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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Passage in a proof of a lemma

Here is a lemma and a proof given to me in class. Lemma If $M$ is a smooth manifold, $K\subseteq M$ a compact subset, $A\subset M$ an open set containing $K$< then there exists a compact-support ...
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271 views

Why can we always take the zero section of a vector bundle?

$\require{AMScd}$ As I understand it, a rank $k$ vector bundle is a pair of topological spaces with a map between them $$ E\xrightarrow{p}B $$ such that there exists an open cover $(U_\alpha)$ of $B$ ...
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1answer
20 views

Are transverse maps in intersection theory local diffeomorphisms?

Suppose that $f: X \rightarrow Y$ and $Z$ is a submanifold of $Y$, all boundaryless. Suppose that $f$ is transverse to $Z$, so that: $$df_x T_xX + T_{f(x)}Z = T_{f(x)}Y$$ for every $x \in f^{-1}(Z)$ ...
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2answers
60 views

$SL(3,\mathbb{R})$ is a smooth manifold?

How do you show $SL(3,\mathbb{R})$ is a smooth manifold? I am thinking to use the preimage theorem, but what kind of thing I need to show first before I can apply the theorem?
4
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1answer
201 views

Intuitive meaning of immersions

I have a hard time understanding the concept of immersions. In my course, it was only introduced by the immersion theorem wich says: Let $f: U \subset \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ be ...
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2answers
81 views

Derivative of determinant at some point

Let $c:\mathbb{R} \rightarrow \mathbb{M}_n(\mathbb{R})$ defined by $$c(t)=A e^{tB}$$ where $A\in GL(n,\mathbb{R})$ and $B \in \mathbb{M}_n(\mathbb{R})$. The question ask me to find $c'(0)$ and ...
4
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1answer
50 views

Want to show two smooth manifolds are diffeomorphic

Consider a smooth manifold $M = \{ (u,v) \in \mathbb{R^3} \times \mathbb{R^3} \mid \|u\|=\|v\|=1 \text{ with } u \perp v \}$, and want to show $M$ is diffeomorphic to $SO(3)$, the rotational group in ...
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1answer
72 views

Are open sets in $R^n$ homeomorphic to $R^n$?

I am working on exercise 1.1 and I think the way to do this would be to show that open sets are homeomorphic to $R^n$ or open balls in $R^n$. Is this even true? I'm not sure how to go about ...
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44 views

a question about finding the points where Df(x) (derivative of f) is an isomorphism.

Let E be the four-dimensional real vector space $M_{2\times 2}$ of real 2$\times$2 matrices. Show that by setting f(X)=X^2 for 2$\times$2 matix X,we define a continously differentiable function f ...
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30 views

Construction of a diffeomorphism handling varying domain

Let $\Omega$ be a strictly convex domain, $\partial\Omega\in C^{2,1}$ We define a foliation $\{\Omega_t\}_{0\leq t\leq 1}\subset\Omega$ as follows. Let $\Omega_0=B_r(x_0)$, a small ball centered at ...
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59 views

Manifolds with smooth structure

One of the remark in my lecture notes said: In dimension $\leq 3$, every topological manifold has a unique smooth structure (up to diffeomorphism.) I don't quite understand what is a structure ...
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23 views

Prove each coordinate is a differentiable function

This is an exercise from a book called "Differential Topology" 2-11: Let $M$ be the sphere $x^2+y^2+z^2=1$ in 3-space. Prove that each of the Euclidean coordinates $x,y,z$ is a differentiable ...
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0answers
45 views

freedom in choosing a smooth function of compact support

Suppose $\Omega$ represents a bounded domain in $\mathbb{R}^n$. For ease, let $\Omega$ be a ball around the origin. Let $\varphi$ be a smooth radial function defined on $\Omega$. I am trying to ...
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1answer
123 views

Explain densities to me please!

