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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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Transversality for Morse homology on manifold with boundary

I'm trying to make my own argument for a situation where $X$ is a smooth manifold with boundary and $f$ a Morse function. The vector field $V=-grad f$ and we denote $D_p, A_q$ as the unstable manifold ...
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How does an intersection survive through (generic) perturbation?

I am looking for the proof of a folklore statement which I know (or heavily suspect) to be true, but haven't been able to find written down yet. I have a (symplectic) manifold $M$ of dimension $2n$, ...
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Typo in Lee's Smooth manifold Theorem 6.30?

The theorem is the statement about transversality; the important bit of the theorem is the statement: Let $F:N\rightarrow M$ be a smooth map transverse to an embedded submanifold $S\subset M$. Then $...
Chris's user avatar
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Confusion about whether or not an embedded submanifold can be written as $f^{-1}(0)$ for some $f:M\rightarrow \mathbb R$

Let $M$ be a smooth manifold, then this post here seems to indicate that embedded submanifolds (that is not an open submanifold) need not be $f^{-1}(0)$ for some smooth function $f:M\rightarrow \...
Chris's user avatar
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Rigorous proof of monotonocity of points on a transversal

I am trying to rigorously prove that if we have a solution to the equation $x’ = F(x)$ and this solution crosses the transversal $\Gamma$ with parametrisation $\Gamma = a + s\vec v$ at times $t_1 < ...
Jayden's user avatar
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Proving a parametrized function cannot generically cover a curve

Let $D$ be a set of infinitely smooth (in $C^\infty$) functions from $\mathbb{R}^2$ to $\mathbb{R}$ that are strictly increasing in both arguments. Let $C$ be a compact subset of $\mathbb{R}^N$ and $...
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What is a transversal intersection? Can it be explained without tangent spaces and tangent bundles?

What is a transversal intersection? Can it be explained without tangent spaces and tangent bundles? Background: Transversal intersection was used to explain If the interior of two convex manifolds ...
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Proof that intersection of manifolds is submanifold

I need some help with understanding this proof. I have 2 questions: What are those functions $f$,$g$? I mean what is the idea behind them? Why do we consider $f^{-1}(0)$ exactly? Where does this $0$ ...
bobby shmurda's user avatar
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Intersection of submanifolds, cup products, and Poincaré duality

Recently I have been thinking and inquiring about how "cup products are dual to intersection of submanifolds", and wanted to verify whether the following is accurate (and to find a source ...
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preimage of oriented submanifold under transversal map is orientable

Consider the map $f: M \to N$, transversal to the regular submanifold $S \subset N$. I.E. $df (T_x M) \oplus T_{f(x)} S = T_{f(x)} N$. We know that $f^{-1} S$ is a regular submanifold. Also we know ...
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When is a ray transversal to a hypersurface? Stuck on a step in a proof of the Jordan-Brouwer separation theorem

$\newcommand{\d}{\mathrm{d}}$I am casually reading Guillemin and Pollack's book "differential topology", which tries its best to present the core theorems with a minimum of machinery. There ...
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About the smoothness of a map

I am currently doing an exercise and I want to show my solution so far and ask for an hint about the last part. Exercise: Let $M$ and $N$ be smooth manifolds and let $S \subset M \times N$ be a ...
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Understanding 3-manifold retraction to graph

I am reading On Fibering Certain 3-Manifolds by Stallings. It's a short paper, and I think I understand at a high level what's happening, but there are a few technical details I don't quite get. This ...
Hempelicious's user avatar
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Is the set of continuous maps on $[0,1]$ with finitely many roots open in the $C^0$ topology?

Let I would like to show that the set $$S=\{f\in\mathcal{C}([0,1],\mathbb{R}):f\text{ has finitely many zeros}\}$$ is open. By considering $f_n(x)=\frac{x}{n}$, we see that $S$ is not closed. By ...
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The Transversality Theorem for C^r manifolds?

I am studying The Transversality Theorem introduced in Sect. 2.3 of the book "Differential Topology" by Guillemin, V and Pollack, A. It seems that the manifolds and submanifolds considered ...
rubik's user avatar
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The Transversality Condition is Generic

Let $M$ and $N$ be submanifolds of $\mathbb{R}^n$. While I understand that the transversality condition $M \pitchfork N$ is stable (assuming $M$ is compact), I want to show the following property (...
LiminalSpace's user avatar
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About the transversality and number of points intersecting a surface in a sequence of half lines starting in a compact surface.

