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Questions tagged [transversality]

In differential topology, transversality formalizes the idea of a generic intersection between two manifolds. It consists in asking an infinitesimal condition, namely on the tangent spaces, to be satisfied everywhere.

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Suppose that $S_1$ and $S_2$ are smooth surfaces in $\mathbb{R}^n$

(a) Let $n = 3$ and suppose that they intersect at a point $p$ and do not have the same tangent plane at that point. Show that $p$ is not an isolated point of $S_1 \cap S_2$. (b) Let $n = 4$ and ...
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Transversality condition for infinite time horizon; Pontryagin Maximum principle

I want to solve a optimal control problem with the Pontryagin-Maximum-Principle, given following differential equation: $\dot x(t)=(a+b-c(t))x-c(t)\\ x(0)=x_0$ Regarding to the conditions: $x\geq0$ ...
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The diagonals $\Delta=\{(v,v)\mid v\in V\}$ is transversal to $W=\{v,Av\mid v\in V\}$ iff $+1$ is not an eigenvalue of A

Learning Differential topology, Sorry for asking anything trivial. I am stuck in this question: Let $V$ be a vector space and let $\Delta$ be the diagonal of $V \times V$ . For a linear map $A : V ...
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Consider the real-valued function $M:=\{(x,y,z)| (2 - (x^2 + y^2)^{1/2})^2 + z^2=1\}$ defined on $\mathbb{R}^3-\{(0, 0, z)\}$.

Show that the manifold $N=\{(x,y,z)\in \mathbb{R}^3|x^2+y^2= 4\}$ is transverse to M. Identify the resulting manifold $N\cap M$. My Attempt: Pardon me for something vacuous, as I am a beginner in ...
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Transverality Condition $u_t+uu_x=-2u$

I am given the quasi-linear PDE, $$u_t+uu_x=-2u,$$ with the boundary condition $u(0,t)=e^{-t}$. I am trying to compute the transversality condition. I proceed as $$\text{det}(J)=\begin{vmatrix} \...
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Show that embeddings, diffeomorphisms, etc. are stable classes of maps

This is part of Problem 16 in Chapter 6 of Lee's Smooth Manifolds. Let $N,M,S$ be smooth manifolds. A smooth family of maps is a collection $\{F_s:N\to M \;|\; s\in S\}$ such that $F_s(x)=F(x,s)$ ...
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Tangent spaces of two transverse subspaces are transverse subspaces

I am very new to differential geometry and was thrown this very long question: Suppose that two subspaces $V$ and $W$ of $\mathbb{R}^n$ are transverse (so $\text{Span}(V,W)=\mathbb{R}^n$). Let $O$ be ...
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Definition of “transversal intersection” for piecewise linear submanifolds

I'm working with knots in the PL category. In "Surface Knots in 4-space" of Seiichi Kamada, the author states on p. 26 that the linking number of two oriented knots $K$ and $J$ in $S^3$ is the ...
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Tangent space of preimage is the preimage of the tangent space

Let $M$ and $N$ be smooth manifolds with $S\subseteq N$ a submanifold, and assume a map $f:M\to N$ is smooth and transverse to $S$. Prove that $T_p(f^{-1}(S)) = (df_p)^{-1}(T_{f(p)}S)$ for some $p\in ...
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A difficulty in understanding the proof of boundary theorem in G&P.

The theorem and its proof is given below: But I could not understand the last line in the proof in particular: Why $F^{-1}(Z)$ is a compact one dimensional manifold with boundary? And why this leads ...
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A difficulty in understanding a case in the intersection theory mod 2(p.80 Guillemin and Pollack)

The following is written just before the boundary theorem in Guillemin & Pollack : But I see that if Z is not transversal to X this is not true,why the book did not consider this case? why the ...
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A difficulty in understanding a part of a paragraph in Guillemin & Pollack p.60

I do not understand the highlighted part of the paragraph given below: Could anyone explain it for me please?
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What does the Jacobian matrix of the projection mapping for Normal bundle look like? (2.3.14 G&P)

I want to solve this question: I feel like the previous question is similar to the one given in this link: Natural projection of tangent bundle is submersion Am I correct? but what does the ...
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If V and W are linear subspaces of $R^n$, then V transversal W means just $V + W = R^n$

If V and W are linear subspaces of $R^n$, then V transversal W means just $V + W = R^n$. Could anyone give me a hint for this exercise please?
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Homotopic transverse extension of a boundary map

I am currently sitting and staring at a proof of the following theorem. Theorem. Let $X$ be a manifold with boundary and $f_0:X \to Y$ a smooth map to a manifold $Y$. Let $g:Z \to Y$ be a closed ...
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$S_1 ,S_2$ submanifolds of $\mathbb{T}^2$. Does there exists a translation $L$, such that $T(S_1)\pitchfork S_2$?

