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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Bott&Tu Definition: "Types of Forms:

In Bott&Tu's well-known book "Differential forms in Algebraic topology", they note -(p34): every form on $\mathbb{R}^n \times \mathbb{R}$ can be decomposed uniquely as a linear combination of two ...
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Tangent bundle of sphere as a complex manifold

I'm trying to show that the tangent bundle, $TS^n$ of the n-sphere $S^n$ is diffeomorphic to the set $\sum z_i^2 = 1$ in $\mathbb{C}^{n+1}$. It's relatively straightforward to see that the tangent ...
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Does every $C^1$ closed differential form differ from some $C^\infty$ closed form by an exact form?

Question: On a $C^\infty$ manifold, does every $C^1$ closed differential form differ from some $C^\infty$ closed form by an exact form? Motivation: This result holds for $C^1$ closed 1-forms on a $C^\...
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Connectedness of the level sets of Mose-Bott functions

I am reading the proof by McDuff and Salamon of the connectedness of the level sets of Morse-Bott functions with index and coindex different from $1$: connectedness of level sets A smooth function $f:...
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Is a translation in a compact Lie group homotopic to the identity?

The following exercise is from Guillemin and Pollack, Differential Topology. Show that the Euler characteristic of the orthogonal group (or any compact Lie group for that matter) is zero. Hint: ...
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Help finding smooth functions that agree on the boundary, but avoid a critical value.

Basically let $U$ be something like a compact neighborhood of $\mathbb{R}^n$ with smooth boundary $\partial U$ and suppose $f:U \rightarrow \mathbb{R}^n$ is smooth. Now fix $x_0 \in \mathbb{R}^n$ with ...
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The quotient of a manifold by a submanifold is never a manifold?

Let $M$ be a connected smooth manifold. Let $S$ be a connected embedded submanifold of positive dimension and co-dimension, which is also a closed subset of $M$. Is it true that the quotient space $...
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Showing deformation retract : $B^3 \setminus T^2$, $R^3 \setminus S^1$, $S^2 \bigvee S^1$

Here what i want to show is $B^3 \setminus T^2$, $R^3 \setminus S^1$, $S^2 \bigvee S^1$, $i.e$, three spaces are deformation retract to each other. Can you give me some hints or concept(?) ...
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Can a connected finite-dimensional manifold have cardinality $>2^{\aleph_0}$?

Can a connected finite-dimensional manifold have cardinality $>2^{\aleph_0}$? I know that if we either impose the condition "Hausdorff" or "second countable", the assertion is false. What if we ...
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Can a connected second countable manifold have cardinality $>2^{\aleph_0}$?

Can a connected second countable manifold have cardinality $>2^{\aleph_0}$? From here, a connected Hausdorff manifold must have cardinality $2^{\aleph_0}$. How about if we change the condition "...
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Every compact hypersurface in $\mathbb{R}^n$ is orientable

Show that every compact hypersurface in $\mathbb{R}^n$ is orientable. HINT: Jordan-Brouwer Separation Theorem. This is an exercise from Guillemin and Pollack. So hypersurface means smooth ...
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Endomorphism vector bundle isomorphic to the adjoint bundle of its frame bundle?

Could somebody help me to prove the following isomorphism (in particular what is the isomorphism)? \begin{equation} End(\xi) \cong ad(E_{\xi}) = E_{\xi} \times_{GL(n,\mathbb{R})} \text{Mat}_n(\...
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Exterior derivative as a special case of covariant derivative?

In terms of local coordinates we can make a covariant derivative exterior-derivative-like (this is actually Levi-Civita connection) \begin{equation} \Gamma_{i,j}^k = \Gamma_{j,i}^k \implies D(dx_k) =...
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Degree 1 map from torus to sphere

I'm trying to find a smooth degree 1 map from the torus $T^2 = S^1 \times S^1$ to the 2-sphere $S^2$. My first thought was to use the two coordinates $(\theta_1,\theta_2)$ to map onto the usual ...
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de Rham cohomology on finitely smooth manifolds

In all of the places I've looked, de Rham cohomology is defined on $\mathcal{C}^\infty$ manifolds with $\mathcal{C}^\infty$ differential forms. What about de Rham cohomology on $\mathcal{C}^r$ ...
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A smooth function $f$, defined on an open ball in $\mathbb{R}^n$, can be written the sum of $n$ smooth functions with a certain property

Let $f: B \to \mathbb{R}$ be a $C^\infty$ function on an open ball $B := B_r(a) \subseteq \mathbb{R}^n$. I want to show that there exist $C^\infty$ functions $g_1, ..., g_n: B \to \mathbb{R}$ with (...
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Discretizations of Differential, Geometric and Topological Notions

I have noticed a recurring theme in Graph Theory / Theoretical Computer Science (abbreviated GT and TCS throughout this post) in that notions typically belonging to differential calculus / geometry / ...
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Homotopy of space of immersions, Smale-Hirsch theorem

If $M$ and $N$ are simply connected manifolds with $\dim M< \dim N$, we denote by $Imm\left(M,N\right)$ the space of immersions of $M$ in $N$. Let $M$ and $M'$ manifolds of dimensions $m>0$. ...
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An orientation on a $(n-1)$-dimensional submanifold.