In the curves and surfaces (Second edition) book from Montiel-Ross, we have the next proposition and proof: Proposition: Let $R^{+}$ be a straight line whose origin is at $\mathbb{R}^3-S$, which ...
Tomas Rodriguez's user avatar
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3264 Example: X non-smooth gives dimensional transversality with intersection in reduced point, but not generic transversality

In Eisenbud-Harris' 3264 and All That, on page 33 they state: Subschemes $Y$ and $Z$ of $X$ have generic transversality iff dimensional transversality and each connected component of $Y\cap Z$ has a ...
locally trivial's user avatar
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Trouble understanding transversality

I'm reading "Differential Topology" by Guillemin, V and Pollack, A. While reading the chapter about transversality, I got through this theorem :https://i.sstatic.net/5wYBo.jpg (I'm not ...
Carlos Cabezas's user avatar
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Transversal intersection of one-dimensional submanifold with $S^{n-1}$

I need some help regarding the following exercise Let $f:\mathbb{R}^n \to S^{n-1}$ be smooth with $f|_{S^{n-1}}=id_{S^{n-1}}$. Show there exist a one-dimensional submanifold $M\subset\mathbb{R}^n$, ...
Schrödinger's cat's user avatar
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Regular value and transversal intersection of submanifolds

I'm not sure how to solve the following exercise. Let $M,N$ be smooth manifolds and $f:M\to N$ smooth. Show that $y\in N$ is a regular value of $f$ iff the submanifolds $G(f) = \{(x,f(x))\in M\times N ...
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Hyperplane such that intersects transversality

Let $M$ be a submanifold of $\mathbb{R}^n$ show that exists a hyperplane $H$ such that intersects transversaly t $M$ First I define $F:M \times \mathbb{S}^{n-11} \rightarrow \mathbb{R}$ given by $F(x,...
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On bounded solutions of a fourth-order linear ODE

Consider the fourth-order linear ODE $$ \label{eq1} v^{(4)} + \frac{-C_2 - 2\alpha \phi}{C_4}v'' + \frac{4\alpha \phi'}{C_4}v' + \frac{k_1 + 3k_3\phi^2 -2\alpha \phi''}{C_4}v = 0. $$ Without getting ...
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Trying to understand transversality

In Jaco's paper "Heegaard Splittings and Splitting Homomorphisms", he defines for a map between a surface and a bouquet of circles the notion of 'transverse to a point x', which is that the ...
nl08's user avatar
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Transverse Intersection of Two Smooth Surfaces

I'm studying for preliminary/qualifying exams, and came across the following problem: Suppose that $S_{1}$ and $S_{2}$ are smooth surfaces in $\mathbb{R}^{3}$ that intersect at a point $p$ and do not ...
LiminalSpace's user avatar
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Regarding Thom's Jet Transversality Theorem proof from Golubitsky-Guillemin (how the Parametric Transversality Theorem is applied)

I am trying to follow the Thom's Transversality Theorem proof from Golubitsky-Guillemin "Stable mappings and Their Singularities" (theorem 4.9, the jet version). The proof is fairly long and ...
Math Dealer's user avatar
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Milnor, Lectures on h-cobordism theorem: Lemma 5.9

In lectures on the h-cobordism theorem, Milnor writes of an isotopy $h_t:\mathbb{R}^n\to\mathbb{R}^n$, with $\mathbb{R}^n=\mathbb{R}^a\oplus\mathbb{R}^b$ in his notation, the following lemma: The ...
horned-sphere's user avatar
3 votes
0 answers
251 views

The transversal pre-image of a manifold with boundary

I am reading "transversal pre-image of a manifold with boundary" from Differential Topology by M. Hirsch. I have some confusion regarding Theorem 4.2. on the page 31. Notice that the first ...
Random's user avatar
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Coordinate representation of the local transversal

Let $S\subset M$ be compact embedded k dimensional submanifold,the embedding $\phi:B(0) \subset \Bbb{R}^{n-k}\to M$ be local transversal of $S$ at $p$(That is $\phi$ transversal to $S$ and $\phi^{-1}(...
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Loop in $S^2\times S^1$ with an orientable compact surface removed

Let $S$ be a closed orientable surface embedded in $S^2\times S^1$ assume that the projection of the surface $S$ on $S^1$ is not a point. I am trying to prove that exist a smooth loop $\gamma: S^1\to ...
Ludovico M. Dziecielski's user avatar
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Existence of a local transversal section of a vector field

I am trying to prove the existence of a local transversal section, i.e. Let $U\subset\mathbb R^n$ be an open, $X:U\to \mathbb R^n$ a vector field of class $C^k$ and $p\in U$ a regular point of $X$, ...
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intersecting *semi-riemannian* manifolds

I read this interesting post: Intersection of manifolds. In geometry/topology one can consider intersection of manifolds (which need not be a manifold). I'm curious if one can do this in Semi-...
zeta space's user avatar
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Morse function induced on fibered product

Let $A,B$ and $C$ be three smooth manifolds. Suppose that $F:A\to C$ and $G:B\to C$ are smooth and transverse functions, making the fibered product $$S=A\underset{F,C,G}{\times}B=\{(x,y)\in A\times B\,...
Balloon's user avatar
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Is there any basis for ‪Transversal‬ matrix space?