Let $\mathbb{T}^2 = [0,1]^2/\sim$ be the torus, and $S_1,S_2 \subset \mathbb{T}^2$ are smooth $1$-manifolds such that $$S_i =\dot{\bigcup}_{j=1}^{n_i} C_{j}^{i}, $$ where each $C_j^{i}$ is a $1$-...
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Showing that the diagonal of $X\times X$ is transversal to the graph of $f$. (1.5.10 Guillemin and Pollack)

The question and its answer is given below: But I am wondering, is it also correct if I showed that graph f is transversal to diagonal of $X\times X$? Also, I can not understand the general ...
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Exercise 1.5.7 in Guillemin and Pollack

The problem and its solution is given below: I am wondering if we can solve it without using 1.5.5 ?
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Transversality as an extension of the notion of regularity p.28 in Guillemin & Pollack.

The book "Differential Topology" book is explaining this in the image below: But I do not understand: In the first paragraph: The 2 statements starting from the fifth line, could anyone ...
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what is the meaning by words of the following transversality problem (GP 1.5.5)

The problem and its solution is given below: But I want to understand what is the problem saying in words, could anyone explain this for me?
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Fundamental Domain and Transversality of Vector Field in Ordinary Differential Equations

Consider the ODE \begin{align} \dot x = f(x) \tag{1} \end{align} Let $x_0$ be a hyperbolic fixed point of $(1)$. Let $V$ be a neighborhood of $x_0$ in $W^s(x_0)$, where $W^s(x_0)$ is the stable ...
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Knots and Transversality: proving a fundamental property of knot diagrams

Let $K:\mathbb{S}^1\times I\to \mathbb{S}^3$ be an isotopy of the knots $K_0 = K(\cdot, 0) $ and $K(\cdot, 1).$ Fix $v \in \mathbb{R}^3\subset \mathbb{S}^3$. I have seen in my knot theory course ...
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Every immersed submanifold can be deformed to have transverse self-intersection

Let $f : M^n \to \overline{M}^{n+k}$ be an immersion between smooth manifolds. Is it true that there exists a smooth map $F : M \times [0,1] \to \overline{M}$ such that the following conditions hold? ...
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$\epsilon$ function of a tubular neighborhood

I´m reading some notes about differential topology. In some point it says the following: if $Y\subset\mathbb{R}^N$ is a embedded submanifold and $U$ is a tubular neighborhood of $Y$, then there exist ...
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Real projective space and transversality problem

I am presented with the following problem: Given $\pi: S^3 \to \mathbb{R}P^3$, the quotient map to the real projective space. For any $t \in [0,1]$, we define $S_t=\{(x,y,z,t): x^2+y^2+z^2+t^2=1\} \...
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A direct sum relation

NOTICE : I deleted my previous version of this question because there were some bad typos. I'm reading some notes online about transversality. For the context, here let $M$ be a (orientable) ...
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Intersection Index invariant under homotopy, counterexample?

I am looking specifically at the theorem in Guillemin and Pollack Page 78, where the intersection number modulo 2 is invariant under homotopy given that X is compact, Z and X are in complementary ...
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Local equation(s) for transversal (surface) singularities

I would like to write down local equation(s) for a surface $S\subset \mathbb C^4$ whose singular locus is itself a singular curve $C$, say $C=\{z=w=x^2-y^2=0\}$, where $(x,y,z,w)$ are the coordinates ...
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Transverse Manifolds. Intersection is a Manifold again.

Suppose we have two manifolds $A,B$ in $\mathbb{R}^n$. I heard that the intersection $A\cap B$ is again a manifold in $\mathbb{R}^n$ if $A$ and $B$ intersect transversally in any point of the ...
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Determining transversality

Consider the function $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ given by $f(x,y)=(x^2+x-2y^2+1,-x^2+y^2+3y-2)$. I am trying to show that the graph of this function is transversal to the diagonal $\Delta=...
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Definition of the preimage orientation

Guillemin and Pollack give quite confusing (at least for me) definition of the preimage orientation (see below). I don't understand the part starting from the last display. Namely: How exactly does ...
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Show that $S^2$ is not diffeomorphic to the Torus.

Show that $S^2$ is not diffeomorphic to the Torus. There are lot of duplications of this question in MSE and I have seen an answer that uses Euler numbers. But let me describe how differ this post ...
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Guillemin-Pollack: application of the Transversality Theorem

I'm working on two exercises from Guillemin-Pollack which have the same flavor: (General Position Lemma) Let $X$ and $Y$ be submanifolds of $\mathbb R^N$. Show that for almost every $a\in \mathbb ...
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Proof of the General Position Lemma (an exercise from Guillemin-Pollack)

Let $X$ and $Y$ be submanifolds of $\mathbb R^N$. Show that for almost every $a\in \mathbb R^N$, the translate $X+a$ intersects $Y$ transversally. Here is my attempt to fill in the details in the ...
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When is the intersection of the hyperboloid with the sphere transverse?