When it comes to integration on manifolds, I speak two languages. The first is of course the language of differential forms, which is something I am relatively well acquinted with. The second ...
1
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1answer
58 views

Isometry of two Euclidean structures on the same vector bundle

I'm reading Milnor & Stasheff's Characteristic Classes, and I was struck by the following problem, which they call the Isometry Theorem: "Let $\mu$ and $\mu'$ be two different Euclidean metrics ...
2
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0answers
75 views

Manifolds and CW-complexes [duplicate]

This is a very naive question. Every manifold (assumed to be paracompact) is a CW-complex? Thanks.
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33 views

Connection between Chladni Plates and Algebraic Topology?

Does anybody know of a connection between Chladni Plates and Algebraic Topology? Any published research on the subject? I am interested to know if Chladni Plates (or surfaces) can be ...
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1answer
30 views

The tangent space of a manifold in some given point.

My question is about the tangent space of a manifold in some given point. Let $M$ be a differential manifold and $(U,\varphi)$ a chart around a given point $p$ of $M$ . My question is : Is that the ...
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59 views

Constructing complex line bundles on orientable smooth manifolds

This questions ask how to construct a complex line bundle over a smooth compact orientable manifold without boundary starting with an n-2 dimensional orientable sub manifold without boundary. The ...
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1answer
71 views

Transverse submanifolds in product manifolds.

Suppose we have smooth manifolds $M,M',N$, a smooth map $f\colon M\rightarrow M'$ and a smooth submanifold $S'\subseteq M'\times N$, such that the projection $\pi_{M'}\colon S'\rightarrow M'$ is a ...
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1answer
120 views

Embedd the Klein bottle into a 3-manifold

Can the Klein bottle $K$ be embedded into $S^{2} \times S^{1}$? If can, how it works. If not, is there an obstruction? Thanks in advance.
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1answer
207 views

Intersection theory proof of the poincare hopf theorem.

Suppose that $M$ is a connected compact oriented smooth manifold, and $X:M\rightarrow TM$ a vector field with isolated zeros. Then if $Z$ is the zero set of $X$ (a $0$-dimensional oriented manifold), ...
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1answer
64 views

$H^{n}(M)$ where $M$ is compact, orientable and connected manifold

I need to show that if $M$ is compact, orientable and connected manifold of dimension $n$, then $H^{n}(M) = \mathbb{R}$. I saw that a possible proof is to take an atlas for $M$, with $U_{\alpha}$, ...
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1answer
99 views

Good text on Differential manifolds?

I am new in field of topology.I am finding a good self readable text on differential manifolds.
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58 views

Analogue of the Euler Class of a Circle Bundle and the Global Angular Form

This is a general question that asks whether there is geometric significant to differential forms that restrict to G-invariant volume elements on the fibers of a principal G-bundle. For an SO(2) ...
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2answers
57 views

Compact universal covering spaces

Let $X$ be a topological compact space admitting a universal covering $C$. When is $C$ again compact? Thanks.
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3answers
295 views

Lie group. How is manifold defined?

If I have the Lie group $SL(2,\mathbb{R})$. Then how is the manifold structure on this algebraic group defined, could anybody explain this to me? I mean this is the group of matrices that determinant ...
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1answer
46 views

Is the Pontrjagin-Thom-map null-homotopic if the normal bundle $N_i$ is trivial?

Let $i\colon X\to Y$ be an embedding of two smooth and compact manifolds (without boundary) and let $N_iX$ be the normal bundle of this embedding. A Pontrjagin-Thom construction is a map $$ c_i\colon ...
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2answers
54 views

The bundle vector $f^\ast(\xi)$ for Moebius over $S^1$

Take the Moebius band like a vector bundle $\xi$ over the circle $S^1$ and the functions $f_n(z)=z^n$ then my question is: how describe the vector bundle define for the pullback $f^\ast(\xi)$ for ...
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1answer
159 views

How equivalent are the homeomorphism and diffeomorphism groups of 3-manifolds?

