Mathematics Stack Exchange
Mathematics Stack Exchange

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.

Definition. Let $f\colon M\rightarrow N$ be a $C^1$ map between two manifolds and let $A$ be a submanifold of $N$. One says that $f$ is transversal over $A$ if and only if for all $x\in f^{-1}(A)$, $\mathrm{d}_xf(T_xM)+T_{f(x)}A=T_{f(x)}N$.

For example, a submersion between two manifolds is transversal over any submanifold of its goal.

References.

  • M. Hirsch, Differential topology, Springer, 1976.

  • V. Guillemin, A. Pollack, Differential topology, American Mathematical Society, 1974

170 questions
10
votes
0 answers

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? $F_0 = f$; $F_t : M \to \overline{M}$ is an…
Eduardo Longa
  • 5,307
10
votes
0 answers

Application of Thoms transversality theorem

I try to verify example 20.4.10 from Wiggins - Introduction to Applied Nonlinear Dynamical Systems and Chaos and I am quite new to the topic so please be patient. In the book is written that the family of maps \begin{equation*} f(x,\mu) = \mu +…
JDoe
  • 724
8
votes
2 answers

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 $p\in f^{-1}(S)$. I have found two instances of this…
Santana Afton
  • 6,978
7
votes
1 answer

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 $F_s(x)=F(x,s)$ for some smooth $F:N\times S\to M.$ A class…
D. Brogan
  • 3,657
6
votes
1 answer

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 of $N$. There is a proof in Lee’s smooth…
user312648
6
votes
1 answer

Why is a transversal intersection of submanifolds a manifold?

Let $N, M \subseteq R^n$be transversal submonifolds of $R^n$ We say that $N$ and $M$ are transversal if $T_pN + T_pM =T_pR^n$ for all $p \in N\cap M$. Why is $N \cap M$ a manifold?
Janusz Tracz
  • 61
6
votes
0 answers

Transversality and homotopic maps

I'm trying to solve some problems in differential topology, and I came across the following: suppose $f:M\times [0,1]\rightarrow N$ is a homotopy, where $M$ is a compact manifold, such that $f_0$ and $f_1$ are $C^r$ functions, both transversal to a…
Pryscilla Silva
  • 843
5
votes
1 answer

A smooth map of a sphere to itself is homotopic to a map with isolated fixed points

Let $v:S^k\to S^k$ be a smooth map of a sphere into itself. Such a map possibly can have nonisolated fixed points (e.g. the identity map of $S^k$). Can we always homotope $v$ to a smooth map $S^k\to S^k$ which has only isolated (hence, finitely…
user302934
  • 1,738
  • 1
  • 9
  • 27
4
votes
0 answers

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 $g:\mathbb{R}^2 \times C \to \mathbb{R}$ be some…
qscty
  • 321
4
votes
2 answers

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 allowed to put pictures yet). After the proof he says that this…
Carlos Cabezas
  • 81
  • 6
4
votes
0 answers

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 $$S=A\underset{F,C,G}{\times}B=\{(x,y)\in A\times B\,;\,F(x)=G(y)\}$$ a manifold with tangent space…
Balloon
  • 8,624
4
votes
0 answers

Genericity of an induced projection map

Let $X,Y$ be smooth manifolds, $S'$ a submanifold of $Y$, and $f:\mathbb{R}\times X\to Y$ a smooth function. Generically, we have that $f$ is transverse to $S'$, which implies that $S:=f^{-1}(S')$ is a smooth submanifold of $\mathbb{R}\times X$. It…
Balloon
  • 8,624
4
votes
1 answer

Transversal and intersection of two foliations

Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two foliations of a manifold. We say that $\mathcal{F}_1\pitchfork \mathcal{F}_2$ if $T_p L^{(1)}+T_pL^{(2)}=T_p M$ for any $p\in M$, where $L^{(1)}$ and $L^{(2)}$ are the leaves trough $p$. Now if we have…
Someone
  • 5,059
4
votes
1 answer

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 $A : V \to V$ consider the graph $W = \{(v, Av) \in V…
Infinity
  • 1,120
4
votes
1 answer

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 Jacobian matrix look like in our situation here? Thanks. …
Idonotknow
  • 89
1
2 3
…
11 12