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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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Difficulty in understanding Guillemin & Pollack Figure 3.7 about degree of a map

I'm having some trouble understanding this picture from Guillemin and Pollack Chapter 3: In G&P chapter 3, they define $\deg(f) = I(f, \{y\})$. In the above image, they claim that they're mapping $S^1$ to a curly circle and then projecting back to…
S.D.
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Any k Symmetric functions possess a common zero (Guillemin-Pollack)

The theorem states that (call it theorem A) Any $k$ smooth functions $f_1,\dots,f_k$ on $S^k$ that satisfy the symmetry condition $f_i(-x) = -f_i(x)$, $i=1,\dots,k$, must possess a common zero. The proof is based on a theorem (call it theorem…
naive101
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A Quick Question about DeRham Cohomology

I was going through some lectures, then I realised that they compute the DeRham Cohomology of $\mathbb{R}^0$ and when they wanted to compute the compactly supported DeRham Cohomology then they used the fact that since $\mathbb{R}^0$ is a compact…
permutation_matrix
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How to show the smooth map $f : T^2 \to S^3$ is an orientation-preserving diffeomorphism?

Let $f : T^2 \to S^3$ be the smooth map of a 2-torus into $S^3$. Let $\omega$ be a closed 2-form on $S^3$. Show that $$\int_{T^2}f^*\omega = 0.$$ So apparently, if I can use the theorem on Guillemin and Pollack's Differential Topology, so I…
WishingFish
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Prove $I(Z_1, Z_2) = 0$.

Define $\mathbb{R}P^k$ as the quotient of $S^k$ by the antipodal map, with smooth structure defined so that the projection $p: S^k \to \mathbb{R}P^k$ is a local diffeomorphism. Suppose that $k$ is odd and $Z_1, Z_2 \subset \mathbb{R}P^k$ are…
WishingFish
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Prove $I(Z,X) = -I(X,Z)$.

I want to show that $I(Z,X) = -I(X,Z)$. So clearly I have two orientations for $X$ and $Z$ each. Do I discuss these four cases, each consider $I(Z,X)$ and $I(X,Z)$? Is this the correct approach and is there a less brute-force way to do so? Thank…
1LiterTears
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Why the degree of $p/|p|$ is zero on $\partial W^\prime?$

Let $W$ be a smooth compact region in $\mathbb{C}$ whose boundary con­tains no zeros of the polynomial $p$. $p$ has only finitely many roots, $z_0, \dots, z_n$ in $W$. Around each $z_i$, circumscribe a small closed disk $D_i$, making the disks…
1LiterTears
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Symmetric functions possess a common zero.

So possess a common zero, means $\exists x \in S^k: f_i(x) = 0$, right? Then I could not follow this brief proof - what is the corollary? Because the information in the proof is so little, I couldn't even guess which. Thanks =) Theorem. Any $k$…
1LiterTears
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$W_2(f,0) = \frac{1}{2} \# f^{-1}(l)$?

There must be some rather straight forward reason for $W_2(f,0) = \frac{1}{2} \# f^{-1}(l)$ but I really get stuck with why. Could someone help me out?
1LiterTears
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Winding number is $1$ or $0$, depending whether inside or outside.

We know that $W_2(f,z) = \#F^{-1}(z)$ mod $2$. That is, $f$ winds $X$ around $z$ as often as $F$ hits $z$, mod $2$, where $\partial F = f$. Then the text says $W_2(X,z)$ must be $1$ or $0$, depending whether $z$ lines inside or outside of $X$. To…
1LiterTears
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mod $2$ self-intersection number

Let $X$ be submanifold inside $Y$. Mod $2$ self-intersection number $I_2(X,X)$ has some strict constrains: Since $X$ and $X$ need to have complementary dimension, $\operatorname{dim}X = \frac{1}{2}\operatorname{dim}Y$. $X$ is a compact…
1LiterTears
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Transversal functions are smooth?

This sounds intuitively true. However, I have some counter claims: Although transversal is defined on smooth manifolds, which implies the image of $df_x$ is smooth. But this does not say if the function $f$ itself is smooth, and the existence of…
1LiterTears
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Let $f:S^1\rightarrow \mathbb{R}^3$ be a smooth map with $df_x \neq 0$, for all $x \in S^1$.

Let $f:S^1\rightarrow \mathbb{R}^3$ be a smooth map with $df_x \neq 0$, for all $x \in S^1$. Show that there is a plane $C$ through the origin of $\mathbb{R}^3$ such that $p\circ f:S^1\rightarrow C$ has $d(p\circ f)_x \neq 0$, for all $x \in S^1$,…
dunkindonuts
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Proving that the circle $S^1$ is a 1 Dimensional Manifold.

Claim: Show that the circle $S^1$ is a one-dimensional manifold. Proof: $(x,y)$ lies on the upper semi-circle, i.e., $y>0$. The upper semi-circle, $V$ is open in $S^1$ containing $(x,y)$, and $U = (-1,1)$ is an open interval in $\mathbb{R}^1$.…
Saikat
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Finite set on compact manifolds

I feel blocked with this claim - it sounds intuitively true, just thinking as a jellyfish entering a real line, the intersection of her legs with the real line is certainly finite since the jellyfish is compact - but I stuck with why. $X$ and $Z$…
1LiterTears
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