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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.

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Transversality at a single point. A stable class.

Claim in a Differential Topology lecture: Consider the collection $G$ of maps from $\mathbb R→\mathbb R^2$ which intersect the $x$ axis transversally at a single point. Then $G$ is a stable class. My Problem: Choose a curve that passes through…
Jale'de jaled
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connection between affine map and differential

Assume that exists $T\in\hom\left(\mathbb{R}^{k},\mathbb{R}^{m}\right)$ such that $\left(Df\right)_{a}=T$ forall $a\in\mathbb{R}^{K}$. Prove that exists $T\in\hom\left(\mathbb{R}^{k},\mathbb{R}^{m}\right)$ such that…
Yechiel Merzbach
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Extension of a self-diffeomorphism of unit ball

Let $B$ be the unit open ball in $\mathbb R^n$ and $\phi$ a self diffeomorphism of $B$. Can we find a self diffeomorphism $f$ of $\mathbb R^n$ such that $$ f=\phi $$ on $B$?
Totoro
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The tangent space to the intersection is the intersection of the tangent spaces; A follow up.

I was reading this proof by Henry T Horton for the problem below: Let $X$ and $Z$ be transversal submanifolds of $Y$. Prove that if $y \in X \cap Z$, then $$T_y(X \cap Z) = T_y(X) \cap T_y(Z).$$ Linked here The tangent space to the intersection is…
user486995
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Number of pre-images of a regular value

I am trying to do the following exercise from Hirsch : Let $f:S^1 \rightarrow \mathbb{R}$ be a $C^1$ continuous map and $y$ a regular value . Then $|f^{-1}(y)|=2n$, where $n\in \mathbb{N}.$ Now I was able to prove that we have …
Someone
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Solid torus with the boundary identified

So I have a solid torus, with the boundary of it ($T^2$) identified. The way I see it is: Since the boundary points are all identified, this manifold then first becomes a "chubby bagel." We can then separate the center of the bagel and deform it to…
user134070
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Exercise: show that if $U$ is a connected open subspace of $ R^2$, then $U$ is path connected.

Show that if $U$ is a connected open subspace of $ R^2$, then $U$ is path connected. The idea was to show that given $x_{0} \in U$, the set of points that can be joined to $x_{0}$ by a path in $U$ is open and closed in $U$, however I have not been…
Curious
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Sards theorem for polynomial

I'm having some struggles with an aspect about something apparently trivial about Sard's theorem, but couldn't find anything online. Let $f$ be a polynomial. According to Sard's theorem, the image $f(Z)$ of the set of critical values $$Z = \{a \in…
Nicola Zaugg
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Every Closed Hypersurface is Orientable

So I would like to show that any closed hypersurface (a smooth, compact, and boundaryless manifold of dimension $n$ embedded in $\mathbb{R}^{n+1}$) is orientable. This is an exercise in Guillemin and Pollack's book and the book suggests that the…
rubikscube09
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When are Linear Operator and Identity transversal in a Vector Space?

Let $A:V \to V$ a Linear Operator on the Vector Space $V$ and $Id:V \to V$ the Identity map, when is $A$ transverse to $Id$ ? My intuition says that $A$ may not have $1$ as eigenvalue but I didn't find out if it is really true Since Two maps $f: X…
BraQuiet
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Gluing two copies of $S^3 \times D^4$ along their boundary

I am trying to visualize a seven sphere here: http://www.youtube.com/watch?v=II-maE5HEj0. What is $(u,v)\to(u, u^hvu^j)$ in the attached picture? Are quaternion maps commonly used to glue boundaries of topological spaces? How do we know that this…
The Substitute
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Connected and compact subset of $R^2$ not a smooth manifold and smooth manifold w/ boundary in $R^2$

Give an example of a connected, compact subset of $R^2$ that is neither a smooth manifold nor a manifold with boundary. Appreciate any help!
Johan Fredrik Agerup
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I do not understand the definition of Hopf fibrations?

How i go about understanding why Hopf fibration is a map,i did not understand the 3-sphere to 2 sphere concept?Could you please explain??
jai ramanathan
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Smooth Map $f: S^k \to \mathbb{R}^n$ Extends to the Disc

I am helping some students study for an exam. We have a two part problem. The first part is to prove that any smooth map $f: S^k \to \mathbb{R}^n$ extends to a smooth map $F: D^{k + 1} \to \mathbb{R}^n$. We tried to define $$ F(x) := |x|f\left(…
J126
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Tangent spaces of compact spaces

In a recent discussion of tangent spaces, it was noted that tangent spaces to a manifold are not compact because by definition they are vector spaces. I was curious as to whether tangent spaces to compact manifolds are always non-compact. It would…
Freedom
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