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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On the “regularity” of the boundary of an open set

Let $M = \mathbb{R}^2$ (or more generally, let $M$ be a topological manifold) and let $\Omega$ be an open set in $M$. I'm considering the following regularity conditions for the boundary of $\Omega$: ...
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Second Stiefel-Whitney Class of a Five Manifold

There is a unique rank 4 nontrivial orientable vector bundle over the 2-torus, denote this by $p:E\rightarrow T^2$. Denote the associated sphere bundle by $S(E)$. Then since $S(E)$ is orientable, the ...
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On the integration of differential forms

In my notes the integration of a differential form on an oriented manifold $M$, with $\{(U_\alpha,\phi_\alpha) \}$ oriented atlas, is defined as: $\int_M \omega = \sum_{i \in \mathbb{N}}\int_M ...
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Proving that charts are related

Let $A$ be an atlas on the set $M$ and let $ x: U \to x(U) $ and $ y : V \to y(V )$ be bijections from subsets $U, V \subset M$ to open sets $x(U), y(V ) \subset \mathbb{R^n} $. Show that if the ...
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prove that this function is an immersion

How I can show that $F: \mathbb{R} \rightarrow \mathbb{R}^2$ defined by $F(t)=(\cos(t),\sin(t))$ is an immersion? In my definition $F$ is an immersion if $\forall p\in\mathbb{R}$, $dF_p$ is injective. ...
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prove that a function is an immersion

How I can show that $F \colon \mathbb{R} \to \mathbb{R}^2$ defined by $F(t)= (\cos (t), \sin(t))$ is an immersion? In my definition $F$ is an immersion if $\forall p$,$dF_p$ is injective. I have ...
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Product manifolds

I have a question on the product of two manifolds. I have $M, N$ two real manifolds (with a smooth differentiable structure), with $\partial M=0$. I have showed that $M\times N$ has a natural induced ...
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homeomorphism between maninifolds

Exist a local homeomorphism between the manifolds with boundary $[0,1) \times [0,1) $ and $\mathbb{R}^{2}_{+}$? I don't think that a local homeomorphism like this can exist..
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Determine $n$ so that manifold is locally homeomorphic to $\mathbb{R}^n$?

I just started studying smooth manifolds. The definition of a topological manifold requires a topological space to be locally Euclidean: homeomorphic to $\mathbb{R}^n$. I know some examples, like how ...
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integration of differential forms on covering space

Let $M_1,M_2$ be $n$-dimensional oriented manifolds. Let $f: M_1\longrightarrow M_2$ be an orientation-preserving diffeomorphism. Then for any $\omega\in \Omega^n_c(M_2)$ we have(page 85 of {Madsen: ...
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What is the best (in terms of effectively building understanding) direction from which to approach manifolds?

Thedore Frankel's book The Geometry of Physics presents Manifolds right away in Chapter 1 in the following manner: Introduce the Euclidean space $\mathbb{R}^N$ only as "the most important manifold". ...
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Source for Differential Manifolds/Geometry Questions?

I'm looking for a good source for a large collection of Differential Manifolds/Geometry questions covering a subset of the following topics: inverse function theorem, local coordinates, induced ...
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$S^2 \times S^2$ is diffeomophic to $G_2(\mathbb{R}^4)$

$G_2(\mathbb{R}^4)$ is the Grassmannian manifold of two-dimensional subspaces of $\mathbb{R}^4$. I would like a detailed proof. Can it be done explicitly? I mean, showing the map and checking its ...
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Is there a topological space and meanwhile a linear space such that its vector addition is discontinuous but scalar multiplication is continuous?

The title is the question. Does there exist a topological space and meanwhile a linear space X such that its vector addition operation is discontinuous but scalar multiplication operation is ...
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Proving Borsuk-Ulam with Stokes

What is the easiest way to deduce the Borsuk-Ulam theorem in the case $n=2$ by using integration on manifolds and Stokes theorem? So I want to prove the following: Given a map $f\colon S^2\rightarrow ...
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Compute $\int_0^{x_0}f^\prime(x)$ and $\int_{x_0}^{x_1}f^\prime(x)$

Suppose $f: S^1 \to S^1$, and $f(x_i) = y$, where $x_i$s are the preimage of a regular value $y$. Then how can I compute $\int_0^{x_0}f^\prime(x), \int_{x_0}^{x_1}f^\prime(x),$? I realize that ...
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Condition for Differential Forms to Pass to the Quotient

everyone: I was reading this question : What do we mean when we say a differential form "descends to the quotient"? which is related to mine. But the reply given did not answer my question ...
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The transverse root for a vector field

