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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De Rham cohomology, and forms on manifolds

In String Theory and M-Theory by Becker, Becker and Schwarz, they introduce a group, $$C^{p}(M)$$ which they denote the group of all closed $p$-forms on the manifold $M$. Furthermore, they state ...
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When $f\colon M\to N\times N$ satisfies that $f^{-1}(\Delta)$ is a ball?

Let $M$, $N$ be smooth manifolds of dimension $m+n$ and $n$, respectively. Suppose that $f\colon M\to N\times N$ is a smooth map and $f$ is transversal to the diagonal $\Delta=\{(x,x)\in N\times ...
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123 views

Reviewing the basics of algebraic topology for further deeper study

I am entering a Ph.D. program in pure math this fall. Over the summer, I am hoping to review the basics of algebraic topology (fundamental group, covering space theory, etc.) and I wondered if you had ...
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119 views

Cauchy's Theorem by Differential Geometry

Is there a prove of Cauchy's theorem footing on the topology of the complex plane (homotopy, differential forms, etc.)? More specific consider a differentiable Banach space valued complex function. ...
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35 views

Making foliations C^1 smooth

I am not an expert in foliations, but I'm interested in the following conjecture of Cantwell and Conlon (from "An interesting class of $C^1$ foliations"): "If $F$ is a transversely orientable ...
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146 views

Is there any embedding theorem for fibre bundles?

I would like to know whether there is an embedding theorem for fibre bundles, like Whitney embedding theorem. When can a given fibre bundle be a subbundle of some higher dimensional bundle?
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147 views

smooth surjective map from lower dimensional manifold onto higher dimensional manifold?

I'm thinking about whether there exists a smooth surjective map $f: M^m \twoheadrightarrow N^n$ where both $M$ and $N$ are smooth manifolds, and $\dim M = m < n = \dim N$. (We might assume that $M$ ...
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42 views

express skew-commutative product of Whitney sum of vector bundles in tensor products

Let $\xi$ and $\eta$ be vector bundles over a paracompact space $B$ and $\xi\oplus\eta$ be their Whitney sum. Can we write $\Lambda(\xi\oplus\eta)\cong \Lambda(\xi)\otimes \Lambda(\eta)$ as (graded) ...
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show that $\omega$ is exact if and only if the integral of $\omega$ over every p-cycle is 0

Let $M$ be an oriented smooth manifold and $\omega$ a closed $p$-form on $M$. Show that $\omega$ is exact if and only if the integral of $\omega$ over every $p$-cycle is $0$. In particular, how to ...
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238 views

show that the compactly supported De Rham cohomology groups $H^{p}_{DR}(\mathbb{R^n})$ are all zero for $0\leq p<n$

I am reading John Lee's book and there is a problem on De Rham cohomology: show that the compactly supported De Rham cohomology groups $H^{p}_{DR}(\mathbb{R^n})$ are all zero for $0\leq p<n$. My ...
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24 views

generation of sub bundle

Let $M$ differentiable manifold with $\dim M=n$. If $(TM,\pi,M)$ be the fiber bundle tangent. Consider the family $E=\lbrace E_x\rbrace _{x\in M}$ such that $E_x \subset T_xM$ and $\dim E_x=k$ for ...
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48 views

Finding zeros of maps between manifolds with different dimensions

One of the questions for which the notion of degree is useful is: does this map have a zero. For example, one can prove the Fundamental Theorem of Algebra using the following fact involving the ...
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124 views

Are $C^{k}$ manifolds the same as $C^{\infty}$ manifolds?

This is a theorem of Hassler Whitney: For $0<k<\infty$ and any $n$-dimensional $C^k$ manifold the maximal atlas contains a $C^\infty$ atlas on the same underlying set. It seems to me ...
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61 views

Constructing maps of degree $k$

One of the common constructions one finds when first learning about the (topological) degree of a map is the construction of maps $f_k:S^n\rightarrow S^n$ of degree $k\in\mathbb{Z}$ (i.e. ...
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28 views

About embeddings of connected sums

Let $M_1$ and $M_2$ be two soomth manifolds who're already embedded in $\mathbf{R}^k$. Can one prove that the connected sum of $M_1$ and $M_2$ can also be embedded into $\mathbf{R}^k$ ?
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175 views

Transverse intersection in a compact manifold

Is it true that if $M$ is a compact manifold and $X,Y$ are submanifolds of $M$ which intersect transversely that the intersection $X\cap Y$ consists of finitely many points? I'm trying to understand ...
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226 views

Is the Hopf fibration the only fibration with total space $S^{3}$?

