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      All Questions

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      6
      votes
      1answer
      134 views

      A coincidence between the Lambert cube, Lobell polyhedron, and hyperbolic 3-manifolds?

      I. Lambert cube $\mathfrak L(\alpha_1,\alpha_2,\alpha_3)$ In this paper (p.8), we find the volume $V$ of the hyperbolic Lambert cube for the special case $\alpha=\alpha_1 = \alpha_2 = \alpha_3$ as $...
      3
      votes
      1answer
      148 views

      Obtaining the bounding 4-manifold from the Heegaard diagram

      It is well known that any orientable closed 3-manifold $M$ admits an Heegaard splitting $M = H_1\cup H_2$ where $H_i$ is an handlebody of genus $g$. It is also well known that such an $M$ is the ...
      4
      votes
      0answers
      106 views

      Are triangulations with common refinements PL-homeomorphic?

      Do there exist simplicial triangulations $K_1$ and $K_2$ of a topological manifold $M$ such that $K_1$ and $K_2$ have a common subdivision but they are not PL-homeomorphic? Ideally, I would like an ...
      4
      votes
      1answer
      121 views

      Circle bundles and surface bundles which admit no strongly irreducible Heegaard splittings

      Let $S$ be a closed connected orientable surface with $g(S)>0$. Jennifer Schultens, in her paper ``The Classification of Heegaard Splittings for (Compact Orientable Surface)$\times S^1$'', proves ...
      2
      votes
      0answers
      103 views

      Existence of smooth structures on topological $3$-manifolds with boundary

      It is said in this thread Unique smooth structure on $3$-manifolds that every topological $3$-manifold admits a smooth structure. However it is not specified whether the manifolds are allowed to have ...
      6
      votes
      3answers
      246 views

      open book decompositions of $\Sigma\times S^1$

      Let $\Sigma$ be a closed orientable surface. Is there a standard open book decomposition on the $3$-manifold $M=\Sigma\times S^1$? If the binding is allowed to be empty in the definition of an open ...
      3
      votes
      2answers
      193 views

      Triangulations of 3-manifolds in Regina and SnapPy

      I have been doing some statistical studies on small 3-manifolds, and I note that one can produce larg-ish censuses of triangulations in Regina. Now, the Regina documentation tells us how to convert a ...
      4
      votes
      1answer
      205 views

      Hyperbolic Dehn surgeries and SU(2)-representations

      Let $S^3-K$ be the complement of the figure eight knot complement. Thurston, in his Lecture Notes, constructed a hyperbolic structure, which comes from a discrete, faithful representation $\pi_1(S^3-K)...
      1
      vote
      1answer
      127 views

      Signature/nullity function for a link obtained by parallel pushoffs of a knot?

      Let $K$ be an oriented knot in $S^3$ together with a framing $n$. Let $K(a,b)$ be the oriented link obtained by taking $a$ copies of the $n$-pushoff of $K$ with the same same orientation as $K$ and $...
      9
      votes
      3answers
      593 views

      Reference request for wild 3-manifolds

      I’m looking for a text on 3-manifolds that focuses on wild/pathological objects, similar to Bing’s work in the field. I know basic algebraic topology (homotopy, homology, cohomology) and have read ...
      5
      votes
      1answer
      185 views

      Is there a generalized Property P - what can we say about framed link descriptions of $S^3$?

      A knot $K$ is said to have Property P if every nontrivial Dehn surgery on $K$ yields a 3-manifold that is not simply connected. It is known that every knot except the unknot has Property P. I am ...
      13
      votes
      0answers
      208 views

      Are there exotic twisted doubles of 4-manifolds?

      Take a smooth 4-manifold $X$ whose boundary has a diffeomorphism $\tau: \partial X \to \partial X$ that extends to a homeomorphism but not a diffeomorphism of $X$. (By Matveyev and Curtis-Freedman-...
      7
      votes
      1answer
      205 views

      Action of diffeomorphism group on non-vanishing vector fields

      Let $M$ denote a closed manifold. Let $\Gamma(TM\setminus 0) $ denote the space of non-vanishing sections of $TM$. Note that the diffeomorphism group $\text{Diff} (M)$ acts on $\Gamma(TM\setminus 0)...
      8
      votes
      2answers
      340 views

      Hyperbolic $3$-manifold groups that embed in compact Lie groups

      Is there a closed hyperbolic $3$-manifold whose fundamental group is isomorphic to a subgroup of some compact Lie group? It is known that every surface group can be embedded into any semisimple ...
      4
      votes
      1answer
      128 views

      Simple invariants to detect concordance in general 3-manifolds

      Let $Y$ be a closed, connected, orientable 3-manifold. We call to oriented knots $K_1, K_2$ in $Y$ (smoothly) concordant if there is a smoothly, properly embedded annulus in $Y \times I$ such that ...

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