Questions tagged [transversality]

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The boundary of the transversal pre-image of a submanifold with boundary

A similar question on MSE without answer. Let $M, N$ be smooth manifolds such that $\partial N=\varnothing$. Let $A$ be a smoothly embedded submanifold of $N$ such that $\partial A\neq \varnothing$. ...
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1 vote
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Transversal pre-image of a small enough trivial tubular neighborhood contains a trivial tubular neighborhood

A similar post on MSE without answer. Let $f\colon M'\to M$ be a smooth map between two orientable closed smooth manifolds and $S$ be a smoothly embedded closed orientable submanifold of $M$ of co-...
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5 votes
1 answer
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Transversality and $C^l$, $C^{\infty}$ spaces of almost complex structures

Recently I have been trying to get a grip on transversality results in Floer homology. That is suppose we the section $\partial_{J,H}: W^{1,p}(u^*(TM))\rightarrow L^{p}(u^*(TM))$ and we want to prove ...
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0 answers
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Mutual transversality in a mapping space

I am confused by Lemma 8, on page 6 in "Perturbative 3-manifold invariants by cut-and-paste topology", by Greg Kuperberg and Dylan Thurston (link). It claims "The system of diagonals $\{...
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  • 177
2 votes
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Genericity of an induced projection map

I am cross-posting a question asked on Math Stackexchange that has not been answered, in which I am still interested in. Let $X,Y$ be smooth manifolds, $S'$ a submanifold of $Y$, and $f:\mathbb{R}\...
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Existence of closed transversals for taut foliations in arbitrary codimension

There are several different definitions of "tautness" for foliations, the most widely know is probably topological tautness, which is specific to codimension one and means that the foliation ...
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1 vote
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Transversality theorem for maps between fiber bundles

I am looking for a possible generalization of the standard Trasversality Theorem which roughly says that transverse maps are generic. For example, see the version below: From page 74, Theorem 2.1 in ...
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3 votes
1 answer
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Almost geodesic on non complete manifolds

Let $M$ be a connected manifold equipped with a connection $\nabla$. By Hopf-Rinow theorem, we know that if $M$ is complete then for any $x,y$ there exist a curve $\gamma:[0,1] \to M$ such that $\...
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1 vote
0 answers
67 views

Stratification of the space of maps transverse to another given one

If $M,N,S$ are manifolds ($M$ and $S$ compact) and $g:S\to N$ is a smooth map, then if we endow $C^\infty(M,N)$ with the strong topology we get that $$\pitchfork(M,N;g):=\{f\in C^\infty(M,N)\,;\,f\...
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9 votes
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131 views

Every immersion can be deformed to have only transverse self-intersections

I asked this question some time ago in MSE, but obtained no answer. Maybe this is the right place to post it. Let $f : M^n \to \overline{M}^{n+k}$ be an immersion between smooth manifolds. Is it true ...
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3 votes
1 answer
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Finite-dimensional argument for Morse-Smale pairs?

Using the Sard-Smale theorem, it is relatively easy to show that Morse-Smale pairs on a manifold $M$ (i.e. pairs $(f,g)$ where $g$ is a metric on $M$, $f$ is a Morse function on $M$, and the stable/...
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2 votes
0 answers
123 views

Transversality for algebraic spaces

$\DeclareMathOperator\dim{dim}$I want to apply EGA IV 4, Proposition 17.13.2 to a cartesian diagram of algebraic spaces over a fixed scheme $S$. I know the relative dimensions $\dim(\mathfrak{X}'/S)...
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Homotopy type of transversal families of submanifolds through deformation

Let $A,B \subset M$ be two transversal submanifolds of a compact manifold $M$. It seems rather intuitive that if $A$ and $B$ are deformed (say smoothly) in a way that they remain transversal to each ...
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7 votes
1 answer
173 views

perturbing one map to be transverse to a second map

Let $f\colon M \to N$ and $g\colon A \to N$ be two smooth maps between manifolds $A,M,N$. Can one perturb $f$ to be transverse to $g$ (without touching $g$)? Transverse meaning: For every $y\in f(M)\...
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1 vote
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Deformation sublevel sets of functions which preserve boundary

I'm interested in proving the following fact, which seems to naturally arise from gradient flow deformations, but appears to be a bit tricky. Consider a smooth family $$f_s : M \to \mathbb{R}, \quad ...
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2 votes
0 answers
144 views

Thom's (parametric) transversality theorem in the complex case

I need a version of the parametric transversality theorem for the complex case. The original one is: Let $M,\ N,\ Z,\ S$ be smooth manifolds. If a smooth map $F\colon M\times S \rightarrow N\supset ...
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4 votes
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237 views

English language and Mathematics

I have a question maybe more relevant to an English language section of StackExchange, but I doubt that anybody but a Mathematician could properly answer my question. Let $\mathcal M$ be a smooth ...
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4 votes
0 answers
176 views

infinite-dimensional transversality theorem and its application on the universal moduli space of pseudo-holomorphic curves

We would like to discuss Proposition 3.2.1 in McDuff&Salamon's book "J-holomorphic Curves and Symplectic Topology(Second Endition)". Let me first remind you some background. Let $\Sigma$ be a ...
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18 votes
1 answer
1k views

Topological transversality

Warmup question: Let us say that two continuous functions $f,g:[0,1]\to \mathbb R$ are topologically transverse if their difference $f-g$ has only finitely many zeros, and each zero separates an ...
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2 votes
1 answer
247 views

Computing tangent spaces of resolutions to Slodowy slices

This question is about (a special case of) the varieties discussed here: Does the preimage of the Slodowy slice in $T^*G/P$ have a name?. Let $G = SL_n(\mathbb{C}), \mathfrak{g} = \mathfrak{sl}_n(\...
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  • 1,894
3 votes
1 answer
319 views

General position argument for reasonable spaces

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of non zero homogeneous degree $d$ polynomials in three variables up to scaling, where $\delta_d:= \frac{d(d+3)}{2} $. Given a point $p \...
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  • 3,195
4 votes
3 answers
552 views

What is the easiest way to show that three lines in two dimensional space do not intersect?

I have two similar questions: 1) Let $X$ and $Y$ be two measure spaces. Suppose for every point $x \in X $ there exists a set $ \mathcal{U}_x \subset Y $ of full measure in $Y$. Suppose $V \subset ...
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