Let $S^{1}$ bundle over $S^{2}$ is homeomorphic (diffeomorphic) to $S^{3}$. Is the Chern class of the fibration 1, i.e. is the Hopf fibration up to isomorphism the only fibre bundle with total space ...
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the knot surgery - from a $6^3_2$ knot to a $3_1$ trefoil knot

It is intuitive that one can simply doing a cut-gluing surgery to make a $6^3_2$ to a $3_1$ trefoil knot: e.g. from to All one needs to do it to cut the three intersections at the angle of ...
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let $\xi$ be an arbitrary vector bundle. Is $\xi\otimes\xi$ always orientable?

Let $\xi=(E,p,B)$ be a line bundle (not nec. orientable). Then the tensor product $\xi\otimes \xi$ is orientable. I obtain this by choosing $b\in U\cap V$, $U,V$ open in $B$ such that ...
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199 views

tensor product of two vector bundles

Let $\xi$ and $\eta$ be two vector bundles over the same base space $B$. Then $\xi\otimes \eta$ is orientable if and only if $\xi$ and $\eta$ are both orientable. How to prove this true or not true? ...
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Three linearly independent vector fields

How can one find three linearly independent vector fields on $S^1\times S^2$? I know that $S^1\times S^2 \cong SO_3( \mathbb{R})$, i.e. the set of orthogonal $3 \times 3$ matrices with determinant ...
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61 views

Structure of level sets of a noncritical point of a smooth function on a two dimensional domain

Let $\psi$ be a smooth function on a two dimensional simply connected domain $\Omega$ such that $\psi=0$ on the boundary $\partial \Omega$. Suppose $\rho$ is not a critical value of $\psi$ then it is ...
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191 views

$U(n)$ and $U(1)\times SU(n)$ are not isomorphic Lie groups

I am reading John Lee and on the chapter about group actions there is a problem that asks me to show that $U(n)$ and $U(1)\times SU(n)$ are not isomorphic Lie groups by showing that they don't have ...
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Is the $C^r(M, N)$ space, with the strong (Whitney) topology, a Fréchet-Urysohn space?

Given smooth, non-compact manifolds $M$ and $N$, consider the function space $C^r(M, N)$. Equipped with the strong (Whitney) topology, this space is Hausdorff and Baire. It is, however, not first ...
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75 views

Orientation of manifolds with boundary

I have an ambiguity about how to orient the boundary of a manifold. In particular : Consider the example $M=B^2 \subset \mathbb{R^2}$ be the manifold with boundary. suppose positive orientation for M ...
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154 views

Picture behind $SO(3)/SO(2)\simeq S^2$

Is there some kind of intuitive/waving hand argument to explain that $$SO(3)/SO(2) \simeq S^2 \; ?$$
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29 views

define distance in a manifold over the reals

G is a Hausdorff manifold over the reals with a finite atlas: $\exists m$ $G=\bigcup_{1 \leq i \leq n}U_i$, $g_i:U_i \to g_i(U_i) \subseteq \mathbb{R}^m$. Can I somehow define a metric inside G, ...
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93 views

Atlas on a smooth manifold that contains 2 charts

How can I show that if an atlas on a smooth manifold has exactly 2 charts then it is orientable? How do I make sure that the Jacobian of the transition map is positive?
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71 views

problem on sheaves

I want to show the following. Suppose $X$ is a smooth manifold and F,G are sheaves of $C^{\infty}_{X}$-modules, then the natural map $Hom(F,G)\to Hom(F(X),G(X))$ is injective. It's easy to see that ...
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Transversality of Subbundles

It is known that transversality of submanifolds is generic in the sense that two submanifolds could be made transversal by small perturbations. I was wondering if the same is true for subbundles of ...
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Sequence of critical values has no cluster point (Milnor, Morse Theory)

The following Claim is used in the proof of Theorem 3.5 in John Milnor's "Morse Theory": Claim: Let $f: M \rightarrow \mathbb{R}$ be a differentiable function on a manifold $M$ with no degenerate ...
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Tangent bundle of $S^1$ is diffeomorphic to the cylinder $S^1\times\Bbb{R}$

How do I construct an explicit diffeomorphism between $TS^1$ and $S^1\times\Bbb{R}$? It will be something like $\phi:TS^1\to S^1\times\Bbb{R}, (x,v)\to(x,...)$. Also we know that for $x=(x_1,x_2)$ ...
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Tangent space to the graph of a function

