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Results tagged with gt.geometric-topology
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user 2051
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).
82
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12
answers
15k
views
Compelling evidence that two basepoints are better than one
This question is inspired by an answer of Tim Porter.
Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his …
- 22.1k
35
votes
4
answers
4k
views
Extending a diffeomorphism of the sphere $S^2$ to the ball $D^3$
A fundamental result in three-dimensional smooth topology, which in computer jargon we might refer to as "a primitive", is the statement that any ($C^\infty$) diffeomorphism of the two-sphere $S^2$ ex …
- 22.1k
31
votes
4
answers
2k
views
Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?
Varieties of representations and characters of $3$-manifold groups in $SL(2,\mathbb{C})$ have been intensively studied. They have provided tools to identify geometric structures on manifolds, and are …
- 22.1k
30
votes
2
answers
3k
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The difference between a handle decomposition and a CW decomposition
Let $M$ be a compact finite-dimensional smooth manifold. I have a question about the relationship between the statements that a Morse function induces a handle decomposition for $M$, and that it induc …
- 22.1k
29
votes
3
answers
4k
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What would the slice-ribbon conjecture imply?
What would the slice-ribbon conjecture imply for 4-dimensional topology?
I've heard people speak of the slice-ribbon conjecture as an approach to the 4-dimensional smooth Poincare conjecture, and to t …
- 22.1k
26
votes
2
answers
3k
views
Why is the volume conjecture important?
The volume conjecture, a formula relating hyperbolic volume of a knot complement with the semiclassical limit of a family of coloured Jones polynomials, is widely considered the biggest open problem i …
- 22.1k
25
votes
3
answers
2k
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What are the implications of the simple loop conjecture?
Drew Zemke recently posted a preprint on arXiv proving the Simple Loop Conjecture for 3-manifolds modeled on Sol.
Simple Loop Conjecture: Consider a 2-sided immersion $F\colon\, \Sigma\rightarrow M …
- 22.1k
24
votes
3
answers
3k
views
Elevator pitch for the Virtual Fibering Theorem
There has been a great deal of excitement among topologists about the proof of the Virtual Haken Theorem, and in fact of the Virtual Fibering Theorem (for closed hyperbolic 3-manifolds, but I'm guess …
- 22.1k
24
votes
1
answer
2k
views
How does the Framed Function Theorem simplify Cerf Theory?
A handle decomposition of a manifold $M$ is a useful structure to carry around. It is induced by a Morse function $f\colon\, M\to \mathbb{R}$.
How are two handle decompositions of $M$ related? The sp …
- 22.1k
23
votes
Accepted
Proofs of Kirby's theorem
There is Bob Craggs' 1974 proof, which was never published. It relies on Wall's result, that any two 2-handle cobordisms between S^3 and itself are stably homeomorphic if the associated bilinear form …
- 22.1k
23
votes
In knot theory: Benefits of working in $S^3$ instead of $\mathbb{R}^3$?
This is a nice question!
Knot theory is in fact knot-complement theory, and a knot complement in S3 is a compact 3-manifold, while a knot complement in R3 is an open 3-manifold. Compact (or closed) 3- …
- 22.1k
22
votes
4
answers
2k
views
Searching for an unabridged proof of "The Basic Theorem of Morse Theory"
Steven Smale labels the following statement "The Basic Theorem of Morse Theory" in A Survey of some Recent Developments in Differential Topology:
Let f be a $C^\infty$ function on a closed manifold …
- 22.1k
22
votes
Status of PL topology
I'd like to address another aspect of your questions. My feeling is that PL topology, or smooth topology, are foundational subjects to the low dimensional topologist, in the sense that set theory is a …
21
votes
1
answer
2k
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How are the Conway polynomial and the Alexander polynomial different?
Background story:
I have just come out from a talk by Misha Polyak on generalizations of an arrow-diagram formula for coefficients of the Conway polynomial by Chmutov, Khoury, and Rossi. In it, he des …
- 22.1k
20
votes
2
answers
1k
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Is there a discrete Cerf theory?
Towards the end of the 1990's, Robin Forman developed a discrete version of Morse theory, which concerns maps from a simplicial complex to $\mathbb{R}$ satisfying a combinatorial analogue to the condi …
- 22.1k