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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.

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Does every specialization of the HOMFLY polynomial produces a knot invariant polynomial?

It's a well known fact that the Alexander and Jones polynomials can be obtained as appropriate substitutions in the HOMFLY polynomial $$V(t) = P(\alpha=t^{-1},z= t^{\frac{1}{2}}-t^{-\frac{-1}{2}}),$$ $...
Ramiro Hum-Sah's user avatar
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What is the use of topology?How Is Topology Applicable to the Real World? [duplicate]

Topology is the study of shapes and spaces, focusing on the structure that remains when geometric objects are stretched or squeezed but not broken. Throughout most of its history, topology has been ...
yinghao luo's user avatar
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How to warp a function $z = f(x,y)$ such that the $xy$-plane will located on the surface $z = x^2$?

The warp should preferably preserve distances. I am not familiar with the branch of mathematics that deals with such problems, so I apologize for not providing an attempt at a solution.
Surzilla's user avatar
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Motivating the second derivative definition given in Audin,Damian Morse Theory text.

I am reading the Morse theory and Floer Homology text written by Audin and Damian.In the first chapter they have made a remark about second order derivatives on manifolds,the remark is quoted verbatim ...
Kishalay Sarkar's user avatar
3 votes
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Topologies on the space of smooth functions

Let $M$ be a compact manifold. Consider $C^{\infty}(M, \mathbb{R})$, the space of smooth, real-valued functions on $M$. I've read about two topologies on $C^{\infty}(M, \mathbb{R})$. One is the [...
Yoona's user avatar
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Continuous injection to manifold with boundary

Let $M^n$ be an n dimensional manifold with boundary. Assume $U$ is open in $M^n$. Let $f:U\rightarrow M^n$ be a continuous injection open map. Then show that $f(U\cap \partial{M})=\partial{M}$ I am ...
monoidaltransform's user avatar
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Do you know if this theorem has a name?

I'd like to ask if the following theorem has a name. Let $M$ be a smooth $n$-dimensional manifold, $\mathcal{P}$ a family of open subsets of $M$ satisfying the following: $(1)$ $\emptyset \in \mathcal{...
Paz's user avatar
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A common misinterpretation of transversality being a stable property.

In the first Chapter in Differential Topology by Guillemin Pollack (Section 6:Homotopy and Stability),the authors have asserted that, Transversality is a stable property in the sense that if $X$ is a ...
Kishalay Sarkar's user avatar
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Grothdieck differential operators - are they all compositions of first order operators?

Let $A$ be a (unital, commutative, associative) algebra over a field $\mathbb{k}$. Then the ring of Grothendieck differential operators on A is defined as: $$ \operatorname{Diff}(A) = \bigcup_{\ell \...
aceincc's user avatar
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Is there a notion of smooth sheaf which corresponds with smooth etale bundles?

It is well-known and quite useful that there is an equivalence of categories between sheaves of sets on a space $X$ and etale bundles over that space, meaning spaces $E$ and continuous maps $\pi:E\to ...
FShrike's user avatar
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Can we pass between sheaves of suitable type and fibre bundles (not ! etale bundles)?

Today I was forced to think about vector bundles (on manifolds), but in the context of a textbook "otherwise" focused on sheaf theory. I'm interested to what extent this really is an "...
FShrike's user avatar
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Why isn't there a nice vector integration theory on smooth manifolds?

I'm very unfamiliar with differential geometry, but recently I'm forced to learn it. One thing I've noticed is it seems differential forms only works nicely for real valued functions defined on smooth ...
user760's user avatar
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Exploring the relationship between Tensorial Techniques and Coordinate-Independence?

I am studying Differential Geometry and Riemannian Geometry.I need some help from my stack exchange community members in this regard. Recently I happened to get a lecture series on Riemannian Geometry ...
Kishalay Sarkar's user avatar
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Partial Derivative of Pullback of Differential form

I'm new to differential forms and the book I'm reading contains a part I don't understand. It states the following: Let $k\geq 1$ and assume that $D \subset \mathbb{R}^k$ and $U \subset \mathbb{R}^n$ ...
Josef K.'s user avatar
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The image of a Riemannian manifold under a distance-based mapping to $\mathbb{R}^n_{\geq 0}$

Consider $n$ points, $p_i\in M,\, i=1,\ldots , n$, in a Riemannian manifold $M$, and define the map \begin{array} &f: M&\to& \mathbb{R}^n_{\geq 0}\\ \qquad q &\mapsto& \left(\begin{...
STU's user avatar
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Degree of a smooth map to a product

Let $f:M\rightarrow S^{n-1}\times [0,1]$ be a smooth map, where $M$ is a $n$-manifold with boundary. Suppose $f(\partial M)\subset S^{n-1}\times \{0,1\}$. Denote $\partial_-M:=f^{-1}(S^{n-1}\times\{0\}...
Jun Xu's user avatar
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Comparing the definitions of Derivative in Guillemin Pollack and in other differential topology books.

