# 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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### 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 $0$ ...
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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 ...
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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 ...
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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 (...
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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 ...
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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 ...
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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 ...
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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 $M\subset\mathbb{R}^n$, ...
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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 ...
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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 ...
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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 ...
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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 ...
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### 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 ...
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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-...
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### 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 $l$ ...
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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.)
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### 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 $k$-...
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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 ...
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### 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 ...
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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 ...
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