# Tagged Questions

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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### Intuitive idea of immersions?

So I understand the definition of immersions and submersions, as well as the motivation for defining such ideas. Not only are they important in understanding properties of mappings of tangent spaces, ...
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### why is there no non-degenerate 2-forms on 4-sphere?

The question is in the title. I have been told that there are actually no non-degenerate 2-forms on $S^{2n}$ for $n \neq 1,3$. I have found the following question: No symplectic structure on ...
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### Invariant forms on principal bundles

Let $\pi:M \to B$ be a principal $G$-bundle and $\xi$ a invarint $k$-form on $M$. Does $k> dimG$ implies that $\xi$ is a basic form (pull back of a $k$-form on the base manifold $B$)?
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### Is this a valid way to show $\chi(SL_n(\mathbb{R}))=0$?

Why does $\chi(SL_n(R))=0$? I'm going about it like this. Let $X:=SL_n(R)$. Define a map $f:X\to X$ such by $A\mapsto BA$, where $B$ is the identity matrix, except with an extra $1$ in the upper ...
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### Given points $x,y$, and tangent vectors $v,w$, is there a diffeomorphism $x\mapsto y$ and $v\mapsto w$?

Suppose $M$ is a smooth manifold, and you're given two points $x,y$, and associated tangent vectors $v$ and $w$. I'm curious, does there exist a diffeomorphism $F$ such that $F(x)=y$ and $dF_x(v)=w$? ...
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### Representative of a cohomology class in once punctured solid torus

Consider a once punctured solid torus $(\mathbb R^2 \times S^1) /\{pt\}$. It is not difficult to see that it is homotopy equivalent to the bouquet of spheres $S^2\vee S^1$. So this guy has a ...
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### Naturality of Lie derivatives

It was left to me as an exercise to show the naturality of the Lie derivative of arbitrary tensors on a smooth manifold. Is the following argument correct (it seems too easy)? Let $X$ be a vector ...
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### Orientable surface bundles over the circle and their structure group

I want to understand whether orientable surface bundles over the circle, i.e. with orientable total space, are always trivial, so I though I would revive an old post and ask for a few clarifications, ...
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### Degree of a vector field on the boundary of a punctured ball?

Suppose $v\colon \mathbb{R}^3\setminus\{x_1,\dots,x_n\}\to S^2$ is a smooth map, where all the points $x_i$ sit inside the unit ball $B^3$, and not on the boundary $S^2$. I suppose this map can be ...
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### rudin's principles of mathematical analysis 10.31

I'm working on rudin's principles of mathematical analysis(3rd edition). There is problem too complex for me to solve.Please help me. The problem is on p270-271 of text. "Let T be 1-1 mapping of ...
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### A doubt from Milnor's “Topology from a Differentiable Viewpoint”.

This is a doubt from Milnor's "Topology from a Differentiable Viewpoint". For a smooth $f:M\to N$, with $M$ compact, and a regular value $y\in N$, we define $n(f^{-1}(y))$ to be the number of ...
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### Is this orientation preserving or reversing?

I am confused about the definition of orientation on manifolds. Let $X=\{(x,y,0)\in \mathbb{R}^3 \mid x^2+y^2=1\}$ and $Y=\{(x,y,1)\in \mathbb{R}^3 \mid x^2+y^2=1\}$ be two one dimensional circles in ...
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### Trivial nature of orientable fibre bundle with cylinder base space

Given a bundle whose base space is a "cylinder", i.e. $\mathbb{S}^1 \times \mathbb{B}^n$ where $\mathbb{B}^n$ is an $n$-ball, is orientability (of the total space) enough to ensure that the bundle is ...
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### Differential Structure of Two Dimensional Manifold

I am a beginner in Differential Topology, and have learnt that Theorem One dimensional differential manifold without boundary is diffeomorphic to $\mathbb R$ or $\mathbb S$. So I am curious ...
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### Is there anything “nice” about the set of normal matrices (over $\Bbb R$ and $\Bbb C$?)

