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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$dg_y$ 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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52 views

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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56 views

Is a Differential 1-form ( on M^3)Dual to a Contact Vector Field a Symplectic Form?

say $w$ is a global contact 1-form on a 3-manifold $M^3$ , meaning $w \wedge dw \neq 0$ at any point in the manifold , and let $X$ be the vector field dual ( under, say, a choice of Riemannian metric ...
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13 views

Persistence of invariant manifolds under noninvertible mappings

According to Fenichel's classic result, a normally hyperbolic invariant manifold of a diffeomorphism persists under perturbations to the diffeomorphism. Normal hyperbolicity requires contraction or ...
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26 views

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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44 views

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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23 views

An involutive property of the quotient bundle coming from a Lie group action.

Assume that a manifold $M$ is equipped with a (locally) free action of a compact Lie group $G$. Then the subbundle $$F = \lbrace X^\ast(x) \in TM \mid X \in \mathrm{Lie}(G), x \in M \rbrace$$ of $TM$ ...
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55 views

How to measure geodesic triangle angles?

Trying to learn some Topology. I understand simple proof in Theorem of Harriot and Girard when geodesic triangle interior angles are $\pi/2$. Is there a way to calculate the geodesic triangle ...
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42 views

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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39 views

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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2answers
167 views

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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47 views

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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1answer
37 views

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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68 views

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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1answer
83 views

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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191 views

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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1answer
30 views

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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57 views

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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65 views

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 ...
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37 views

Some questions on definition of Grassman manifolds

In W.M. Boothby, An introduction to differentiable manifolds and Riemannian geometry , Page 64. We use coordinate correspondences $\phi_j : U_j \to R^{k(n-k)}$ in order to define Grassman manifolds ...
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114 views

Splittings of 4 manifolds

I am wondering why a copy of $CP^2$ may be split off from $CP^2\mathbin\#9\overline{CP^2}$ to leave $\overline M_{E_8}\mathbin\#\overline{CP^2}$(«The wild world of 4-manifolds» by Alexandru Scorpan, ...
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42 views

Homological Interpretation of the intersection number

Let $M^r$ and $N^s$ be smooth submanifolds of the smooth manifold $V^{r+s}$ and $M,N,V$ compact, connected and orientable. The Thom-Isomorphism together with the Tubular Neighbourhood Theorem gives me ...
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62 views

$\Bbb{R}P^1$ bundle isomorphic to the Mobius bundle

This was asked several times on math.se but none of them was answered. I'm trying to construct an explicit isomorphism from $E = \{([x], v) : [x] ∈ \Bbb{R}P^1, v ∈ [x]\}$ to $T = [0, 1] × R/ ∼$ where ...
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53 views

Uniqueness of smooth manifold obtained from glueing two manifolds along their boundary

I have a question about a result in Lee's Introduction to Smooth Manifold(2nd Edition). Theorem 9.29 (Attaching Smooth Manifolds Along Their Boundaries). Let $M$ and $N$ be smooth $n$-manifolds with ...
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1answer
108 views

Question about image of proper smooth map of constant rank. (Undergraduate)

I have to find a proof of the following theorem Given a smooth function f between smooth manifolds X and Y that: has constant rank is proper the preimage of every point in f(X) is connected and ...
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2answers
77 views

differmorphism and homeomorphism for manifolds

For two abstract manifolds that are differmorphic, why are they always homeomorphic? Why does differentiability imply continuity for abstract manifolds? (for $R^n$ this is certainly clear)
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132 views

Reference request : Study of Differential topology post Milnor's book

I am just about to finish my study of Milnor's book 'Topology from the Differentiable Viewpoint' and I really love the subject. I would like to continue my study of Differential Topology and am ...
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38 views

Definition of linking number for disjoint submanifolds of the sphere- A problem from Milnor's book

In problem number 13 of Milnor's 'Topology from the Differentiable Viewpoint', the linking number for two compact boundary-less manifolds $M,N \subset \mathbb{R}^{k+1}$ of dimensions $m,n$ such that ...
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1answer
53 views

Real line bundle is smoothly isomorphic to Möbius bundle

I'm stuck on this question and tried to follow the partial answer of Neal. Erno's answer is fine too but it seems like I need to find the local trivializations of the Mobius bundle, which requires a ...
3
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2answers
53 views

zero of vector field with index 0

I'm currently studying vector fields on surfaces in the $\mathbb R^3$ and I currently I am doing some reading on the index of zeros of vector fields, which got me wondering: Is it possible to find a ...
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1answer
60 views

Is the classical derivative the weak derivative on any domain?

