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Results tagged with gt.geometric-topology
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user 36108
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
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Branched Coverings of the $4$-sphere branched along a knotted surface
Anton's comments above answer the question.
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Is there a knotted torus in 4-sphere whose complement's fundamental group is infinite cyclic...
I have been avoiding addressing this since almost all that I know about the question is in that book. I don't recall exactly, but I think that Kawauchi showed that a torus with the fundamental group o …
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Knots and their Morse functions
The answer to your question is a qualified, yes. The full reference is this article by Cooper, Mond and Wit Atique. In it they describe complex multi-germs of functions. This singularity theory is a …
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4
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answer
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Branched Coverings of the $4$-sphere branched along a knotted surface
This question is related to, but apparently not exactly the same as, Ramified cover of the $4$-sphere. Piergallini, et al. have singular points on their branch loci.
Which closed orientable $4$-dim …
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3
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Which closed orientable $4$-dimensional manifolds cannot be embedded in $6$-space?
This question is a follow-up to my previous question . The statement of the question is the title.
Note that the $4$-dimensional real projective space is non-orientable and a characteristic class a …
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7
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Generating ribbon diagrams for knots known to be ribbon knots
I think Kawauchi's book has tables that include ribbon diagrams, but I don't have a copy with me. Look at
Livingston and Cha . It is not hard to get a ribbon disk from this diagram: add a handle bet …
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Surface Eversions: Generalizing from Sphere and Torus Eversions
My answers should be comments, but I am, unfortunately, more verbose. For a single handle, consider making the double cover of a Klein bottle as follows. Take an immersed Mobius band which is half the …
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Knotted projective planes and fake complex projective space
Paul Melvin gave a talk at Knots in Washington last year in which he asked whether the connected sum of an odd twist-spin of a classical knot and a standard cross-cap embedding of ${\mathbb R}P^2$ is …
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Is there a combinatorial version of PL ambient isotopy in dimension $>3$?
Unfortunately, this is not quite an answer. So for knotted surfaces in 4-space, there is Roseman's Theorem. I think that the context of Dennis's proof is in the smooth category. Or certainly the proof …
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3
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Visualising locally flat embeddings of surfaces in R^4
I have been thinking about this as well. My approach would be to construct a broken surface diagram or chart of an immersed disk that a classical knot bounds. For simplicity take the untwisted double …
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3
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Application of a quandle cocycle invariant for virtual knots
To find the answer to your question, I would look through papers of Sam Nelson and his students, all of which are available on the arXiv. It is true that most of his work has to do with using quandles …
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Source needed (at final-year undergrad level) for the double cover of SO(3) by SU(2)
I agree that Artin has this, but also "Group Theory and Physics" by Sternberg has nice discussions early on (circa page 8-15).
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What is an immersed submanifold?
Bing's house with two rooms is the image of an immersed sphere that is not in general position.
General position immersions are easy to build out of local pictures --- well sort of easy to build. Co …
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8
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Explicit embeddings of Cappell-Shaneson knots
There is a paper by Iain Aitcheson (possible mis-spelling of the last name) and Hyam Rubenstein published in a Contemporary Mathematics Series of the AMS (Conference Proceedings) that is the most expl …
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2-tangles and quantum groups and 2-groups
There are many more details that are needed in Eitan's answer. Since weak 2-groups are determined by group 3-cocycles, one would think that group cocycle conditions are related to 2-knots. They are, b …
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