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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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There is no $f:\mathbb{S}^1 \times \mathbb{S}^1 \to \mathbb{S}^1$ satisfying $f(x,x)=x$ and $f(x,y)=f(y,x)$

I am trying to solve the following exercise I've found in a qualifying exam in Differential Topology: Show that there is no $f:\mathbb{S}^1 \times \mathbb{S}^1 \to \mathbb{S}^1$ satisfying $f(x,x)=x$ and $f(x,y)=f(y,x)$. I think the word "smooth"…
User
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Does "Add up" just means oriented counterclockwisely?

$f: X \to Y$ and $Z$ are appropriate for intersection theory $X,Y,Z$ are boundaryless oriented manifolds, $X$ is compact, $Z$ is closed submanifold of $Y$, and $\dim X + \dim Z = \dim Y$), $f$ is transversal to $Z$. According to the text: The…
1LiterTears
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An image manifold that is a diffeomorphic copy of $X$ adjacent to the original.

The entire content is rather drafty, but I am especially baffled with the last comment "and thus produces an image manifold that is a diffeomorphic copy of $X$ adjacent to the original." This sentence does not make sense to me, like why we suddenly…
1LiterTears
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For what values of $r$ and $b$ is $X ∩ Y$ a smooth manifold? When it is a manifold, what is its dimension?

Consider the hyperbolic paraboloid $X$ contained in $\mathbb{R}^3$, and the sphere $Y$ in $\mathbb{R}^3$ given by the equations $x^2 −y^2 =z$ and $x^2+y^2+(z−1)2=r^2.$ For what values of $r$ and $b$ is $X ∩ Y$ a smooth manifold? When it is a…
dunkindonuts
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Transverse a single point

I got very confused with understanding this theorem. So $\{y\}$ is a point, how could it be transversed by $f$? Proof: Given any $y \in Y.$ alter $f$ homotopically to make it transversal to $\{y\}$. Thank you for your help~~
1LiterTears
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$f$ is either regular or $df_x = 0$.

In the text, it says: Consider smooth functions on a manifold $X$: $f: X \to \mathbb{R}$, at a particular $x \in X$, $f$ is either regular or $df_x = 0$. So I am not certain here: if $df_x \neq 0$, then all we know here is $df_x$ is injective.…
1LiterTears
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$\mathbb{R}^k$ and $\mathbb{R}^k$ are trivially diffeomorphic.

Is this claim correct? If so, is it because identity is the diffeomorphism?
WishingFish
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What is $g$ in Guillemin and Pollack's Differential Topology?

Is it canonical immersion when it appears on Page 15, and cannonical submersion on Page 20? I never really see where it is defined, except for Page 15: "Define $G$ s that $g = G \circ$ (canonical immersion)" - which does not really defined $g$…
WishingFish
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The solution set in $\mathbb{R}^3 - \{0\}$ of $x^d +y^d= z^d$ has the form $p^{-1}(X_d)$.

I am wondering if my proof is legit? The ending looks rather soft. I don't know whether it is correct, or how to rephrase it if it is correct. Let $p: \mathbb{R}^3 - \{0\} \to \mathbb{R}P^2$ be the usual projection. Prove that the solution set in…
WishingFish
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Relate to GP 1.3.9 - Differentiating $x_{i_1}, \dots, x_{i_k}$ result span($e_{i_1}, \dots, e_{i_k}$)?

I start to think of this is question when I attempt exercise 1.3.9 on Guillemin and Pollack's Differential Topology Consider the projection $\varphi: \mathbb{R}^N \rightarrow \mathbb{R}^k$: $$(x_1, \dots, x_N) \mapsto (x_{i_1}, \dots,…
WishingFish
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Restriction of an immersion to any submanifold is still an immersion.

If $f$ is an immersion, prove its restriction to any submanifold of its domain is an immersion. Consider a submanifold $\tilde{X}$ of $X$, and take any point $p \in \tilde{X}$. Then when $d\tilde{f}_p(\tilde{x}) = 0$... I was not able to show…
1LiterTears
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The tangent space $T_{(x,x)}(\triangle)$ is the diagonal of $T_x(X) \times T_x(X).$ - Is this proof legit?

If $\triangle$ is the diagonal of $X \times X$, show that its tangent space $T_{(x,x)}(\triangle)$ is the diagonal of $T_x(X) \times T_x(X).$ Is the following proof legit? $T_{(x,x)} \Delta \subseteq \Delta \subseteq T_x X \times T_x X$ All…
WishingFish
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The image of $I$ is an open interval and maps $\mathbb{R}$ diffeomorphically onto $\mathbb{R}$?

This is Problem 3 in Guillemin & Pallock's Differential Topology on Page 18. So that means I just started and am struggling with the beginning. So I would be expecting a less involved proof: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a local…
WishingFish
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GP 1.2.10(b) The tangent space $T_{(x,x)}(\triangle)$ is the diagonal of $T_x(X) \times T_x(X).$

If $\triangle$ is the diagonal of $X \times X$, show that its tangent space $T_{(x,x)}(\triangle)$ is the diagonal of $T_x(X) \times T_x(X).$ I don't have the slightest idea on how to do this. By definition, the tangent space of $X$ at $x$ is the…
WishingFish
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Intuition behind proof that open subsets of $\mathbb{R}^n$ are spectra of their real algebras

Let $U$ be an open subset of $\mathbb{R}^n$. Given a unital $\mathbb{R}$-algebra $\mathcal{F}$, let $|\mathcal{F}|$ be the set of surjective algebra homomorphisms $\mathcal{F}\rightarrow\mathbb{R}$. I am trying to understand a proof for bijectivity…
P-addict
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