diffeomorphisms, leading to infinitely many properly embedded separating 3-balls that are non-isotopic relative to the boundary.