In standard differential geometry and topology books I have seen the authors defining the derivative/differential in the following way: Let $f:M\to N$ be a smooth map between two smooth manifolds.Then ...
Kishalay Sarkar's user avatar
3 votes
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Kahler differentials and etale fundamental group

In Algebraic geometry we can define two notions that generalize tools from topology/differential geometry: the etale fundamental group and the module/sheaf of Kähler differentials. In the complex ...
user720386's user avatar
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Proof of section is smooth iff component functions are smooth

I'm trying to understand the proof of this lemma in "An introduction to manifolds" 2ed by Loring W.Tu. (p. 138) Lemma 12.11 Let $\phi:E|_U \longrightarrow U \times \mathbb{R}^r$ be a ...
MLe's user avatar
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A question about Jordan Curve Theorem

I have a "separation of a surface" question. Assume that $N^2$ is a connected regular surface in $\mathbb R^3$ containing an infinite funnel. Let $L$ be the boundary of the funnel in $N$, ...
Geomancer's user avatar
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Is the derivative at a point x of a smooth real-valued map linear?

I am currently reading "Differential Topology" by Victor Guillemin and Alan Pollack. They are in the process of explaining the preimage theorem in terms of a set of common zeroes (to show ...
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Generalizations of Jordan Curve Theorem

In Dale Rolfsen's Knots and links it is proved that If $L$ is a closed subset of $\mathbb R^2$ which is homeomorphic with $\mathbb R$, then $\mathbb R^2 \backslash L$ has two connected components and $...
Geomancer's user avatar
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Tu, Exercise 18.5

In Exercise 18.5 Tu asks the reader to find the transition formula of a 2-form on an n-manifold. He provides the formula below before asking the reader to find $a_ij$ in terms of $b_{kl}$ $\omega = \...
EEH's user avatar
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Hirsch tangent bundle definition

I am new to differential topology, and I attempt to study it using Hirsch book "Differential Topology". In the first chapter, after introducing the definition of manifold, he talks about ...
Anna  Vakarova's user avatar
1 vote
1 answer
57 views

Open subset of paracompact manifold is paracompact?

I assume a smooth manifold is paracompact (including Hausdorff) instead of Hausdorff and second countable. We can show that a connected topological manifold is paracompact iff it's second countable. ...
wsz_fantasy's user avatar
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What's the continuous $\mathbb C^0$ version for immersion? I'm trying to see if topological embeddings are 'topological immersions.'

A 'smooth embedding' $f: M \to N$ between smooth ($m$,$n$)-manifolds ($M$,$N$) is a smooth map that is both a smooth immersion and a topological embedding, which is simply defined as that the range-...
BCLC's user avatar
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Can locally Euclidean Hausdorff 2nd countable topological space always be made into topological manifolds (as in continuous or $C^0$)?

For any locally Euclidean Hausdorff 2nd countable topological space $M$, can $M$ always be made into a topological manifold (not necessarily of some uniform dimension $n$ ... but if ever I assume ...
BCLC's user avatar
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Do smooth manifolds care about the continuous atlas?

A smooth manifold $M$ can be defined as a pair (topological manifold $X$, smooth, as in $C^{\infty}$, atlas $\mathscr M$), where a topological manifold is defined as a locally Euclidean Hausdorff 2nd ...
BCLC's user avatar
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Example of Critical Path

If $F$ takes on its minimum at a path $\omega_0$, and if the derivatives $\frac{\mathrm{d}F\left(\bar{\alpha}(u)\right)}{\mathrm{d}u}$ are all defined, Milnor states in his Morse Theory that clearly $\...
一団和気's user avatar
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Is every simple path in Euclidean space homotopy equivalent to its $\varepsilon$-neighborhood when $\varepsilon$ is small?

I would like to formulate the problem in the simplest situation at first. Given a path $\gamma: I\to \mathbb{R}^n$ and suppose $\gamma$ has no self-intersection, namely, injective. Now we can see $\...
Absurdus's user avatar
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De Rham Cohomology of $M\times S^n$

I was recently asked to prove that $$H^k(M\times S^n)\cong H^k(M)\oplus H^{n-k}(M)$$ for all $k\in\mathbb{Z}$ and $M$ a compact $C^\infty$ manifold without boundary. At the time, we had nothing ...
Dowdow's user avatar
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3 votes
2 answers
106 views

Give X a differentiable structure such that the inclusion map is smooth

Let $X = \{(x,y,z) \in R^3: z^4 = x^2+y^2, z \geq 0\}$. First I'm trying to prove that this is not a smooth submanifold of $R^3$. Clearly the problematic point is $(0,0,0)$ however I haven't been able ...
H4z3's user avatar
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2 votes
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Generic intersections of the preimage and a submanifold

Let $f: X \to Y$ be a smooth map between manifolds of the same dimension. Let $Z$ be a submanifold of $X$ of codimension 1. By Sard's theorem, for almost every $y \in Y$, $f^{-1}(y)$ is a 0-...
Yoona's user avatar
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Do orthogonal trivializations induce orthonormal frames?