This question goes on where this question ended. Given is an non-empty $(n-1)$-dimensional ($n\ge2$) differentiable submanifold $X\subset\mathbb{R}^n$ such that there exists an open $U\subset\mathbb{R}...
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Topological boundary of a specific manifold.

Let $\emptyset\neq X\subset\mathbb{R}^n$, $n\ge2$, be an $(n-1)$-dimensional differentiable submanifold, i.e. for every $p\in X$ there is an open $U_p\subset\mathbb{R}^n$ with $p\in U_p$, and a ...
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Is $S^3\times S^2$ orientable?

The question comes from another question, I am asked to calculate the dimension and check orientability of the manifold $$ V_2(\mathbb{R}^4) = \{(v_1,v_2) \in \mathbb{R}^4\times \mathbb{R^4} \mid |...
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Why are tangent vectors coordinate-dependent?

Why does the coordinate basis for $T_pM$ depend on the coordinate chart we are using? Any two charts containing $p$ agree on some neighborhood of $p$, so shouldn't we be able to find a basis for $T_pM$...
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Can we lower bound the volume of the image of a ball under a diffeomorphism?

Apologies if this question is overly simple, I'm new to differential geometry. Suppose I have two Riemannian manifolds $M_1$ and $M_2$, along with a diffeomorphism $f:M_1\to M_2$ between them. Let $d$...
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Finding all $2$-forms in the right half-plane that are invariant under glide transformations

I'm trying to find all 2-forms $\omega$ that are invariant under glide transformations in the right half-plane, $X = \{ (x,y) \in \mathbb{R}^2 : x > 0\}$. To do this, we can write the vector field ...
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Proving that $N$ is a manifold.

I'm dealing with the following exercise from Munkres' "Analysis on Manifolds": Let $f:\mathbb R^{n+k}\rightarrow \mathbb R^n$ be of class $C^r$. Let $M$ be the set of all $x$ such that $f(x)=0$. ...
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$A\cup \text{Fr}A$ is a manifold

I'm dealing with the following exercise from Spivak's "Calculus on Manifolds": Let $A\subset \mathbb R^n$ be an open set such that the (topological) boundary of $A$, $\text{Fr}A$ is an $n-1$ ...
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Every open cover of a smooth Manifold has a regular refinement

I am trying to understand the proof of Let M be a smooth manifold. Every open cover of M has a regular refinement. The proof begins as follows [Lee] : Let $X$ ...
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Is there exist a smooth function $f:\mathbb{R}^{m}\to M$ such that $f|_{U′}=\phi^{-1}$?

Let $M$ topological smooth manifold and $(U,\phi)$ chart fixed with $\phi(U)=U′$ open in $\mathbb{R}^{m}$. Is there exist a smooth function $f:\mathbb{R}^{m}\to M$ such that $f|_{U′}=\phi^{-1}$? I ...
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How these charts are written?

The spherical coordinate map$$σ(u, v) = (\cos u \cos v, \cos u \sin v,\sin u), −π/2 < u < π/2, −π < v < π,$$ and its variation $$σ˜(u, v) = (\cos u \cos v,\sin u, \cos u \sin v), −π/2 < ...
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Prove that integration of a differential $k$-form is independent of choice of basis

This is Exercise 4 of Section 33 of Munkres' "Analysis on Manifolds" book: (Let $A$ be an open set in $\mathbb{R}^k$.) If $\eta$ is a $k$-form in $\mathbb{R}^k$ and if $\mathbf{a}_1,\ldots,\mathbf{a}...
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Reference about the space of closed curves in Riemann manifold

Some days ago, I listened a report about the width of Riemann manifold. I am interesting in the space of closed curves in Riemann manifold. It seemly has good topology and different construction.For ...
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Smoothing a continuous section in a fibre bundle.

