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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Smoothness independent of chart

Given a continuous map $f:M_{1}\rightarrow M_{2}$ between differentiable manifolds, a map is smooth if for all $p\in M_{1}$ with there exist charts $\varphi_{1}:U_{1}\rightarrow V_{1}$ and ...
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124 views

What is the the fundamental group of $ H_{\mathbb{R}}/H_{\mathbb{Z}}$

Consider $M = H_{\mathbb{R}}/H_{\mathbb{Z}}$, where $H_{\mathbb{R}} = \lbrace\begin{bmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1\\ \end{bmatrix}|x,y,z \in ...
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291 views

Algebraist's definition of the tangent space of a manifold

By the "algebraist's definition" of the tangent space of manifolds, can we say that the partial derivative $d/dx$ belongs to the the tangent space of $S^1$? It feels strange, but I can't see why it ...
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Preimages of 0 in antipode preserving maps on $S^n$

Attempting to find an inductive argument for the Borsuk-Ulam theorem led me to another question, which I found interesting in its own right but am stuck on. Let $g:S^n\rightarrow \mathbb{R}$ be a ...
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A simple explanation of differential calculus and its link to geometry?

The wikipedia articles on differential calculus and differential geometry are quite long and not so straightforward for a layman like me. Is there a master of math vulgarization out there that could ...
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Commuting smooth maps

Suppose $f:A\to B$ and $g:C\to D$ are smooth embeddings, $h:B\to D$ is a smooth map, and $i:A\to C$ is a continuous map such that $g(i(x))=h(f(x))$. Then, how to show that $i$ is smooth? An ...
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70 views

Maslov Index product property.

I am not sure how to show the following property of the Maslov Index, which in McDuff and Salamon's Introduction to Symplectic Topology, Theorem 2.35, is called the product property. Let $\Lambda: ...
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310 views

Higher-order derivatives in manifolds

If $E, F$ are real finite dimensional vector spaces and $\mu\colon E \to F$, we can speak of a (total) derivative of $\mu$ in Fréchet sense: $D\mu$, if it exists, is the unique mapping from $E$ to ...
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Openness and differentiation

Given that $A$ is an open set in $\mathbb R^n$ and $f:A \to \mathbb R^n$ is differentiable, and its derivative is non-singular at every point in $A$, prove that $f(A)$ is open in $\mathbb R^n$ Note ...
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132 views

Differential forms on tensors

With T: R^m -->R^n be linear transformation T(x) = B*x and if psi sub I is an elementary alternating k-tensor on R^n, then T*psisub I has the form: $$ T^**\psi_I $$ = sigma sub [J] cJ*psi[J] where ...
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130 views

Defining a Differentiable Function in a Non-Pointwise Manner

Is there a way to define a differentiable function in a non-pointwise manner? That is without defining function differentiable at a point first. Just like we define a continuous function through open ...
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114 views

Tubular neighbourhood style theorem reference request

Let $X$ be a smooth manifold and $Y$ be a closed submanifold. Then there exists a neighbourhood $U$ of $Y$ in $X$ such that $Y$ is a deformation retract of $U$ right? I can only find (stronger forms ...
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71 views

classifying vortices whose base space is $S^{3}$ or $S^{7}$

On $S^{1}$ there are three choices for a vector pointed inward to the same angle at the tangent line at every point. The vector is either in the opposite direction as the normal, or to the left of ...
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167 views

Self-intersection of parametric surface using Gauss-Bonnet theorem

I am trying to detect when a closed parametric surface intersect itself. My surface is described as a triplet of parametric functions $x(u,v)$, $y(u,v)$ and $z(u,v)$ where $u,v\in[0,1]$. For that ...
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176 views

Conditions for left-invariant one-forms to be closed.

Let $G$ be a connected (semisimple) Lie group with Lie algebra $\frak{g}$. For $\omega \in \frak{g}^*$, we may define a left invariant one-form on $G$ by $\left[ \omega (g)\right] (v)=\omega \left( ...
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246 views

Surgery, framing and Dehn twist

Let $L$ be a framed knot in $S^3$. Let $U$ be a closed regular neighborhood of $L$ in $S^3$. How can I interpretate the following sentence? "We identify $U$ with $S^1 \times B^2$ so that $L$ is ...
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149 views

reference for poincare-hopf theorem

I am an graduate student interested in fluid dynamics and have almost zero background in differential and algebraic topology. I must say that I do know some analysis (Lebesgue integration plus basics ...
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380 views

