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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Continuous map to quotient space

Let $X,Y,Z$ be topological spaces, and suppose $\pi:Y\rightarrow Z$ is a quotient map. Is a continuous map $f:X\rightarrow Z$ necessarily the composition of a continuous map $g:X\rightarrow Y$ with ...
3
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1answer
336 views

How does handle attachment work in Morse Theory

I am reading R.E.Gompf and A.I. Stipsicz, 4-Manifolds and Kirby Calculus. I can't understand the 2nd-paragraph of p.101, where they explain framings on the attaching sphere. In particular I cannot ...
5
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1answer
760 views

The winding number and index of curve

If $\gamma $ is a smooth closed curve in $\mathbb R^2-\{0\}$, I want to know whether the winding number of $\gamma $ about $0$, i.e., $\frac{1}{{2\pi i}}\int\limits_\gamma {\frac{1}{z}} dz$ is equal ...
2
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1answer
128 views

Various types of TQFTs

I am interested in topological quantum field theory (TQFT). It seems that there are many types of TQFTs. The first book I pick up is "Quantum invariants of knots and 3-manifolds" by Turaev. But it ...
3
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1answer
198 views

Embedding/Submersion Properties of Cotangent Maps (Pullbacks)

Let $M$ and $N$ be smooth manifolds and $f: M \to N$ a smooth map. Define the pullback bundle $\pi_f^*:f^*(T^*N) \to M$ as usual by $ f^*(T^*N) = \{(x,j^1_{f(x)}g) \in M \times T^*N \}$ with ...
3
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1answer
457 views

Showing a bijective, continuous function between connected, locally euclidean spaces is a homeomorphism.

This question comes from Conlon's Differentiable Manifolds (it's Exercise 1.1.13). Let $X$ and $Y$ be connected, locally Euclidean spaces of the same dimension. If $f:X \rightarrow Y$ is bijective ...
2
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1answer
122 views

Showing $[f^{-1}(y)]$ is Poincare dual to $f^*(\operatorname{vol})$.

Let $f: N^n \to M^m$ be a smooth map between closed oriented manifolds. Then I'm trying to show that for almost all $y \in M$, the homology class $[f^{-1}(y)] \in H_{n-m}(N)$ is Poincare dual to $f^* ...
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1answer
464 views

Proof that two spaces that are homotopic have the same de Rham cohomology

I know this is true, but how do I prove it? Specifically, I'm trying to calculate the de Rham cohomology of the 3-sphere by using the Mayer-Vietoris sequence and covering $S^3$ with two hemispherical ...
2
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0answers
95 views

A Heegaard splitting of $S^2\times S^1 \# S^2\times S^1$.

For a Heegaard splitting of $S^2 \times S^1$, we can take two copies of genus 1 handlebodies and glue boundaries with the identity map. I want to generalize this a little bit. In the case of ...
1
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1answer
62 views

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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1answer
125 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 ...
5
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1answer
296 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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0answers
74 views

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 ...
3
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2answers
1k views

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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0answers
101 views

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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0answers
74 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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0answers
394 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 ...
4
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2answers
104 views

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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1answer
135 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 ...
4
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2answers
132 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 ...
4
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1answer
118 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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1answer
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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2answers
191 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 ...
3
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1answer
228 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( ...
3
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1answer
284 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 ...
2
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1answer
162 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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1answer
517 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 ...
0
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2answers
118 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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0answers
179 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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3answers
544 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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3answers
722 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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0answers
97 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 ...
2
votes
1answer
740 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 ...
3
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0answers
113 views

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 ...
2
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1answer
217 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 ...
2
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0answers
127 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 ...
8
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1answer
287 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 ...
5
votes
3answers
606 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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1answer
113 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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1answer
90 views

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 ...
5
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1answer
257 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 ...
0
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1answer
165 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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1answer
370 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 ...
3
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1answer
263 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 ...
4
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2answers
163 views

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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0answers
180 views

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 ...
0
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1answer
204 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 ...
4
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1answer
161 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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0answers
178 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 ...
4
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1answer
258 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 ...