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      All Questions

      2
      votes
      0answers
      60 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
      87 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 ...
      8
      votes
      3answers
      228 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
      178 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
      184 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
      118 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
      575 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
      177 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 ...
      12
      votes
      0answers
      182 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
      201 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
      325 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
      123 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 ...
      9
      votes
      1answer
      432 views

      Which 3-manifolds are known to admit exotic pairs of bounding 4-manifolds?

      Let $M$ be a compact connected three manifold. By an exotic pair of bounding 4-manifolds, I mean two smooth 4-manifolds $X_1,X_2$ such that $X_1$ and $X_2$ are homeomorphic but not diffeomorphic, and ...
      4
      votes
      2answers
      258 views

      Open book decompositions of $T^3$

      Please pardon my ignorance on the subject of open books, I'm a noob. I would like to know some explicit descriptions of open book decompositions of the three torus $T^3$. Are there examples with ...
      5
      votes
      2answers
      595 views

      Classification of closed 3-manifolds with finite first homology group?

      I am interested in a topological classification of connected closed 3-manifold $M$ that have finite homology group $H_1(M)$. Since $H_1(M)$ is the abelization of the fundamental group $\pi_1(M)$, ...

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