# 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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### 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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### Exercise of transversality on Hirsch's Differential topology

This is the second exercise of the chapter III (Transversality), where the principal theorems are about transversality of maps and jets, and an example of how this can be used to prove density of ...
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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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### 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 ...
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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: ...
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### Show that there exists a line that intersects with all lines $l$.

Consider a line $k$ in the projective space $P(\mathbb{R}^4)=\mathbb{R}P^3$ and 3 points $A_0,A_1,A_2\in k$. Let $B$ be an arbitrary point in $P(\mathbb{R}^4)$ but not on $k$ and $l$ be a "...
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### $I_2(X,Z) = 0$ for $X,Z$ manifolds in $\mathbb S^k$ of complementary dimension

Suppose $f : X → \mathbb{S}^k$ is smooth, where $X$ is compact and $0 < \text{dim } X < k$. Then for all closed $Z \subset \mathbb{S}^k$ of dimension complementary to $X$, $I_2(X,Z) = 0$. The ...
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### If $\alpha$ is an arbitrary simple curve that cuts transversaly a compact surface the set of common points it can be infinity?

MY ATTEMPT: Let $S^2=\{(x, y, z)\in \mathbb{R}^3| x^2+y^2+z^2=1\}$ be a compact surface. Defining for every $n\in \mathbb{N}$ a regular parametrized curve $\alpha_n:\mathbb{R}\rightarrow\mathbb{R}^3$ ...
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### Orientability of a submanifold which is a preimage of a submanifold

I am reading Milnor & Stacheff, Characteristic Classes, Chapter 18. There is a short review of smooth manifolds, and there is a following statement: Suppose $f:M\to N$ is a smooth map between ...
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In Chapter 2, Section 3 of the book, most of the theorems requires the codomain $Y$ to be a manifold without the boundary and the submanifold $Z$ to be boundaryless as well. But I don't see why the ...