Let $E$ be a real vector bundle with a bundle metric $g$ over $M$, and $\{U_i,\phi_i\}$ an orthogonal trivialization; that is for all $i$ and $j$ we have that the map $\phi_{ij}:U_i\cap U_j\rightarrow ...
Chris's user avatar
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4 votes
1 answer
107 views

Linking number of irregular curves

Let $f,g \colon S^1 \to \Bbb R^3$ be two continuous functions (not necessarily embeddings) whose images are disjoint. We define the linking number of these closed curves to be $$ L(f,g) = \text{deg}(...
BigbearZzz's user avatar
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In how far is the Lyapunov spectrum characteristic for a dynamic system?

If two dynamic systems have the same Lyapunov spectrum on their respective attractor (of equal dimension, of course), which results relate to the properties of these two dynamic systems on those ...
algebruh's user avatar
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About regular value theorem.

Let $\lambda\in\mathbb{R}^{+}\setminus\mathbb{Q}$ and $C\in(\mathbb{R}\setminus\{0\})$. Consider $$\Sigma=\{(x_1 , x_2 , y_1 , y_2)\in\mathbb{R}^{4} : y^{2}_{1}+y^{2}_{2}=C^{2}(x^{2}_{1}+x^{2}_{2})^{\...
user 987's user avatar
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Relation between two theories of degree of maps

For a holomorphic map between 2 Riemann surfaces, we can define its degree to be the sum of ramification index of all preimages of a point, or simply the cardinality of the preimage of a regular(not ...
minukesis's user avatar
2 votes
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Do diffeomorphisms map increasingly smaller sets to increasingly "smaller" sets?

Let $\Omega \subset \mathbb R^n$ be an open set. Let $\phi: \Omega \rightarrow \Omega$ be a diffeomorphism. Suppose $y \in \Omega$ is such that for each $n \in \mathbb N$, $\overline{B(y, 1/n)} \...
rosecabbage's user avatar
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1 answer
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From degree theory on sub-manifolds to abstract smooth manifolds

I am studying the degree theory on smooth manifolds of $\mathbb{R}^n$ and I would like to see how bridge this theory in a more general setting. In a previous post, a comment proposed 2 ways to define ...
G2MWF's user avatar
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2 votes
1 answer
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The natural map from compact vertical cohomology to de Rham cohomology is not injective

Let $\pi:E\to M$ be an oriented vector bundle. In Bott-Tu's book Differential Forms in Algebraic Topology, the compact vertical cohomology $H^*_{cv}(E)$ is defined by using differential forms $\omega$ ...
user302934's user avatar
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Brouwer fixed point theorem and degree theory

I have heard that a proof of the Brouwer fixed point theorem can be given using the degree of a continuous map. I would like to know if the one I have in mind is correct please. Let $f:B^n\to B^n$ be ...
G2MWF's user avatar
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Degree of a smooth map is nonzero implies that it is surjective

I would like to prove that if $f: M\to N$ is a smooth map between oriented manifolds with $M$ compact, then $\text{deg}f\neq0$ implies that $f$ is surjective. Here is my attempt : Let $y\in N\setminus ...
G2MWF's user avatar
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1 vote
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On mod 2 intersection number

On page 78 of the book Differential Topology by Guillemin and Pollack is presented the following Theorem: If $f_0 , f_1 : X\to Y$ are homotopic and both transversal to $Z$, then $I_{2}(f_0 , Z)=I_{2}(...
user 987's user avatar
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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 ...
cheeseboardqueen's user avatar
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44 views

Pre image orientation and smooth map

I consider a smooth map $ F: X\to N$ where $X$ and $N$ are smooth, oriented manifolds of dimension $m+1$ and $m$ respectively, $X$ being compact and with boundary $\partial X=M$. For a regular value $...
G2MWF's user avatar
  • 1,381
3 votes
0 answers
27 views

Vector field with almost non-periodic orbits

Let $M$ be an n-dimensional smooth manifold (Open or compact). I want to know if it is possible to construct an smooth vector field with exactly one singularity, such that the set of periodic integral ...
Pablo Cid's user avatar
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46 views

How to construct the monotone union of disks

In proof of the Brown-Stallings theorem, Milnor mentioned that $M$ is a monotone union of disks. But the author didn't mention how to construct such monotone disks $W_1 \subset W_2 \subset \cdots \...
Leilei Cui's user avatar
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0 answers
26 views

Generalizing Reidemeister-Singer Theorem to manifolds with boundary

I was studying some Heegaard splittings theory for a course; we saw the Reidemeister-Singer theorem, which states that for a closed, connected 3-manifold $M$, any two Heegaard splittings share a same ...
Nennee's user avatar
  • 158
3 votes
1 answer
78 views

Lie group structure on exotic $\mathbb{R}^4$

Are there Lie group structures on exotic $\mathbb{R}^4$s? By the theorem that every continuous group homomorphism of two Lie groups is smooth, we can conclude that if $G$ and $H$ are two Lie groups ...
Strichcoder's user avatar
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