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Results tagged with differential-topology
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user 3460
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
8
votes
Accepted
Open non-parallelizable 4-manifolds
Yes to your first question; the Stiefel-Whitney classes obstruct parallelizability, even for open manifolds. So for instance a non-orientable manifold (eg a Mobius band cross R^2) is not parallelizabl …
- 19.4k
3
votes
manifolds with unusual rational cohomology rings
Doesn't Sullivan's paper On the intersection ring of compact three manifolds, Topology 1975; 14(3):275-277 characterize the cohomology ring? I think he shows that any skew tri-linear form is realized …
- 19.4k
15
votes
Homeomorphisms of $S^n\times S^1$
Some older results reduce the problem to a calculation. Browder (Diffeomorphisms of 1-Connected Manifolds, Transactions of the American Mathematical Society
Vol. 128, No. 1 (Jul., 1967), pp. 155-163) …
- 19.4k
3
votes
Examples of Self-Maps of E8-Manifold
This is more like a long comment than a real answer.
This seems like it's an algebra problem that is probably hard to solve. If you had such a map, of degree $d$, then you would get the following. C …
- 19.4k
5
votes
Open book decompositions in dimension 4
Nice question; there's not much known in general. The signature obstruction you mention goes back to Winkelnkemper (Bull. Amer. Math. Soc. 79 (1973), 45–51) and is sufficient for simply connected $4n$ …
- 19.4k
10
votes
Is $\mathrm{Diff}_0(S_g)$ torsion-free?
There's another proof of this that is well worth knowing, using the Lefschetz fixed point theorem (for the surface, not the hyperbolic plane as above). It's apparently due to Serre, and is nicely expl …
- 19.4k
2
votes
Existence of a continuous map of a disk with a given boundary image on a surface to its comp...
You basically answered your own question in the last paragraph. Take an unknotted torus, with meridian $\mu$ and longitude $\lambda$, and write $S^3- \nu(T) = A \cup B$. Then $\mu$ is null homologous …
- 19.4k
7
votes
Homotopy type of an oriented, closed, simply connected manifold
Essentially the same question has been answered; see Generalizations of the handle trading techniques. For simply-connected manifolds, Smale showed that you can get a handle decomposition that is as …
- 19.4k
8
votes
Accepted
Why does the Gluck twist on a spun knot give the standard $S^4$?
This was shown by Gluck, in the cited paper; see section 22. The basic point is explained in section 17. (What we now call) the Gluck twist will produce an equivalent knot if the circle action on the …
- 19.4k
12
votes
When a homeomorphism is a diffeomorphism w.r.t to a suitable smooth structure?
I will address your first question: Is it always possible to construct a smooth structure on M w.r.t to it $\phi$ will be a diffeomorphism?
It is not always possible, even with a change of smooth str …
- 19.4k
3
votes
Novikov-Wall non-additivity theorem with twisted coefficients
In a paper of W. Neumann, I noticed a reference to the thesis of W. Meyer, Die Signatur von lokalen Koeffizientensystemen und Faserbündeln. Neumann says that Meyer discusses some details Wall's non-ad …
- 19.4k
17
votes
Accepted
Handlebody decomposition of an open 4-manifold
There are not that many explicit handlebody pictures of exotic open 4-manifolds, because they get awfully complex in short order. The ones that I know of are in work of Žarko Bižaca from the mid-90's …
- 19.4k
7
votes
Accepted
Embedding problem for 3-manifolds attacked via 4-manifolds
Since you've asked for an opinion, my answer is also an opinion. I would say that the embedding in $R^5$ is not going to be helpful. You are seeking to take $M = \partial N_i$ and `improve' the $N_i$ …
- 19.4k
4
votes
Smooth embedding of space forms in the Euclidean space
You can certainly show that many such space forms do not embed. The simplest version would be that a lens space ($L =S^n/\mathbb{Z}_k$; here $n$ should be odd) whose fundamental group has order $k=p^r …
- 19.4k
3
votes
Wildness of codimension 1 submanifolds of euclidean space
This may be overkill but you could argue as follows for the special case of $M = \mathbb{R}^n$. A theorem of Kirby (On the set of non-locally flat points of a submanifold of codimension one. Annals, 8 …
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