Normal matrices are of course useful to any linear algebra buff, not least because of the spectral theorem. However, taken as a whole, they tend to have some not-so-nice properties. For example: ...
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### Orientability of space with only even dimensional cells.

I have the following question: Suppose a compact $\mathbb{R}$-manifold has finite cell decomposition with only even dimensional cells. Then $M$ is orientable. It's a theorem that a closed ...
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### Transitivity of smooth submanifolds

I was reading through Guillemin and Pollack and was having trouble verifying this for myself. Given $M \subset N$ and $N \subset P$, where $M$ is a submanifold of $N$, and $N$ a submanifold of $P$, ...
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### How can we prove that $GL(2,\mathbb{R})$ is a topological group in $R^{4}$

A group is called topological group if it satisfies three properties 1) G is a Hausdorff space in K (Here we want to prove that if $A,B \in GL(2,\mathbb{R})$ then we can find two disjoint open sets ...
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Let $\xi_1,\cdots,\xi_n$ be vector bundles over $M$. Let $\omega$ be the Stiefel-Whitney class and $\bar\omega$ be the inverse. By an exercise in Milnor's book, $$w(\Pi_{j=1}^a ... 1answer 136 views ### Questions about simplex and affine space For the sake of completeness, I would like to give you some concepts before asking the questions: For every simplex S=<<x^{0},x^{1},...,x^{k}>> in \Bbb R^{n}, denote by H_s, the ... 1answer 73 views ### Is the image under a homeomorphism of the cut locus C_p a null-set? Let M be a complete Riemannian manifold with a point p \in M and let U \subset T_pM be an open disk containing 0_p in the tangent space to p. By C_p we denote the image of the boundary of ... 1answer 36 views ### Taylor development on manifolds and Manifolds of differentiable Mappings? I'm trying to study the book "Manifolds of Differentiable Mappings" written by Michor and I came across the following: He considers two smooth manifolds M and N and define an equivalence relation ... 1answer 44 views ### The space of collars of a manifold is contractible Theorem: Let M be a smooth manifold with boundary \partial M. Let e_0,e_1 : \partial M\times [0,1]\rightarrow M be collars of M, i.e. e_i are embeddings such that e_i(x,0)=x for each ... 0answers 47 views ### How to get a Kirby diagram of S^1 \times M^3 if M^3 is given by a surgery diagram? In "4-manifolds and Kirby Calculus" by Gompf and Stipsicz, there is a nice description of how to get the Kirby diagram of S^1 \times M^3, given a Kirby diagram of M^3. Basically, one thickens the ... 1answer 106 views ### Transversality, mod 2 degree, Winding numbers in differential topology From Chapter 2 Section 5 of Guillemin and Pollack, Differential Topology, \mathbf{X} is a compact connected manifold, and f:\mathbf{X}\rightarrow \mathbb{R}^n a smooth map and \dim{X}=n-1. ... 1answer 143 views ### Minkowski space is locally Euclidean? The Minkowski spacetime \mathbb{R}^{1,3} is said to be a manifold (isomorphic to SO^{1,3}. But according to the definition of a manifold it should be locally euclidean. However, this seems to be ... 1answer 528 views ### Analytic approximations of the step function Consider the Heaviside step function:$$H:\mathbb{R}\to \mathbb{R} $$defined by$$H(x)=\begin{cases} 0 & \mbox{if } x<0 \\ 1 & \mbox{if } x\geq 0\end{cases}$$Fix any \delta>0. Given ... 1answer 279 views ### C^{k}-manifolds: how and why? First of all, I have a specific question. Suppose M is an m-dimensional C^k-manifold, for 1 \leq k < \infty. Is the tangent space to a point defined as the space of C^k derivations on the ... 1answer 36 views ### In uniform circular motion in R^2, is acceleration in the normal bundle? In physics we learn that accleration is a vector quantity parallel to the radius and orthogonal to the velocity. With the embedding \mathbb{S}^1 \hookrightarrow \mathbb{R}^2 and the induced ... 