Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [differential-topology]

Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

Differential topology is the field dealing with differentiable functions on differentiable manifolds. Somewhat simplified view of this field is that it describes the setting to which the notion of differentiable function can be generalized from the more familiar case of functions $\mathbb R^n\to\mathbb R^k$. Differential topology is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

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Submersion and immersion

I googled wiki about submersion and immersion. Wiki states that submersion is dual to immersion. I wonder where this duality relationship comes from.
user45765
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Representative of a cohomology class in once punctured solid torus

Consider a once punctured solid torus $(\mathbb R^2 \times S^1) /\{pt\}$. It is not difficult to see that it is homotopy equivalent to the bouquet of spheres $S^2\vee S^1$. So this guy has a non-trivial second de Rham cohomology. How can one…
user79456
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Why are the basis elements of $T_pM$ a vector field if we let $p$ vary?

At the end of this article on tangent vectors, they say each $\frac{\partial}{\partial x_i}$ is a vector field, if we let the point $p$ vary. However, they are only defined in a particular coordinate chart. I know that $\{ \frac{\partial}{\partial…
clueless
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Is there a rule for computing the differential of a product of maps?

A lot of partition of unity arguments have some map of the form $f=\sum_i \psi_if_i$. Is there a formula for the differential $df_p$ in terms of its summands? For instance, suppose $f_i:U_i\to V_i$ are a family of maps where the $U_i$ cover a smooth…
Leliana
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Doubt regarding proof from Milnor's notes

I was reading Milnor's 1958 notes in differential topology and came across the following theorem Theorem: Let $U$ be an open set in $\mathbb{R}^n$ and let $f;U \to \mathbb{R}^p$ be differentiable, where $p \geq 2n$. Given $\epsilon$ > $0$, there…
Sandeep Thilakan
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The transverse root for a vector field

I encountered the term, so may I ask - what is the transverse root for a vector field? Thank you~
1LiterTears
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A homework problem about Tangent space of Lie groups at the identity element

This is one of our homework problem. Let $G$ be a Lie group be defined as a manifold with group structure such that the map $F:G\times G \mapsto G, F(a, b)=ab^{-1}$ is smooth. Show that $$dF_e(a,b)=a-b,$$ where $e$ is the identity. I assume the…
user110373
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transverse homotopy

Let $f,g:M \rightarrow N$ two smooth maps between smooth manifolds that are smoothly homotopic by $F$. Suppose also that $f$ and $g$ are transverse to a submanifold $A$ of $N$. I know that transverse maps are dense in the weak and strong topologies…
Vincent L.
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Show $\lambda$ is smooth

Let D be the closed unit disc in $\mathbb R^2$ and $S^1= \partial D$. Let $f$, $g$: $S^1 \rightarrow \mathbb R^3$ be smooth embeddings s.t. $f(S^1) \cap g(S^1) = \emptyset$. Define $\lambda: S^1 \times S^1 \rightarrow S^2 $…
WWK
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Multiplication in homotopy groups and cobordism

Each homotopy class of a map of an n-sphere into the Thom space of the universal vector bundle determines a cobordism class of embedded smooth manifolds in Euclidean space. How do the cobordism classes change when two elements of the homotopy group…
lavinia
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Computing the coordinate representation of a vector field

Let $V=x\frac{∂}{∂x}+y\frac{∂}{dy}$ be a vector field on the plane. Compute its coordinate representation in polar coordinates on the right half-plane $\{(x,y):x>0\}$. What I got so far: The question asks for a coordinate representation, which can…
figura
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Transversality and intersection mod.2

I am troubled by the following fact: If $f,g : X\to Y$ are homotopic and both transversal to $Z$ then the mod.$2$ intersection numbers are equal $I_2(f,Z)=I_2(g(Z)$. (the book by Guillemin and Pollack, Differential Topology, page 78). If $f,g :…
user102980
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Normal bundle is locally trivial

Could someone tell me how to prove the following result? Let $Z$ be a submanifold of codimension $k$ in $Y$. Prove that the normal bundle $N(Z;Y)=\left\{(z,v):z\in Z, v\in T_{z}(Y)\text{ and } v\bot T_{z}(Z)\right\}$ is locally trivial, that is,…
R.Shaug
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Extending a function defined on an arbitrary subset of $\mathbb{R}^n$

This question appeared on an old qualifying exam: Let $X$ be any subset of $\mathbb{R}^n$ and $f\colon X\to\mathbb{R}$ a function with the following property. For every $x\in X$ there is a neighborhood $U_x$ of $x$ in $\mathbb{R}^n$ and a smooth…
D Wiggles
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a condition for smooth vector field

Let $M$ be a Hausdorff manifold. I'm trying to prove that a vector field $Y:M\to TM$ is smooth if and only if the derivation induced by $Y$ for all globablly defined smooth functions is smooth. That is, $Yf:M\to \mathbb{R}$ is smooth for all $f\in…
D. Huang
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