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      Questions tagged [3-manifolds]

      A 3-manifold is a space that locally looks like Euclidean 3-dimensional space

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      92 views

      Book recommendations beyond an introduction [on hold]

      So I’ve scraped the surface of many topics, but I would like to go further. Can anyone recommend some continuations to the following introductory books? It’s okay if necessarily it needs to be a ...
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      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 ...
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      votes
      1answer
      110 views

      SnapPy isometry routine

      Dear Colleagues and Friends, Here's a question that I hope some of you, more experienced in programming, can answer. Once SnapPy is used to compute the symmetry group of a hyperbolic manifold by ...
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      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 ...
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      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 ...
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      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 ...
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      votes
      0answers
      126 views

      P-adic Volume Conjecture

      Let $M$ be a closed hyperbolic 3-manifold. One can use hyperbolic structure on $M$ to define hyperbolic volume $Vol(M)$. Thanks to Mostow's rigidity theorem the volume depends only on the topology of ...
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      44 views

      Surgery (Kirby) diagram of connected sum of lens spaces?

      What is the kirby/surgery diagram for a connected sum of two lens spaces? Question 1: is it just the unlink with $p/q$ coefficients? Question 2: if not, what manifold is the unlink with $p_i/q_i$ ...
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      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)...
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      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 $...
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      votes
      0answers
      92 views

      3d Chern-Simons TQFT of gauge group (E8)$_1$ = SO(16)$_1 \otimes$ a trivial spin TQFT = Cartan E$_8$ matrix

      In this post, we like to relate the following 3 bosonic TQFTs that can be defined on generic non-spin manifold $M^3$. Given a non-abelian Chern-Simons (CS) TQFT of a gauge group $G$ and the $k$ named ...
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      vote
      0answers
      82 views

      Manifold with no closed components?

      Let $M$ be a manifold with boundary. Reading some papers on $3$-manifolds I have come across some statements where they require that: ”$M$ has no closed components.” What does this mean? The ...
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      votes
      0answers
      157 views

      What is variation of the Chern-Simons functional, and why can it be calculated as follows?

      Let $G$ be a Lie group. Assume that we have an Ad-invariant bilinear symmetric form $$\langle-,-\rangle : \mathfrak{g} \times \mathfrak{g} \to \mathbb{C}.$$ Given a smooth manifold $X,$ we let $\...
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      votes
      1answer
      137 views

      Pre-images of Seifert surfaces are incompressible?

      Consider a knot $K \subset S^3$ and let $M_K$ be the associated double branched cover. The pre-image $S$ of a Seifert surface is a surface without boundary inside $M_K$. Can $S$ be incompressible? If ...
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      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 ...

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