A difficult question about diffeomorphism about submanifold

Let $M$ and $N$ be two smooth manifolds, and $f: M \to N$ be a submersion , ${{f}^{-1}}(y)$ is compact for all $y$ in $N$. Then prove for any $x$ in $N$ there is an open neighborhood $U$ of $x$ such ...
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111 views

diffeomorphism invariance of characteristic classes

I read everywhere :"By definition, the characteristic classes of smooth manifolds are invariant under diffeomorphisms." Does it follow from de Rham cohomology? If this is so, then what about ...
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171 views

Bundles over glued n-balls, homotopy classes

This is a homework assignment. I'm not sure I even understand the question fully, as the parametrization seems slightly wrong. Over an $n$-ball, let $r \in [0,1)$ denote a radial coordinate, and let ...
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456 views

Boundaries of Manifolds Necessarily Orientable

Let $M$ be a smooth manifold (not necessarily orientable) and let $N=\partial M$. Is $N$ necessarily orientable? I have no particular reason to believe that this is the case, but I wasn't able to ...
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647 views

Sanity check about Wikipedia definition of differentiable manifold as a locally ringed space

Most textbooks introduce differentiable manifolds via atlases and charts. This has the advantage of being concrete, but the disadvantage that the local coordinates are usually completely irrelevant- ...
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92 views

Another time on jets and composition

Suppose we have four smooth maps between smoot manifolds: $$f: M \rightarrow X$$ $$g: X \rightarrow N$$ $$h: M \rightarrow Y$$ $$i: Y \rightarrow N$$ an the equation on compositions of jets $$j_m(g ...
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577 views

Is a cube a smooth manifold?

Is the unit square $\partial I^2$ (i.e. the square with vertices $(0,0), (0,1), (1,0), (1,1) \in \mathbb R^2$) a smooth manifold? I guess it shouldn't be smooth because it has "corners", but i have ...
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Proof of Cellular Approximation from Sard's Theorem

I'd like to prove the cellular approximation theorem using Sard's theorem. The only hard part is the induction step: Let $f: \mathbb{D}^k\rightarrow Y$ be a map from the closed $k$-disk to a ...
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1answer
180 views

Differential forms on a $S^1$-manifold

I am reading about differential forms on manifolds with group actions and there is an 'obvious' formula which I don't quite understand. Let $X$ be a manifold with a smooth circle action, that is a ...
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116 views

Thom-Pontryagin construction of manifolds with boundary

Thom-Pontryagin construction gives the 1-1 correspondence between framed cobordism classes of $k$-dimensioanl sub-manifolds of $S^{n+k}$ and homotopy classes of maps from $S^{n+k}$ to $S^n$. Are ...
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263 views

Is an immersed submanifold second-countable?

By general manifold I mean Hausdorff differential manifold not necessarily second-countable. By standard manifold I mean Hausdorff, second-countable differential manifold. So my question is, we have ...
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521 views

On the Use of the Topology on Tangent Bundles

On learning about how to define smooth vector fields on a manifold $M$, I learned that one should first define a tangent bundle , $T(M)$, as $\cup T_p(M)$ together with a topology(smooth structure). ...
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110 views

On Tangent vectors as jets & submanifolds

Here is my second question on understanding jets better: For a smooth manifold $M$ the set of jets $J^1_0(\mathbb{R},M)$ is the same as the Tangent bundle $TM$. This implies that any equivalence ...
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Is there any connection between partial derivative and matrices?

I can see in some texts and books that the authors use big letters in order to describe partial derivative of function in $\mathbb{R^n}$ similar to the way we write matrices in linear algebra, for ...
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224 views

Morse theory and surfaces

I have heard that one can prove the classical classification of surfaces theorem using Morse theory. I am planning on learning this approach as a way to motivate and get comfortable with Morse ...
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163 views

Let $\mathbb{R}^m_+ = \lbrace x \in \mathbb{R}^m : x \geqslant 0 \rbrace.$ What is the boundary set of the set $ \mathbb{R}^m_+$?

Let $ x \in \mathbb{R}^m,$ $ x = [x_i], i = 1,2, \dotsm , m.$ Define $\mathbb{R}^m_+ = \lbrace x \in \mathbb{R}^m : x_i \geqslant 0, 1 \leqslant i \leqslant m \rbrace.$ What the boundary set $ ...
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313 views

Stokes for integration along the fiber

I want to use a version of Stokes theorem for integration along the fiber and I need some help in proving a general statement. Let $F$ be a $k$-manifold with boundary and let $E \to M$ be a smooth ...
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227 views

Knot with genus $1$ and trivial Alexander polynomial?