Let $\pi: X \rightarrow M$ be a smooth fibre bundle and let $p^{1}_{0} : X^{(1)} \rightarrow X$ be its 1-jet bundle. Suppose there is a $\mathcal{C}^{1}$ section $h: M \rightarrow X$ such that it is ...
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On lifts of a trajectory of a quadratic differential

Let $X$ be a Riemann surface of genus $g\ge 2$ and $q$ a holomorphic quadratic differential on $X$. The differential $q$ defines a flat metric with conical singularities on $X$: if $q=f(z)dz^2$ ...
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Cobordism Groups an the Pontryagin-Thom Construction

I am confused by the statement that $\Omega^\text{framed}_1(S^3) \cong \mathbb{Z}$ which I came across as an application of the Pontryagin-Thom construction for showing that $\pi_3(S^2) \cong \mathbb{...
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Generate Lie-Algebra $su(N)$

Let $A \in \mathbb{C}^{n \times n}$ be a diagonal matrix with $Tr(A)=0$ and all eigenvalues (purely imaginary eigenvalues) are mutually different from each other. Hence, in particular $A \in \mathfrak{...
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Does there exist a non-parallelisable manifold with exactly $k$ linearly independent vector fields?

For any positive integer $k$, is there a smooth, closed, non-parallelisable manifold $M$ such that the maximum number of linearly independent vector fields on $M$ is $k$? Note that any such $M$, ...
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Computing the degree of a one-variable map

Edit: This question originally contained a typo where the function $f$ specified below was equal to $x$, not $x^2$ as currently written, outside an interval $[-T,T]$, and the accepted answer was ...
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Can we recover a compact smooth manifold from its ring of smooth functions?

It is well-known that if $X$ is a reasonably nice topological space (compact Hausdorff, say) then we can recover $X$ from the ring $C(X)$ of continuous functions $X\to\mathbb R$; see this MSE question ...
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Example of a diffeomorphism from all of $\mathbb{R}$ to itself

I can think of diffeomorphisms from an interval to $(a,b)\rightarrow \mathbb{R}$, scaling the tangent function, and from the punctured plane, polar coordinates, or some odd polynomial, but does anyone ...
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O.D.E. in Homogeneity Lemma

Let $\psi: \mathbb{R}^{n} \to \mathbb{R}$ smooth such that $\psi(x) > 0$ for $x \in B(0,1)$ and $\psi(x) = 0$ for $x \notin B(0,1)$. Let $c \in S^{n-1}$ fix and arbitrary and consider the O.D.E. ...
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Computing relative de Rham cohomology?

Let $f: N \to M$ be a smooth map. Then in Bott and Tu they define a relative de Rham complex $\Omega^{\bullet}(f)$ where $\Omega^k(f) = \Omega^k(M) \oplus \Omega^{k-1}(N)$ with coboundary map given in ...
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Which constructions on vector bundles satisfy a universal property?

I'm wondering whether constructions on vector bundles, such as the Whitney sum or tensor product of two bundles, satisfy some kind of universal property. What I mean by "some kind of universal ...
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How to keep symplecticity of a diffeomorphism after a coordinate rescaling and Taylor series expansion?

Let's take some, implicitly written, symplectic diffeomorphism $F$ of $\mathbb{R}^{2n}$, let it preserve the (possibly non-standard) symplectic form $\omega$. Suppose I introduce a small parameter $\...
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Why do we need the $S \otimes L^{1/2}$ bundle product to determine a $Spin_c$ structure?

I am reading Marino's book on topological field theory and 4-manifolds and I am very confused in the construction of the $Spin_c$ structures for manifolds that do not admit a $Spin$ structure. In the ...
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Construct a lifting from the the total space of a fiber bundle to its spoace of 1-jets

I am reading the book "Convex Integrtion Theory by D.Spring" and need some help. Let $\pi:X \rightarrow V$ be a smooth fiber bundle over a smooth base manifold $V$. Let $h\in\Gamma^{0}(X)$ and $\phi\...
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What does $C^k$ at a single point mean?

I'm reading textbooks on manifolds. I usually see a saying that "function $f$ is $C^k$ at a point $p \in M$". I am confused with this. What does this mean? Furthermore if $f$ is $C^k$ at $p$, is it ...
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Derivative of vector field on a manifold sends the tangent plane to itself

A vector field $\vec{v}$ on a (smooth) manifold $X \subset \mathbb{R}^N$ is a map $\vec{v}:X \to \mathbb{R}^N$ such that $\vec{v}(x)\in T_x(X)$ for every $x \in X$. Suppose $\vec{v}(x)=0$, show that ...
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Let $f : M\to N$ be a submersion with $M$ compact and $N$ connected. The $f$ is surjective.

I have no idea how to do this. I tried think in, once $N$ is connected and locally path connected it has to be path connected, but, this does not help. Any hints, solutions, will be very appreciate... ...
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Example of a disconnected manifold where the tangent space is not the dimension of the manifold?

Wikipedia says that the tangent spaces of a connected manifold all have the same dimension, equal to that of the manifold. Well, is there an example of a simple disconnected manifold that doesn't ...
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De Rham cohomology ring of flag bundles/manifolds in Bott and Tu

I'm trying to understand the result for the de Rham cohomology ring of flag manifolds in Differential Forms in Algebraic Topology by Bott and Tu. I'm sort of starting from the result and working ...