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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”.
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
22
votes
Very particular kind of 4-manifolds. Classification
I would say no. If M is simply connected, then it is contractible and hence determined topologically by its boundary. But there's no current smooth classification; the case when the boundary is $S^3$ …
- 19.4k
20
votes
Accepted
Is the normal bundle of a torus trivial?
You should be able to prove that the normal bundles in codimension $2$ are trivial as well. This is a little harder than codimension $1$; you need to know that such bundles are determined by their Eul …
- 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
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
15
votes
Accepted
What are Kirby diagrams of candidate exotic 4-manifolds?
There is no comprehensive list in the format you ask about; you will probably want to look at the original papers. Searching for "exotic" and "4-manifold" on Mathscinet gives > 100 responses, and pro …
- 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
12
votes
The convex hull of a manifold whose cobordism class is trivial
Implicit in the other responses is the fact that if $M$ bounds a convex manifold $W$, then $W$ is contractible and so M has the homology of a sphere. So any null-cobordant manifold that is not a homol …
- 19.4k
11
votes
Accepted
Smooth structure on direct product
Extending Michael Albanese's answer above, $M \times N^k$ will never be smoothable. For if it were then choose a point $p\in N$ and a chart U around $p$. Then $M \times U$ is an open subset of $M \tim …
- 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
9
votes
Smooth complex projective surface as the total space of a Serre fibration
I don't think so. (There are the exceptions you mention: the two $S^2$ bundles over $S^2$.) By Thomas's comment, let's assume that d = 2.
If M is a fibration of a manifold with finite complexes for b …
- 19.4k
9
votes
Accepted
Surgery along an embedded surface in a 4-manifold
For $g\geq 2$, this construction doesn't yield anything new. In this situation, any self-diffeomorphism of $\Sigma_g \times S^1$ extends to a self-diffeomorphism of $\Sigma_g \times D^2$, and hence yo …
- 19.4k
9
votes
Godbillon–Vey invariant and leaf space of foliations
You should check out Thurston, William, Noncobordant foliations of $S^3$. Bull. Amer. Math. Soc. 78 (1972), 511–514.
He constructs foliations with arbitrary real-valued GV invariants. In this paper, h …
- 19.4k
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
8
votes
Generalizations of the handle trading techniques
Yes; see Smale, On the structure of manifolds (Amer. J. Math. 84 1962 387–399) where it's shown that in high dimensions, you can eliminate handles under various connectivity assumptions. The h-cobordi …
- 19.4k