1answer 147 views ### Homeomorphism of a Genus-2 Surface Does there exist a homeomorphism from a genus-2 surface, the connected sum of 2 tori, to two circles, S^1, intersecting at a point? Intuitively it seems that the double torus can be squeezed into ... 1answer 57 views ### Number of intersections of two closed loops on a genus zero surface I have stumbled onto the following fact and I am quite helpless in seeing why this is true (although I can agree intuitively). Let M be a surface of genus zero (open or closed, with or without ... 3answers 109 views ### T^2\times S^n is parallelizable This is taken from a UCLA Geometry/Topology qualifying exam. How would one prove that T^2\times S^n is parallelizable for all n\geq 1? Is there a way to find n+2 linearly independent vector ... 4answers 1k views ### Why is there no natural metric on manifolds? One of the things that always bothered me after learning introductory differential geometry (as a physics student) and then delving deeper into this field on my own is that, the usual construction of ... 1answer 187 views ### Why the whole theory about differentiable manifolds is based on open sets? I only have studied basic topology, which means i haven't studied any about differentiable manifolds. I just skimmed pages on wikipedia. Here is a simple illustration on a basic situation. Let E ... 2answers 146 views ### How to prove that all smooth vector bundles on a given vector bundle are the pull back of a vector bundle on the base Recently, during a conversation, I heard about the result (previously mentioned also here on MO), whose statement is reported below. Not having the specific background necessary to reconstruct a proof ... 1answer 124 views ### Why does every noncompact orientable surface have a complex structure? There is a high-powered proof of the fact that orientable noncompact surfaces have free fundamental group here that invokes the ability to put a complex structure on any such surface. But why should ... 1answer 201 views ### Are definitions for smooth map between manifolds are equivalent? There are two ways to define a smooth map between manifolds. The 1st way (for example, Lee): f:M\rightarrow N is smooth iff for every p\in M there exist charts (U,\varphi) at p and (V,\psi) ... 0answers 69 views ### How does a differential act when we identify T_p(M\times N)\cong T_{p_1}M\times T_{p_2}N? It's fairly common to identify the tangent space of a product manifold as$$ T_p(M\times N)\cong T_{p_1}M\times T_{p_2}N $$where p=(p_1,p_2), and the actual isomorphism is given by v\in ... 1answer 93 views ### If two Curves in \Bbb R^3 are transversal then they do not intersect Proving this is easy: Let X and Z be the two curves in \Bbb R^3. Assume X and Z intersect at a point y. Then, at most \dim(T_y(X) + T_y(Z)) = 2, where T_x(X) and T_z(Z) denotes the ... 1answer 151 views ### If S\subseteq M\times N is embedded, and S and \{p\}\times N intersect transversely in one point, then \pi_M|_S is a diffeomorphism? I'm trying to prove the equivalence of the following statements: Suppose M^m and N^n are smooth manifolds, S\subseteq M\times N immersed, and \pi_M and \pi_N the projection maps. TFAE: ... 0answers 109 views ### Transition Functions for Cartesian Coordinate Systems This is my first time using Mathematics SE (I've only used Physics and Astronomy before), so I apologize if this question is awkwardly phrased or incorrectly presented. I welcome any and all edits and ... 2answers 71 views ### Proving that a form is exact Maybe this question is rather obvious but I didn't manage to solve it myself. Assume M is a closed, oriented manifold. take$$ \Omega^k(M)\ni \omega = \begin{cases} d\beta~,~~~ in~ U\\ 0~,~~~ ...
If $M$ is a smooth $n$-manifold, the famous Whitney embedding theorems show that we can view $M$ as an embedded submanifold of some Euclidean space $\mathbb{R}^N$. Does the Heine-Borel theorem still ...
It is well known that every closed oriented 3dimensional manifold can be obtained from a framed link in $S^3$. Let $M$ and $N$ be topological 3-manifolds with boundary such that boundaries \$\partial ...