Given That An N-By-N Matrix A, Is A Set Of Numbers With One From Each Row And Each Column Is Called A Transversal Matrix Given that an n-by-n matrix A, is a set of numbers with one from each row and ...
Amirhosein Mohammadi's user avatar
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67 views

Transversality of stable and unstable manifolds along a homoclinic orbit

Consider the system $$ \begin{cases} \dot{x}=f(x,\alpha), \\ \dot{\alpha}=0, \end{cases} $$ with $(x,\alpha)^T\in\mathbb{R}^{n+1}$. The system has a homoclinic orbit at $\alpha=0$ and we assume ...
Minonimo's user avatar
3 votes
1 answer
176 views

Intersection of lines with submanifold

I come across an interesting problem below which confuses me a lot. Let $L$ be a submanifold of $\mathbb{R}^n$ with codimension > 1 . Prove that: (1) if $x$ $\notin L$, for almost every line $l$ ...
Mark Joe's user avatar
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$L$ is submanifold of $\mathbb{R}^n$ with $\dim L\leq n-2$. [duplicate]

Suppose $L$ is submanifold of $\mathbb{R}^n$ with $\dim L\leq n-2$. Prove that If $x\notin L$, then for almost every line $l$ passing $x$, $l \cap L=\phi$; If $x\in L$, then for almost every line $l$ ...
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Transversality just on a single point but not in any neighborhood

I just learned the concept of Transversality. Briefly, let $f:M \to N$ be a smooth map and $A \subset N$ be a submanifold. If $K \subset M$ we write $f \pitchfork_K A$ to mean that $f$ is transverse ...
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John Lee's proof of Transversality Theorem

In John Lee's proof, I cannot understand why the equality holds in (6.9). (and I think only subset relation is needed in the proof.)
Dyne Simian's user avatar
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1 answer
473 views

How to show that the set of hyperplanes that are transverse to some projective variety is Zariski-open?

Let $V$ be a analytic/algebraic variety in $\mathbb{CP}^n$, that is $V$ is the vanishing locus of some homogeneous polynomial. Let $d = \dim V$. Everyone knows$^\mathrm{TM}$ that a "generic $k$-...
Carlos Esparza's user avatar
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question about global nonzero section of bundle

Suppose $M$ a manifold with dimension $m$ and $E$ a vector bundle whose based manifold is $M$. Assume $\mathrm{dim} E$ is $2m+1$, by Transversality theorem, seems I can obtain a global section without ...
taiat's user avatar
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3 votes
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238 views

Dropping closed condition in Transversality Theorem

Most theorems stating that transversality is an open condition look like: We have manifolds $M$, $N$, and a submanifold $A\subseteq N$. Then assuming $A$ is closed in $N$, in the appropriate topology ...
horned-sphere's user avatar
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Understanding $T_{p}\mathbb{RP}^{2}$ ,tangent planes of his opens sets and transverse

I have to find two submanifolds $N_{1}$ and $N_{2}$ of dimension 1 of $\mathbb{RP}^{2}$ the real projective space such that $N_{1}\cap N_{2}\neq \emptyset$ and $N_{1}\pitchfork N_{2}$ . For this I ...
angie duque's user avatar
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163 views

Transversality if and only if a regular value

Let $f:M^{m}\rightarrow N^{n}$ a differential application and $S^{k}$ a submanifold of $N$. We say that $f$ is transversal to $S$ in $p\in f^{-1}(S)$ if $df_{p}(T_{p}M)+T_{f(p)}S= T_{f(p)}N$. If $S^{0}...
angie duque's user avatar
1 vote
1 answer
526 views

Transversality in $2$-manifold

Let $X, Y$ be $2$-dimensional smooth manifolds without boundary, and $P$ be a disconnected $1$-dimensional boundaryless embedded submanifold of $Y$. Let $f\colon X\to Y$ be a smooth map that is ...
Random's user avatar
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How is The Number of Self-Intersections of a Compact Connected Riemann Surface Embedded in a Complex Surface defined?

Complex Algebraic Foliations, Book by A. Lins Neto and Bruno Scárdua, Chapter 3, Page 82. Let $M$ be a Complex Manifold of complex dimension two (i.e., $M$ is a Complex Surface). Let $S$ be a Compact ...
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Minimal sets, Perfect sets, Exceptional sets and Foliations

The first image is taken from Geometric Theory of Foliations, Book by A. Lins Neto and César Camacho, Chapter 3, Page 53. The second image is taken from Geometry, Dynamics And Topology Of Foliations: ...
Neil hawking's user avatar
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why does $ f| int(D)$ have only finitely many intersections with the knot?

We define a compressing disk of the knot $K$ (a smooth emmbedding from $S^1$ to $S^3$) to be a smooth map $f: D → S^3$ such that $f|∂D = K$ and such that $f| int(D)$ is transverse to $K$. Then $f| int(...
xupeng duan's user avatar
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121 views

The set of pseudo-gradient vector fields with the Smale property and a given set of trajectories is open (Proposition 3.4.3 in Audin-Damian)

In Audin Damian's proof of the Invariance of Morse Homology from vector field and Morse function, p.71, there's a proposition, where the Smale property is that the stable and unstable manifolds of the ...
horned-sphere's user avatar
1 vote
2 answers
147 views

Transversality Condition and the Proof of Smale Theorem (Audin-Damian), Lemma 2.2.8 (Part 2)

In Audin and Damian, p.44-45, there seems to be a claim that one can prove transversality without directly showing the tangent spaces span the tangent space of the ambient manifold. In particular, in ...
horned-sphere's user avatar