I encountered the term, so may I ask - what is the transverse root for a vector field? Thank you~
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The piecewise-smooth homotopy

Let $M$ be a smooth manifold. If $\gamma_1:I \rightarrow M$ and $\gamma_2:I\rightarrow M$ are two piecewise-smooth homotopic curves (rel to endpoints), then can we find a map $f:I\times I\rightarrow ...
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A homework problem about Tangent space of Lie groups at the identity element

This is one of our homework problem. Let $G$ be a Lie group be defined as a manifold with group structure such that the map $F:G\times G \mapsto G, F(a, b)=ab^{-1}$ is smooth. Show that ...
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Section of a vector bundle as a submanifold

I am currently working on one part of a problem surrounding sections and submanifolds. Given a real vector bundle $\pi: E\rightarrow M$ of rank k, with a smooth global section $s:M\rightarrow E$, can ...
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Crosscap function in $\mathbb{R}^4$ - and how to show it is proper?

I found the Cross-cap function in $\mathbb{R}^3$ as follows: $$f(x,y,z)=(yz,2xy,x^2-y^2),$$ My questions are (I couldn't show any progress for Q1,2.I have thought hard but had no clue): Q1: Is ...
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Is this curve a minimal geodesic?

Please help me identify this curve. I strongly feel that it is a minimal geodesic, but I cannot show it. Suppose that $M$ is a Hadamard manifold (i.e., complete, simply-connected smooth Riemannian ...
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Diffeomorphism $S^1\to S^1$ extends to a diffeomorphism $D^2\to D^2$

Suppose $\phi:S^1\to S^1$ is a diffeomorphism. We can think of $D^2$ as the half sphere: Actually this half sphere is just the rotation of the semicircle $C$ from $0$ to $\pi$ (I mean we rotate ...
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transverse homotopy

Let $f,g:M \rightarrow N$ two smooth maps between smooth manifolds that are smoothly homotopic by $F$. Suppose also that $f$ and $g$ are transverse to a submanifold $A$ of $N$. I know that transverse ...
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225 views

a Problem about Degree of Map between Spheres

When I read the book "Algebra Topology-A First Course", I find a problem. It is on the Page 97, Exercise (16.15). Problem We define $f$, $g\colon \mathbb{S}^{n}\to\mathbb{S}^{n}$ to be orthogonal ...
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Are there degree-1 maps from $S^2 \times S^3 \rightarrow S^5$ or from $S^5 \rightarrow S^2\times S^3$?

This is a question from a past qualifying exam I am stuck on: For a smooth map $f:M\rightarrow N$ between smooth, compact, oriented $n$-manifolds, the degree of $f$ is the unique integer $k$ such ...
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$GL(n,\mathbb{C})$ is semi-locally simply connected.

How can we show that $GL(n,\mathbb{C})$ is semi-locally simply connected.
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Loops in $RP^2$

We know that $\pi_1(RP^2)=Z_2.$ How do non-trivial loops in $RP^2$ look like? (If $RP^2$ is the upper hemisphere $D$ with antipodal points of $\partial D$ identified)
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Intersection number on $S^k$

Is the intersection number on $S^k$ is always zero? Choose $X$ a compact submanifold of $S^k$, and $Z$ a closed submanifold of complementary dimension, viewing $I_2(X,Z) = I_2(i, Z)$ where $i: X ...
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Moebius strip as a fibre bundle

I've alrealdy asked this question, but now I have more clear ideas, so I'm going to ask again and see if I'll understand a bit more. It's about the trivialisation of the Moebius strip as a bundle on ...
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Submersions and complex structure

Let $f : \Lambda \rightarrow X$ be a continuous surjective map, where $\Lambda$ is a complex manifold and $X$ a topological space. Suppose that for all $x \in X$, there is a neighborhood $U_x$ of $X$ ...
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Show $\exists p \notin f(X) \cup Z$ by Sard

Guillemin & Pallack P83, Ex 2.4.9: Suppose $X$ compact with $0< \dim(X) < k$ and $f: X \to S^k$. Suppose $Z \subset S^k$ a closed submanifold with $\dim(X) + \dim(Z) = k.$ Show that ...
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$dg_y$ carries a subspace of $T_y(Y)$ onto $\mathbb{R}^l$

The question arises from Guillemin and Pallack Page 28 above the frame: $dg_y$ carries a subspace of $T_y(Y)$ onto $\mathbb{R}^l$ precisely if that subspace and $T_y(Z)$ span all of $T_y(Y)$. I ...
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Are space of paths between two different points and space of pointed loops only homotopy equivalent? What about smooth case?

