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Results tagged with gt.geometric-topology
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user 1465
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).
89
votes
Accepted
Can every manifold be given an analytic structure?
(similar to Mariano's post)
Q1: no. There are topological manifolds that don't admit triangulations, let alone smooth structures. All smooth manifolds admit triangulations, this is a theorem of White …
- 44.4k
29
votes
Classification problem for non-compact manifolds
In a sense you more or less know what non-compact manifolds must look like provided you have a classification of the compact manifolds. The starting point is the basic observation (Whitney) that a no …
- 44.4k
28
votes
Accepted
Can we decompose Diff(MxN)?
The homotopy-type of the group of diffeomorphisms of a manifold are fairly well understood in dimensions $1$, $2$ and $3$. For a sketch of what's known see Hatcher's "Linearization in three-dimension …
- 44.4k
28
votes
Square roots of $\mathbb R^{2n}$
I'm pretty certain the answer is no provided $n \geq 3$. Let $X$ be the Whitehead manifold: http://en.wikipedia.org/wiki/Whitehead_manifold
It's a contractible open 3-manifold which is not homeomorp …
- 44.4k
28
votes
What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$
No, not really. In dimension 4, for example, an open subset of $\mathbb{R}^4$ can be homeomorphic to $\mathbb{R}^4$ but not diffeomorphic, as there are exotic smooth $\mathbb{R}^4$'s that embed smoot …
- 44.4k
26
votes
Accepted
Strong Whitney embedding theorem for non-compact manifolds
Regarding question 1, yes you can always ensure the image is closed. You prove the strong Whitney by perturbing a generic map $M \to \mathbb R^{2m}$ to an immersion, and then doing a local double-poi …
- 44.4k
25
votes
Is the universal covering of an open subset of $\mathbb{R}^n$ diffeomorphic to an open subse...
The answer is no, and there is a counter-example in dimension $4$.
A theorem of Whitney and Massey states that the total space of a disc-bundle over a non-orientable surface $\Sigma$ embeds in $S^4$ …
- 44.4k
25
votes
Classification of homology 3-spheres?
Certainly. There's a general description of all compact 3-manifolds now that geometrization is about.
So for homology 3-sphere's you have the essentially unique connect sum decomposition into primes. …
- 44.4k
24
votes
Examples of non-diffeomorphic smooth manifolds with diffeomorphic tangent bundle
For $k = \infty$ (a continuum, to be precise), the continuum of non-diffeomorphic smooth structures on $\mathbb R^4$ would suffice. The tangent bundle of any $\mathbb R^4$ is trivial (since $\mathbb R …
- 44.4k
22
votes
Accepted
What manifolds are boundaries of euclidian spaces ?
$N$ has to be a homotopy-sphere. So as long as it's dimension isn't $4$, there's a proof that it has to be the standard $S^{n-1}$.
These arguments appear in the Kosinski book on smooth manifolds. T …
- 44.4k
22
votes
Are there any very hard unknots?
I believe it's not known whether or not the uniform electrostatic charge potential function (as studied by Bryson, Freedman, He and Wang: Möbius invariance of knot energy) has any critical points othe …
- 44.4k
22
votes
How to see isometries of figure 8 knot complement
If you're interested in the involution only defined on the complement, Igor's answer does a fine job.
But the involution extends to an involution of $S^3$ and perhaps you'd like to see that?
I thi …
- 44.4k
20
votes
Accepted
Is there a 2 component link with full symmetry?
I think there is a non-hyperbolic link that does the job.
The link that I'm thinking of could be called the splice of two Bing doubles of a figure-8 knot. Another way to describe this link is to sta …
- 44.4k
20
votes
Accepted
Pseudoisotopy in low dimensions
When $M$ is a compact $2$-manifold, with or without boundary, $P(M)$ is known. When $M$ is a 3-manifold there's bits and pieces known, especially once you get to more fine detail like pseudo-isotopy …
- 44.4k
19
votes
Applications of knot theory
I think historically one of the big formal motivations for knot theory were things like the Brieskorn varieties, i.e. looking at solutions to equations of the form
$$z_1^{p_1}+z_2^{p_2}+\cdots+z_n^{p …