For which values of $a$ does the hyperboloid $x^2+y^2-z^2=1$ intersect the sphere $x^2+y^2+z^2=a$ transversally? What does the intersection look like for different values of $a$? What I can see ...
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Intersection of non-transverse manifolds

Suppose that $X$ and $Z$ do not intersect transversally in $Y$. May $X\cap Z$ still be a manifold? If so, must its codimension still be $\operatorname{codim}X+\operatorname{codim}Z$? (Can it be?) ...
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Transversality of maps

Let $X\xrightarrow{f} Y \xrightarrow{g} Z$ be a sequence of smooth maps of manifolds, and assume that $g$ is transverse to a submanifold $W$ of $Z$. Show $f\pitchfork g^{-1}(W) \leftrightarrow g\circ ...
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Do these linear spaces intersect transversally?

Do these linear spaces intersect transversally? (For a,b, find a relationship between $n,l$ and $k$ that do so. a. $\mathbb{R^k}\times \{0\}$ and $\{0\}\times \mathbb{R^l} $ in $\mathbb{R^n}$ ...
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Transversal Intersections - Some Examples

Here is a part of an exercise from Guillemin-Pollack: Which of the following linear subspaces intersect transversally? (d) $R^k\times \{0\}$ and $\{0\} \times R^l$ in $R^n$. (Depends on $k,l,...
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Guillemin-Pollack Exercise 1.5.3: Normal Intersections [duplicate]

Let $V_1, V_2, V_3$ be linear subspaces of $\mathbb R^n$. One says they have 'normal intersection' if $V_i\pitchfork(V_j\cap V_k)$ whenever $i\ne j$ and $i\ne k$. Prove that this holds iff $$\...
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Proof of uniform matrix being transversal matrix by selecting singletons

I am doing an assignment where I have to proof that every uniform matroid is a transversal matroid. My solution is a lot simpler than in the solutions manual, and normally this means that my solution ...
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Inverse image of a sub manifold - Transversal intersection

Suppose $N,M$ are smooth manifolds and $f:N\rightarrow M$ is a smooth map intersecting transversally with a submanifold $S$ of $M$. The question is to prove that $f^{-1}(S)$ is a smooth submanifold ...
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Self intersections of a smooth closed curve being deformed

Let $\gamma_0$ be a smooth closed curve in $\mathbb{R}^2$, such that it has no self intersection. You can smoothly deform it in order to create a (still smooth) curve $\gamma_1$ that have some self ...
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General position : Why is a picewise linear embedding necessary?

I am currently studying a little bit of general position and transversality theory and I was told that it is necessary to consider at least a picewise linear embedding of two submanifolds to be able ...
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Transverse intersection of two fixed point submanifolds?

Let a finite group $G$ act smoothly on a smooth manifold $M$ and $H_1, H_2$ be subgroups of $G$ such that the set $H_1\cup H_2$ generates $G$. Moreover, assume that $$ \dim(M^{H_1})+\dim(M^{H_2})-\...
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How to derive the transversality condition of a free-endpoint optimal control problem?

I've been studying the calculus of variations and optimal control theory only for fixed-endpoint problems now. Liberzon has a somewhat complex explanation of transversality conditions, that I don't ...
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Transverse submanifolds?

Let $M_1$ and $M_2$ be submanifolds of a manifold $M$. Assume that $M_1\cap M_2$ is also a submanifold of $M$ and that dim$(M_1)$+dim$(M_2)$-dim$(M_1\cap M_2)$=dim$(M) $. Is it true that $M_1$ and $...
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Intersection of fixed point subspaces

Let a finite group G act smoothly on a smooth manifold M of dimension n. Let $H_1$ and $H_2$ be subgroups of G and assume that dim$(M^{H_1})+$dim$(M^{H_2})=n$. I have two questions: How to show that $...
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A problem about coset and transversal. [duplicate]

For a finite group $G$ and its subgroup $H$. Show that there is a subset $T$ which is simultaneously a transversal for the left and right cosets of $H$. Further ,if we consider any two partition of $...
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Transversality and regular values

Suppose we are given three smooth manifolds $V\subset U$ and $M$, and a smooth function $f\colon U\to M$. Suppose also that $\epsilon\in M$ is a regular value for $f$ (so $W=f^{-1}(\epsilon)$ is a ...