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Results tagged with differential-topology
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user 11142
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”.
22
votes
Accepted
A manifold is a homotopy type and _what_ extra structure?
You are talking about the (much studied) Poincare duality spaces. For a survey, see the very nice one by John Klein: (seems to be unpublished, but dates to April 2010).
- 96.4k
18
votes
Homeomorphisms of $S^n\times S^1$
This is false for $n=1.$ The mapping class group of the torus is $SL(2, Z),$ of which the homeomorphisms you describe are but a small part - the parabolic matrices $\begin{pmatrix}1 & n\\ 0 &1\end{pma …
- 96.4k
15
votes
Can an oriented closed $n(\geq 2)$-dimensional manifold be smoothly embedded in $\mathbb{R}^...
Every closed orientable (smooth) $n$-manifold (smoothly) embeds in $\mathbb{R}^{2n-1}.$ This is true for $n> 4$ by Haefliger-Hirsch, $n=3$ by C.T.C. Wall (Wall's paper has the reference to Haefliger-H …
- 96.4k
12
votes
Were 3-manifolds with $\sec>0$ known to be space forms before Ricci flow?
No, this was not known before Hamilton's paper. I think a Ricci-flow-free argument would still be of interest, so if you know how to do it, by all means...
- 96.4k
11
votes
Accepted
is there a diffeomorphism with only finite orbits but of infinite order?
No, there is no such animal. This is due to Montgomery (1938), see my answer to this question: Nonperiodic points of homeomorphisms of a ball
- 96.4k
11
votes
What is the weakest negative curvature condition ensuring a manifold is a $K(G,1)$?
For 1-3, yes, by the Cartan-Hadamard Theorem.5-... No. For example, every 3-manifold admits a metric of negative scalar curvature (I think this is actually true for any manifold, due to Lohkamp).
- 96.4k
10
votes
Accepted
Is a compact, connected, orientable 3-manifold with $\mathbb{Z}^K$ fundemental group uniquel...
NO, since the three-torus $T^3$ does not have this form.
EDIT if the OP really means a free product of $\mathbb{Z}$s, so the free group $F_k,$ then the answer is YES. It is a fact (see Hempel's book, …
- 96.4k
10
votes
Accepted
A question on continuous maps from Möbius to itself
I might be missing something, but for $f = g\circ h,$ where $h$ is the retraction onto the core circle (see this question), and $g$ is the rotation of the circle by, say, 1 radian there appears to be …
- 96.4k
9
votes
Can homologous submanifolds be connected by an immersed manifold with boundary?
Presumably, the answer is NO, since every manifold $K$ embeds in $\mathbb{S}^n$ of high dimension (where it is then null-homologous), but not every manifold is null-cobordant, in other words, there ar …
- 96.4k
9
votes
Extending a diffeomorphism of the sphere $S^2$ to the ball $D^3$
A combinatorial proof of Smale's theorem (actually a stronger theorem, which at the time of the paper cited was a conjecture of N. Kuiper) is given in the paper
The space of simplexwise linear homeom …
- 96.4k
9
votes
Accepted
Charts needed for an atlas
You are looking for the smallest number of open contractible sets needed to cover $M.$ This is so well studied, that it has a name: *the Lyusternik-Shnirelman category of $M$." For references, you can …
- 96.4k
7
votes
A riemannian manifold with finitely many closed contractible geodesics
Ellipsoids with almost but not quite equal axes have exactly three simple closed geodesics.
- 96.4k
6
votes
Accepted
Computing the Euler characteristic of the complex projective plane using differential topology
There is a canonical way to construct holomorphic vector fields on $\mathbb{C}P^2,$ and that way is described in Zoladek's "Monodromy Group", page 335. If you read the description, it will be pretty c …
- 96.4k
6
votes
Obstruction to a general S^1-action
It is a result of Atiyah-Hirzebruch (1970) that the $\hat{A}$ genus of a spin manifold with a nontrivial $S^1$ action vanishes, and a result of Herrera and Herrera that the same result, if the manifo …
- 96.4k
5
votes
Computing the Euler characteristic of the complex projective plane using differential topology
By the way, a very cool (to my mind) way of computing the Euler characteristic of $\mathbb{C}P^n$ is to treat it as the $n$-fold symmetric product of $\mathbb{C}P^1 = \mathbb{S}^2$ with itself. Then, …
- 96.4k