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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 and Morse function

Let $f:\mathbb{R}^{k} \to \mathbb{R}^{k}$, and, for each $a \in \mathbb{R}^{k}$, define $$f_a(x)=f(x) + a_1x_1+...+a_kx_k$$ Prove that for almost all $a\in \Re^k$, $f_a$ is a Morse function.
Paola
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Are all latitudes of $S^2$ closed submanifolds of $S^2$?

Are all latitudes of $S^2$ closed submanifolds of $S^2$(closed subspace rather than compact)? I think it should be true, because latitudes can be viewed as $S^1$.
koch
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Exercise on linking number on Milnor's Topology from the differentiable viewpoint

It's problem 13 on the book. For disjoint compact manifold $M, N$ with no boundary in $R^{k+1}, m+n=k$, define their linking number $l(M,N)$ by the degree of mapping$\lambda (x,y)=\frac{x-y}{||x-y||}$. If M is the boundary of an oriented manifold…
Kirby Lee
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Why is the $f^{-1}(1)$ transversal with $\{1\}$?

From 4a of this released practice final it asks: Let $f : H^3 \to R^3$ be given by $x^2 + y^2 + xz$. Show that $f^{-1}(1)$ is a manifold with boundary, and determine its boundary. In the proof she states: [...] If both [$x$ and $y$] are zero,…
Dair
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Is this also a valid alternative to showing $R^2$ is not diffeomorphic to $S^1 \times R$

The solution to number 5 of this released exam seems rather sophisticated to me. I would have said: The dimension of $S^1 \times R$ is $3$ as the $\dim(S^1 \times R) = \dim(S^1) + \dim(R) = 2 + 1 = 3$ but $\dim(R^2) = 2$, as they do not have the…
Dair
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Why is support of this function compact?

In these notes under Lemma 4.10, the paper makes the claim: Lemma 4.10. Suppose $M$ is a manifold. Then there is a basis $\{U_α| α ∈ A\}$ such that: $\overline{U_{\alpha}}$ is compact and For each $\alpha \in A$ there is a smooth function $\phi_α…
Dair
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Differential vector field that vanish in the sphere.

Let $L$ be a regular closed curve on the sphere $S^{2}.$ Let $\vec{V}$ be a differential vector field in $S^{2}$ such that $\vec{V}$ is not tangent to $L.$ Show that each of the two regions determined by $L$ have at least one point where $\vec{V}$…
HelaHelheim
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Compute the integral $\int_{S^2} zdx\wedge dy$ in two ways

Compute the integral $\int_{S^2} zdx\wedge dy$ in two ways: First using spherical coordinates and second using the parameterization of the hemispheres using square roots.
user316016
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Characterization of linear flows in $\mathbb R^n$

Let $\varphi:\mathbb R\times \mathbb R^n\to \mathbb R^n$ be a flow, that is, a $C^\infty$ map such that $\varphi(0,p) = p$ for every $p\in\mathbb R^n$ and $\varphi(t+s,p) = \varphi(s,\varphi(t,p))$ for every $p\in \mathbb R^n$ and $t,s\in\mathbb…
user55268
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Connected sum of pairs

Let $M_1^m,M_2^m$ be two smooth closed connected manifolds and $N_1^n,N_2^n$ their submanifolds respectively. Choose embeddings $(D^m,D^n)\to(M_i,N_i),i=1,2$, we can use them to define $(M_1,N_1)\#(M_2,N_2)$. Is the diffeomorphism type depend on the…
Ivy
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Submanifold and Regular Value problem.

Let $Y\subset X$ a submanifold. Show that there is a manifold $Z$ with a regular value $z_0\in Z$ and a map $C^r, r\ge 1$, $f:X\longrightarrow Z$ such that $Y=f^{-1}(z_0)$.
Matematico
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Why should we consider $\sqrt{b/a}$ instead of $\frac{b}{a}$?

To paraphrase (only slightly) Guillemin and Pollack asks: Given the function $f : R^3 \to R^1$ defined by: $$f(x,y,z) = x^2 + y^2 - z^2$$ Prove that if $a$ and $b$ are both positive or both negative then: $f^{-1}(a)$ is diffeomorphic to…
Dair
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How do I know that $h(0)$ is $0$?

I'm a bit lost in Differential Topology by Victor Guillemin and Alan Pollack. The book states in Chapter 2: Before proceeding, we must resolve an ambiguity in the definition of $T_x(X)$; will another choice of local parametrization produce the same…
Dair
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Thom Transversality Theorem for non smooth manifolds?

I currently study the above mentioned theorem for jets. Just for convenience here is the statement: Let $X,Y$ be smooth manifolds and $W \subset J^k(X,Y)$ a submanifold of the k-jet space. Define $T_W = \{f \in C^{\infty}(X,Y) ~|~ j^kf \pitchfork…
JDoe
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Is true that a closed $k-$form on $T^n$, the torus, is exact if, and only if, the integral of $\omega$ over every compact submanifold is zero?

Is true that a closed $k-$form on $T^n$, the torus, is exact if, and only if, the integral of $\omega$ over every compact and oriented submanifold is zero?
L.F. Cavenaghi
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