Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [differential-topology]

Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

Differential topology is the field dealing with differentiable functions on differentiable manifolds. Somewhat simplified view of this field is that it describes the setting to which the notion of differentiable function can be generalized from the more familiar case of functions $\mathbb R^n\to\mathbb R^k$. Differential topology is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

7287 questions
1
vote
0 answers

Getting a smooth structure on a vector bundle over a smooth manifold

If $M^n$ is a smooth manifold, the tangent bundle $TM$ is defined to be $$TM = \bigcup_{p\in M} \{p\}\times T_pM$$ where $T_pM$ is the tangent space at $p$. In order to talk about a vector field (a map $v: M\to TM$ with $\pi(v(p)) =p$) being…
florence
  • 12,819
1
vote
2 answers

Is there a compact, connected manifold $M$ with boundary $S^2$ not diffeomorphic to $D^3$?

Is there a compact, connected, smooth 3-manifold $M$ with boundary $S^2$ not diffeomorphic to $D^3$ (the closed unit ball)? If so, what is it? The compactness condition rules out the complement of the open ball in $\mathbb{R}^3$, and the…
AnonymousCoward
  • 5,548
1
vote
1 answer

Closed hypersurfaces are global level sets?

Let $M$ be a compact ($\mathcal{C}^2$) hypersurface of $\mathbb{R}^k$. Then $M$ is the global level set of a function $f$ having $0$ as a regular value (see this related question). I think that $f$ can be constructed using the Jordan-Brouwer theorem…
Olivier
  • 3,893
1
vote
1 answer

Show $f: \mathbb{R} \rightarrow S^1$ given by $f(p) = (cos(p), sin(p))$ is a smooth map.

I have no idea how to show that $f: \mathbb{R} \rightarrow S^1$ given by $f(p) = (cos(p), sin(p))$ is a smooth map. The definition of smooth that we have been given is: A continuous map $f: M \rightarrow N$ is smooth at $p$ if for any chart $\phi: U…
user263626
1
vote
0 answers

Can someone elaborate on this proposition about creating manifolds with boundary?

In Guillemin and Pollack they state: Proposition. The product of a manifold without boundary, $X$ and a manifold with boundary $Y$ is another manifold with boundary. Furthermore: $$ \partial(X \times Y) = X \times \partial Y$$ and: $$ \dim ( X…
Dair
  • 3,064
1
vote
2 answers

Euler characteristic of a surface

I'm suppose to find the Euler characteristic of the surface $$M = \{(x,y,z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^6 =1. \}$$ I know I have to triangulate the surface and $\chi(M) = V-E+F$ with V=vertices, E=edges and F=faces, of the triangles, but…
HelaHelheim
  • 169
1
vote
1 answer

Property of non-transversal manifold

In Guillemin and Pollack they ask: Suppose that X and Z do not intersect transversally in $Y$. May $X \cap Z$ still be a manifold? If so, must its codimensions be $\text{codim}\space{X} + \text{codim}\space{Z}$? (Can it be?) Answer with…
Dair
  • 3,064
1
vote
1 answer

Local compactness of $C^k(M,N)$ strong space.

Let $M$ and $N$ be two $C^k$ manifolds with $k\geq 1$, with $M$ non compact. I know that $C^k(M,N)$ with its strong (Whitney) topology isn't metrizable and that it's a Baire space. Can I prove that it's a locally compact and/or Hausdorff space or do…
Philippe Malot
  • 2,667
1
vote
2 answers

Proof of the composition of smooth functions is smooth.

I have seen a couple questions involving the composition of smooth maps is smooth, but the proof I came up with does not look quite like the proofs given and I want to know if there is anything wrong with my proof. Let $X \subset R^k$, $Y \subset…
Dair
  • 3,064
1
vote
1 answer

Differentiable Structure of Atlas and its Relationship to Topology

If two atlases have the same differential structure (they're both $C^r$) do they necessarily have the same topology? My thought is, since an atlas induces both the differentiable structure and the topology then the answer to my question might be…
Bob
  • 1,462
1
vote
1 answer

Does the Poincaré-Hopf index theorem imply that on the sphere every tangent vector field has a singularity?

We know that $\chi(M) = 2 = \sum_{x_0 \in \text{Sing}(X)}I(x_o,X)$, where $X$ is any tangent vector and $x_0$ is a singularity. I mean, the sum is over all singularities of $X$, an arbitrary tangent vector field. Then, once this sum is equal $2$…
L.F. Cavenaghi
  • 4,651
1
vote
1 answer

Why is this map transversal

I am just learning about the concept of transversality so please be patient with me. So far I understand that a map $f$ is transversal to a submanifold $Z$ if $\text{Im}(d_xf) + T_yZ = T_yY$. Now I consider this example: $f:\mathbb{R} \to…
JDoe
  • 704
1
vote
1 answer

Are $C^r$ diffeomorphisms between $C^r$ manifolds always smooth?

I'm starting studying differential topology and I somehow arrived to the following (which I feel is wrong). If $M,N$ are $C^r$ manifolds and $f:M\to N$ is a $C^r$ diffeomorphism then $f$ is actually a $C^\infty$ diffeomorphism (*) ($r>0$) I…
Zero
  • 3,413
  • 23
  • 45
1
vote
2 answers

Degree of square map in $\mathbb{P}_{\mathbb{R}}^3$

How can I compute the degree of the map $f\colon \mathbb{P}_{\mathbb{R}}^3 \rightarrow \mathbb{P}_{\mathbb{R}}^3$ given by $f([x_0:x_1:x_2:x_3])=[x_0^2: x_1^2: x_2^2 :x_3^2]$? Clearly, a general point in the image has $8$ pre-images and the map is a…
abrax
  • 261
1
vote
1 answer

Is it true that a compact non-orientable manifold $M^n$ must have $H_{dR}^n(M)=0$?

I have seen this statement assumed to be true several times --- I just can't find a reference, and now I'm starting to suspect there can be catch somewhere.
Francisco
  • 363
1 2 3
…
40 41