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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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How to show that is a submanifold or How to derivate the determinant function?

I am trying to show that the space of $2\times 2$ matrix with rank equals $1$ is a submanifold of $\mathbb{R}^4 - \{0\}$ whoose the dimension equals $3$. To do this, I have defined $\det : \mathbb{R}^4 \to \mathbb{R}$, then the result will follows…
L.F. Cavenaghi
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Proving that $I \times I$ is not a manifold

I want to prove that $I \times I$ is not a manifold where $I=[0,1]$. But I believe this is false. Here we will denote closed unit disc as $\overline{D}^2$.Since we can take a homeomorphism $\phi : I \times I \to \overline{D}^2$ and then if $\{\psi…
happymath
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Regular values and maps of degree 0

There is an elementary phenomenon in differential topology, that I've never quite understood: It is well known that the mapping degree (Brouwer degree) $\operatorname{deg}(f) = \operatorname{deg}(f;y)$, $y \in N$, of a smooth map $f \colon M^n \to…
difftop
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Showing $\mathbb R^n$ is a smooth premanifold

I am attempting to fill in a proof that $\mathbb R^n$ is a smooth premanifold and its smooth functions are what one would expect: the infinitely differentiable functions from $\mathbb R^n$ to $\mathbb R$. I have the following. Pick a basis $\mathbf…
Benjamin
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Exercise 1.8.6 - Differential topology (Guillemin and Pollack)

Here the problem : 1.8.6 - A vector field $\vec{v}$ on a manifold $X$ in $\mathbb{R}^N$ is a smooth map $\vec{v}:X \to \mathbb{R}^N$ such that $\vec{v}(x)$ is always tangent to X at x. Verify that the following definition (which does not explicitly…
user230283
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Fixed points of diffeomorphisms: eigenvalues of the pushforward

I want to answer this question: Let $M$ be a smooth manifold, and let $f: M \rightarrow M$ be a diffeomorphism. Let $\mathrm{Fix}_f$ be the fixed points of $f$, and suppose that $x \in \mathrm{Fix}_f$ is not isolated. Show that $df_x$ has an…
user31800
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Restriction of the projection from compact manifold onto hyperplane is a smooth embedding

Problem: Let $M \subset \mathbb{R}^{n+1}$ be a compact submanifold ($\dim M=k$) and $n \geq 2k + 1$. Show that, for the projection $\pi : \mathbb{R}^{n+1} \longrightarrow H^n$ onto a suitable hyperplane of $\mathbb{R}^{n+1}$, the restriction $\pi|M…
user178318
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Difficulties on proof of $\epsilon $-Neighborhood Theorem.

I'm trying to proof the $\epsilon$-Neighborhood Theorem from Guillemin and Pollack's book. I'm not good at topology, and I'm having some difficulties to completely understand the theorem. For the proof is necessary do some exercises, and the first…
Keith Nale
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Is the disjoint union of submanifolds a submanifold?

Let $M$ a manifold and $X \subset M$. Let $N \subset X$ such that $N, X \backslash N$ are submanifolds of $M$. Can I conclude that $X$ is a submanifold of $M$?
ted
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Thom-Pontryagin construction of manifolds with boundary

Thom-Pontryagin construction gives the 1-1 correspondence between framed cobordism classes of $k$-dimensioanl sub-manifolds of $S^{n+k}$ and homotopy classes of maps from $S^{n+k}$ to $S^n$. Are there any analogous theorem for the $k$-dimensional…
Cobordism123
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Suppose that $Z$ is a hyperspace in the oriented manifold $Y$, as sub manifold of codimension $1$. Prove that following statement are equivalent

Suppose that $Z$ is a hyperspace in the oriented manifold $Y$, as sub manifold of codimension $1$. Prove that following statement are equivalent a) $Z$ is orientable b) There exists a smooth field of normal vector $\vec {n}(z)$ along $Z$ in $Y$ c)…
Diane Vanderwaif
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Prove that the Borsuk-Ulam theorem and following ttheorem are equivalent

Prove that the Borsuk-Ulam theorem and following ttheorem are equivalent Borsuk-Ulam theorem: Let $f:S^k \to R^{n+1}$ be a smooth map whos image does not contain the origin, and supposed that $f$ satisfies the symmetry condition $f(-x)=-f(x)$ for…
Diane Vanderwaif
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Can a smooth function $f\colon\partial D^n\to\partial D^n$ be extended to a smooth function $\hat{f}\colon D^n\to D^n$?

Suppose for $n\geq 1$, you have a smooth map $f\colon S^{n-1}\to S^{n-1}$. Viewing $S^{n-1}=\partial D^n$, is it possible to extend $f$ to a smooth map $\hat{f}\colon D^n\to D^n$, $D^n$ being the closed $n$-ball? I noticed it extends it to the…
Yong Pan
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Show that if $f_0$ is Morse in some neighborhood of a compact set $K$, then so is every $f_t$ for $t$ is sufficiently small.

Suppose that $f_t$ is a homotopic family of function on $R^k$. Show that if $f_0$ is Morse in some neighborhood of a compact set $K$, then so is every $f_t$ for $t$ is sufficiently small. I know that $f_t$ is a homotopic family of function on $R^k$,…
Diane Vanderwaif
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A question from Milnor's "Topology from a differentiable viewpoint"

Milnor's "Topology from a Differentiable Viewpoint" says the following: Let $f:M\to N$ be a smooth mapping, where $M$ is $m$ dimensional and $N$ is $n$ dimensional. Moreover, $m\geq n$. If $y\in N$ is a regular value, both for $f$ and the…
user67803
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