I was looking at the following definition of the weak derivative: Let $\Omega$ be a domain (ie an open connected subset of $\mathbb{R}^n$). Suppose $u,v \in L_{1,loc}(\Omega)$ and \begin{equation} ...
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1answer
50 views

open sets in a regular surface in $R^3$

I am reading Differential geometry of curves and surfaces by Do Carmo. Let S be a regular surface in $R^3$. I wonder How is open set of S defined? Is a subset of S open if and only if it is the ...
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1answer
46 views

Morse Sard Theorem

I'm reading the Morse-Sard's theorem in the book Differential Topology by Hirsch, and I wonder if anyone has a paper on the theorem with more details? I'm not proving it for manifolds, just open sets ...
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1answer
132 views

Is it possible to learn differential topology before analysis?

Currently I'm self studying for my own enjoyment topology and algebra (munkres and herstein). Since I start at the university next year everything I'm learning now is for my own enjoyment and I will ...
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1answer
24 views

zeroes of vector fields on surfaces

I know that for compact (smooth) surfaces in $\mathbb R^3$ the 2-torus is the only surface that has vector fields with no zeroes. What happens if we take the compactness of the surface away? Does this ...
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0answers
63 views

Confusing remark in Guillemin and Pollack

On page 23 of Guillemin and Pollack they say the following. Suppose $g_1,\dots,g_l$ are smooth, real-valued functions on a manifold $X$ of dimension $k\geq l$. Under what conditions is the set ...
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30 views

Cylinder as Fibre bundles

I have to show that the cylinder C is a fibre bundle over $S^1$ with fibre an open interval and I have to write a trivialization and the cocycles. I think that this is a trivial bundle, because I can ...
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25 views

Multiplication in homotopy groups and cobordism

Each homotopy class of a map of an n-sphere into the Thom space of the universal vector bundle determines a cobordism class of embedded smooth manifolds in Euclidean space. How do the cobordism ...
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2answers
96 views

How does one prove that the Klein bottle cannot be embedded in $R^3$?

How does one prove that the Klein bottle cannot be embedded in $R^3$? I'm talking about smooth embeddings.
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3answers
166 views

Application of Hodge decomposition

Hodge decomposition states any $p$ form can be decomposed into three orthogonal $L^2$ components: exact form, co-exact form and hamonic form. But actually we don't know how to decompose a general one. ...
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Locally Euclidean can be defined for whole of $\mathbb{R}^n$, or an open set or open ball of $\mathbb{R}^n$?

Topological manifolds are defined to be locally Euclidean (e.g. John Lee). That is, any point is in an open set that is homeomorphic to either $\mathbb{R}^n$, an open ball in $\mathbb{R}^n$ or an open ...
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5answers
179 views

Trivial tangent bundle of sphere with handles

I am wondering if there is a simple proof of this statement: A sphere with $g$ handles has trivial tangent bundle iff $g=1$ I know that it is a corollary of Poincaré-Hopf theorem, but it seems ...
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1answer
32 views

isometric imbedding of the projective plane

How to isometrically imbed the projective plane (identifying antipodal points of the unit sphere) in $\mathbb{R^5}$? My textbook indicates that there is an isometric imbedding of the projective ...
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33 views

Reference: forms invariant under Lie group action give the de Rham cohomology?

I'm looking for a reference for a proof of the following fact: Let $G$ be a compact, connected Lie group acting on a smooth manifold $M$. Then inclusion of the differential forms invariant under the ...
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1answer
64 views

Meaning of differentiability

Could anyone give an intuitive idea of the meaning of differentiability in general in any dimension and any space?
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1answer
48 views

Product of manifolds with boundary

If $M$ and $N$ are manifolds with boundaries and $\{(U_a,f_a)\}$ and $\{(V_a,g_a)\}$ are their respectives $C^r$ atlas, why $\{(U_a \times V_b,f_a \times g_b)\}$ isn't an $C^r$ atlas for $M \times ...
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2answers
57 views

Isometry of surfaces in $\mathbb{R}^3$

Let $F$ be an isometry of the Euclidean space $\mathbb{R}^3$. Hence $F$ is orthogonoal transform followed by translation by a constant vector. Let M be a surface of $\mathbb{R}^3$ that is connected, ...
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1answer
117 views

About Stokes' theorem

I noticed that in the proof of Stokes' theorem on manifolds, the condition that the form $\omega$ is compactly supported ensures that the sum is finite so that we can change the order of sum and ...