I would like to know whether there exists a knot $K$ with genus $g(K)=1$ and trivial Alexander polynomial $\Delta(K) \doteq 1$. A linked question could be: does there exist a Whitehead double with ...
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Why is the pullback completely determined by $d f^\ast = f^\ast d$ in de Rham cohomology?

Fix a smooth map $f : \mathbb{R}^m \rightarrow \mathbb{R}^n$. Clearly this induces a pullback $f^\ast : C^\infty(\mathbb{R}^n) \rightarrow C^\infty(\mathbb{R}^m)$. Since $C^\infty(\mathbb{R}^n) = ...
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What's so special about a homotopy $15$-sphere?

I just saw a table which counts diffeomorphism classes of homotopy $n$-spheres (that is, spaces homotopy equivalent to $n$-dimensional spheres). Such a table can be seen on the first page of this ...
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186 views

Preimage of submanifold under an embedding

Suppose we have two smooth manifolds $M_1$ and $M_2$ and a smooth map $i:M_1 \rightarrow M_2$ that is an embedding of $M_1$ into $M_2$. Moreover we have another submanifold $N \subset M_2$ that has a ...
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139 views

Image of smooth vector bundle morphism

Let $\pi_1:V_1 \rightarrow B_1$ and $\pi_2:V_2 \rightarrow B_2$ be smooth vector bundles and write $\phi_V: V_1 \rightarrow V_2$ as well as $\phi_B:B_1 \rightarrow B_2$ respectively for the total and ...
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172 views

How does a geodesic equation on an n-manifold deal with singularities?

My general premise is that I want to investigate the transformations between two distinct sets of vertices on n-dimensional manifolds by: Minimalizing the change in the fundamental shape of the ...
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249 views

A theorem in Morse theory

If $M$ is a smooth manifold on which there is a smooth function $f:M \to ( - 1,2)$ such that all $[0,1]$ are regular values of $f$ and ${f^{ - 1}}(s)$ is a compact set for all $s \in [0,1]$,then is ...
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190 views

$\{\gamma \in C^0([0,1],M): \gamma (0)=p, \gamma (1)=q\}$ with the compact-open top has the homotopy type of a CW complex

...where $M$ is a smooth manifold and $p, q \in M$. Does anyone know of any slick or accessible proofs of this? I was referred to Milnor's "On Spaces Having the Homotopy Type of a CW Complex" which ...
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Action of U(1) on a sphere bundle

Suppose $N$ is a closed, $n$-dimensional, orientable smooth manifold. Moreover, $n$ is odd. Consider the tangent bundle $TN$ of $N$. By adding a point at infinity to each tangent space I define a ...
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98 views

Non-intersecting smooth paths in the plane and the relation to curvature

I'm interested in a problem and have no idea how to approach solving it. Could you please point me in the right direction. Given 3 smooth paths $\varphi_{1}:[0,1]\to \mathbb{R}^{2}$, ...
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491 views

What can Lefschetz fixed point theorem tell us about the group of automorphisms of a compact Riemann surface?

Let $X$ be a compact Riemann surface of genus $g>1$, $f\in Aut(X)$, a biholomorphism of $X$ onto itself, $x\in X$ a fixed point of $f$. Since tangent map of a holomorphic map (on the real tangent ...
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“Immediate” Applications of Differential Geometry

My professor asked us to find and make a list of things/facts from real life which have a differential geometry interpretation or justification. One example is this older question of mine. Another ...
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Diffeomorphism group of the unit circle

I am given to understand that the group of diffeomorphisms of the unit circle, $\operatorname{Diff}(\mathbb{S}^1)$, has two connected components, $\operatorname{Diff}^+(\mathbb{S}^1)$ and ...
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171 views

Real analytic diffeomorphisms of the disk

Is there any real analytic diffeomorphism from two dimensional disk to itself, except to the identity, such that whose restriction to the boundary is identity?
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82 views

extension to the ball of sphere immersion

What are the constraints to extend an immersion of the sphere $S^2$ into $\mathbb{R^3}$ to an immersion of the closed unit ball $B(0,1)$ to $\mathbb{R}^3$? Suppose, I get an immersion of $S^2$ into ...
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Properties preserved by diffeomorphisms but not by homeomorphisms

Diffeomorphisms between manifolds are particular homeomorphisms, so each property preserved by homeomorphisms is preserved by diffeomorphisms. Can you show me some examples of properties preserved by ...