Let $X$ be a path-connected CW-complex and $x$, $y$ points in $X$. Any choice of a path between $x$ and $y$ provides maps (in both directions) between the space $L(x, y)$ of paths from $x$ to $y$ and ...
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Prove where exp: Skew($3\times 3$) $\rightarrow SO(3)$ is local homeomorphism

The matrix exponential on skew-symmetric $3\times3$ matrices onto $SO(3)$ is not local homeomorphism everywhere. I have been instructed that one problem is with the spheres of radius $2n\pi$ ...
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Does 2 manifolds can be “isotoped away”?

Let $M,N\subset P$ be two manifolds such that dim($M$) + dim($N$) < dim($P$), suppose that $M$ is compact and $N$ is closed, is it true that there exists an isotopy $F$ of $M$ such that $F(M,1)\cap ...
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Topological property of some manifold

I am provided with a smooth map $g : N \rightarrow N^\prime $ between differentiable manifolds. $N$ is assumed to be compact and connected. Moreover, the differential $dg_x: T_x N \rightarrow ...
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Show $\lambda$ is smooth

Let D be the closed unit disc in $\mathbb R^2$ and $S^1= \partial D$. Let $f$, $g$: $S^1 \rightarrow \mathbb R^3$ be smooth embeddings s.t. $f(S^1) \cap g(S^1) = \emptyset$. Define $\lambda: S^1 ...
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Confused on Guillemin and Pollack's proof of the $\epsilon$-Neighborhood theorem.

On pg. 71 of Guillemin and Pollack they prove the $\epsilon$-Neighborhood theorem. Here $Y$ is a compact boundaryless manifold in $\mathbb{R}^M$. They say Proof: Let $h:N(Y)\to\mathbb{R}^M$ be ...
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Restriction of a line bundle to a a two-cycle

I am reading a paper on Chiral Differential Operators http://arxiv.org/pdf/hep-th/0604179v3.pdf and it says on page 23 that a line bundle over a manifold $C$ can be characterized completely by its ...
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Hairy ball theorem : a counter example ?

Recall the hairy ball theorem : any continuous vector field on a sphere $S^n$ of even dimension must vanish at least once. Now consider the Riemann sphere $\mathbb{C} \cup \infty \simeq S^2$. Let ...
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When can a connection be lifted?

Let $P \rightarrow X$ be a principal $G$-bundle, and $P' \rightarrow X'$ be a principal $G'$-bundle. Let $(f',f'')$ be a morphism from $P'$ to $P$, i.e., a pair of maps $f': P' \rightarrow P$ and ...
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Assumptions required for an implicitely defined surface/manifold to have a specified dimension

What are some normal assumptions made on implicitly defined manifolds? More specifically, by implicitly defined manifold, I mean the definition of a surface such as $g(x,y)=x^2+y^2-1=0$ for the ...
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Does a “typical” reproducing kernel on a manifold generate an infinite-dimensional RKHS?

Let $M$ be a finite dimensional manifold and $k:M\times M\rightarrow\mathbb{R}$ a smooth reproducing kernel. Is the set of all such $k$ with the property that the reproducing kernel Hilbert space ...
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Question about a lemma from Milnor's Topology from the Differentiable Viewpoint

I've been reading John W. Milnor's Topology from the Differentiable Viewpoint for some time and currently I'm stuck at a little lemma. I would appreciate if someone can clarify it to me. The details ...
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Preparing for “differential forms in algebraic topology”?

I'd very much like to read "differential forms in algebraic topology". Apart from background in calculus and linear algbra I've thoroughly went through the first 5 chapters of Munkres. I'm thinking ...
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Submersion implies every point is in the image of a local section

I want to show the following: Let $\pi:M\to N$ be a submersion. Then, every point of $M$ is in the image of a smooth local section of $\pi$. Since $\pi$ is a submersion, it is also an immersion ...
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Transversality of Vector Fields Defined in terms of Diff. Forms and Open Books.

All: Sorry for the length of the post, but I think it is necessary to set things up so that the post is understandable. I'm trying to understand how it is that the transversality (in this case , the ...
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Concrete non trivial computation of Morse homology

I am studying Morse homology and have found only examples on spheres and tori so far. Of course the homology of these manifolds is better understood by other more standard methods, so I am having ...