$X$ and $Y$ are smooth manifolds and let $f:X\to Y$ be a map. Let $\Gamma$ be the graph of $f$ in $X\times Y$. Prove that $T_{(x,y)}\Gamma$ is the graph of $df(x):T_xX\to T_yY$. $\Gamma$ need not be ...
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100 views

Finding the critical points of a map on the torus

Consider the $2$-torus $T$ obtained by revolving about the $z$-axis the circle $(x-2)^2+z^2=1$. I want to find the critical points of the map $f:T\to\Bbb{R}$ defined by $f(x,y,z)=x$. The equation of ...
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104 views

About “good” covers of manifolds

Let $M$ be compact manifold of dimension $n$. I am wondering if a certain type of covering of M by coordinate charts exists (this is not the usual good covering theorem but seems related to it). I ...
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Are there any results linking hyperbolic operators and Pseudo-Riemaniann geometry?

In the case of a $n$-dimensional pseudo-riemannian manifold one has a symmetric bilinear form, $h$, with signature $(1,n)$ and the analogous operator to the Laplace -Beltrami operator,$\Delta$, is the ...
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99 views

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, ...
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77 views

Symplectic submanifolds in $\mathbb{R}^{4}$

Which symplectic submanifolds can be realized in $\mathbb{R}^{4}$? It easy to show that such submanifolds aren't compact. So, they are spheres with some handles and holes. Which relations between the ...
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250 views

Definition of smooth maps between manifolds

Here is a page from Guillemin-Pollack's differential topology: My question is: At the bottom he defines $df=d\psi\circ dh\circ d\phi^{-1}$. Why doesn't he just define $df=dh$, like here: ...
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141 views

Upper-half space of a manifold with boundary

Suppose that $X$ is a manifold with boundary and suppose that $x$ is a boundary point. Define the upper half space $H_x(X)$ in $T_x(X)$ to be the image of $\mathbb{H^k}$ under $d \phi_0:\mathbb{R^k} ...
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submanifold of Euclidean space is oriented if and only if normal bundle is an oriented vector bundle.

Let $f:M\longrightarrow \mathbb{R}^{n+k}$ be an immersion of $n$-dimensional manifold $M$ into $\mathbb{R}^n$. Let $\nu(M)$ be the normal bundle of $M$. Prove that $M$ is oriented if and only if ...
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Regular values and manifolds with boundary

Question: Let $X^m$ and $Y^n$ differentiable manifolds. $f:X\rightarrow Y$ a differentiable map. Show that if $\partial X=\emptyset$, $y\in Reg(f)$ and $f^{-1}(y)\neq\emptyset$, then $y\not\in\partial ...
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145 views

Definition for Euler characteristic without CW-complexes

It is possible to have a definition of the Euler characteristic without using CW-complexes? (I'm referring to the definition given by Wikipedia : ...
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151 views

measure on non-oriented Riemannian manifold

Let $M$ be a non-oriented Riemannian manifold of dimension $m$. Nash embedding theorem implies that there exists an isometric embedding $\phi: M\longrightarrow \mathbb{R}^n$ for $n$ sufficiently ...
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84 views

Tangent bundle of a manifold [duplicate]

can anyone help me with this problem: Show that for a manifold $M$, the tangent bundle $TM$ also has the structure of a manifold. If $M$ is an n-manifold, what is the dimension of $TM$? for the 1st ...
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1answer
125 views

Construction of Hodge decomposition

We know Hodge decomposition splits any $k$-form into three $L^2$ components. And I see some proofs, none of them provide an explicit constructive method. Is there any general method to construct one? ...
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85 views

handle moves: proof

In several 4-manifold textbooks, when handle moves (creation, cancellation, sliding) are discussed, they are explained using very helpful drawings. However, I would like to know if there is a ...
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43 views

Proof that I can always get a height function that is Morse.

So a height function $h(x_{1},...,x_{m})=x_{k}$ for mfld $M^{m}\subset \mathbb{R}^m$. I proved that Morse functions are dense in $C^{\infty}(M,\mathbb{R})$. So I can approximate h by Morse functions, ...
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154 views

Contractibility of the space of sections of a fiber bundle

Let $\pi: E \to M$ a fiber bundle and $\Gamma(M,E)$ the space of smooth sections of the bundle with topology induced by the Whitney topology on $C^{\infty}(M,E)$. Assume that each fiber is ...
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Proving that a regular value of a smooth function isn't in the boundary of the counter-domain

Suppose $X$ is a manifold without boundary and $Y$ is a manifold. Suppose there is a smooth function $f: X \rightarrow Y$ and we are given a $y \in Y$ such that $y$ is a regular value of $f$ and ...