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Results tagged with gt.geometric-topology
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user 3460
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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The Jones polynomial at specific values of $t$
Not exactly a value of the Jones polynomial, but I think in the spirit of the question: The logarithmic derivative of the Jones polynomial, evaluated at -1, is related to the Casson-Walker invariant o …
- 19.4k
4
votes
Handle decompositions subordinate to an open cover
In an appendix to a paper of Tsuboi, On the uniform perfectness of the groups of diffeomorphisms of even-dimensional manifolds, you can find something rather close to what you are asking for. He build …
- 19.4k
6
votes
Residual finiteness of hyperbolic 3-manifold groups
Here's another negative answer for Q2. I'm assuming (as in Sam Nead's answer) that the covering should be locally isometric. By Ahlfors-Bers, a tame infinite volume hyperbolic manifold with ends of th …
- 19.4k
7
votes
Accepted
Slice knots in 3-manifolds
Suppose you know that the universal cover of $Y$ embeds in $S^3$, i.e. is $S^3-A$ for some $A$. For example, this happens when $Y$ is a connected sum of lens spaces. (I'm thinking this is always true …
- 19.4k
23
votes
Accepted
What can we say about the Cartesian product of a manifold with its exotic copy?
Your question seems to be about simply connected exotic 4-manifolds, for which the answer is yes. That's because $M$ and $M^E$ are h-cobordant (by Wall), say via an h-cobordism W. Then $M \times W$ is …
- 19.4k
6
votes
Kirby diagrams of Mazur manifolds
The paper of Fickle (not the paper of Gordon to which you link) has an explicit construction for $\Sigma(3,5,19)$. It's behind a paywall but maybe you can get it from interlibrary loan or some kind so …
- 19.4k
1
vote
First usage of the terms pseudo-isotopy and concordance in manifold theory
Going along with Gael Meigniez's reference, but presumably more accessible, is an article by Hudson and Zeeman, On combinatorial isotopy,
Publications mathématiques de l’I.H.É.S., tome 19 (1964), p. 6 …
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14
votes
4-manifold $M$ with intersection form of Leech lattice
If you're assuming $M$ is simply-connected, then it would be spin (since the Leech lattice is even). So a smooth manifold would violate Rokhlin's theorem. In the topological case (still assuming simpl …
- 19.4k
13
votes
Exotic smooth structure
If you take an exotic $\mathbb{R}^4$ and remove an open ball lying in a chart, the result is an exotic smooth structure on $S^3 \times [0,\infty)$ (with smooth boundary). As Anubhav mentions, you can …
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8
votes
Accepted
Hyperbolic homology spheres with infinite $\mathrm{SL}_2(\mathbb{C})$ character variety
Take two knot complements (say of $K,K'$) and glue them together, interchanging meridians and longitudes. This is called splicing and produces a homology sphere $S(K,K')$; if both knots are non-trivi …
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7
votes
In knot theory, what is this link property and how to detect it: "linkings between component...
Nice question--I'm not sure if this already has a name.
Here is one way to show that a link is not a necklace, that applies to the Borromean rings. Suppose that the link has components $(R,G,B)$, and …
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7
votes
Accepted
Lens space bounding a topological, simply-connected 4-manifold with $b_2=1$
This follows from the relationship between the $\mathbb{Q}/\mathbb{Z}$ linking form of a $3$-manifold and the intersection form of a $4$-manifold that it bounds. Suppose that $W$ is 1-connected ($H_1= …
- 19.4k
11
votes
Accepted
Small knots becoming isotopic after connect sum
I believe can happen in dimension $4$, and probably in all higher dimensions. Take two inequivalent knots $K, K'$ with the same exterior (Cappell-Shaneson; Gordon). Then $K'$ is obtained from $K$ by a …
- 19.4k
7
votes
Dual surfaces of a first cohomology class of a 3-manifold
Some things are known in the non-orientable case (in orientable 3-manifolds). Bredon and Wood work out the genus in lens spaces $L(2n,q)$ (and some other manifolds) in their paper Non-orientable surfa …
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12
votes
Accepted
Kervaire-Milnor group of homotopy spheres and smooth Poincaré conjecture
That a homotopy 4-sphere is h-cobordant to $S^4$ is in principle a step towards proving the 4-dimensional Poincaré conjecture. But it's known from Donaldson's work that the h-cobordism theorem is fals …
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