Every topological 3-manifold carries a smooth structure, unique up to diffeomorphism. In addition, "up to homotopy", the maps between them are the same: every continuous map is homotopic to a smooth ...
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If $M$ is a manifold and $\dim M\geq n-2$, is $\mathbb{R}^n\setminus M$ never connected and simply connected?

Suppose $M$ is a smooth submanifold of $\mathbb{R}^n$. Through some transversality tricks, I was able to prove that if $\dim M<n-2$, then $\mathbb{R}^n\setminus M$ is always connected and simply ...
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48 views

Isomorphism of Hom?

How do you show that $$\operatorname{Hom} (\mathbb{R}^n, \mathbb{R}) \simeq \mathbb{R}^n$$? What is the explicit isomorphism? I'm trying to understand the concept of the cotangent bundle ...
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34 views

Definition of Diffeomorphism for Arbitrary Subsets of Euclidean Spaces

In pg 1 of Chapter 1 of Milnor's Topology From the Differentiable Viewpoint, it is defined that Definition. Let $f:X\to Y$ be a function from a subset $X$ of $\mathbf R^k$ to a subset $Y$ of ...
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1answer
21 views

Expressing derivative as linear combination of derivatives of coordinate functions?

This is an old exam problem at my school: Let $F\colon M\to\mathbb{R}^k$ be a smooth map of smooth manifolds, with coordinate functions $F^1,\dots,F^k$. Let $c\in\mathbb{R}^k$ be a regular value ...
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1answer
37 views

Examples of mod 2 degree of f.

I'm reading Milnor's Topology from the Differentiable Viewpoint and I've just finished the chapter about degree mod 2. Since every two smooth homotopic maps $f,g$ must have the same degree mod 2 ...
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22 views

Reference on Loop Space

I need to study the foundations of the theory of closed loop spaces. I have been referenced to Klingenberg's "Lectures on Closed Geodesics", but found it a dry and difficult reading. Is there some ...
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1answer
39 views

Euler charcteristic of the intersection of hyperplanes with a pointed cone

Consider a pointed closed convex polyhedral cone $C$ in $\mathbb R^n$. Let $S_1,..,S_n$ be hyperplanes in $\mathbb R^n$. Consider $S=C \cap \left(\bigcap\limits_{i=1}^n S_i\right)$. If $S$ is ...
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1answer
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Tangent space of tangent vector

Let $M$ be a smooth manifold. There's a (split) short exact sequence $$0\to T_aM\to T_v(TM)\stackrel {D_v}{\to} T_aM \to 0,$$ where $v\in TM$ and $a=\pi(v)\in M$. I'm trying to understand what this ...
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0answers
23 views

The negative of a vector field and its flow

I have a relatively short question about vector fields. Let $G$ be a Lie Group, and $X$ a smooth vector field on it. If its flow is $\left\{\phi_t\right\}$ what is the corresponding flow for $-X$? ...
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Local submersion theorem and $O_{3}(\mathbb{R})$

I was attempting to follow the proof of the local submersion theorem given in Differential Topology by Guillemin & Pollack in the case that $X = O_{3}(\mathbb{R})$ and $f(A) = AA^{T}$. I worked ...
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1answer
45 views

How do geometers define “locally looks like” in differential geometry?

From reading some introductory texts on differential geometry, the author would usually invoke the phrase "locally looks like" when it comes to defining a manifold. For example, the real line is a ...
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1answer
29 views

Trivial tangent bundle of manifolds with boundary

In the Lee‘s book there is a proposition stating: If $M$ is a smooth $n$-manifold with or without boundary, and $M$ can be covered by a single smooth chart, then $TM$ is diffeomorphic to $M\times ...
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2answers
45 views

Problem with the concept of connection

I've been told that there is only a canonical way for doing the vertical subspace of the tangent bundle of a manifold and in order to do the horizontal subspace you need a connection. These are very ...
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continuity of exterior derivative

Stimulated by an exercise in da Silva: For a time-dependent vector field $v:M\times \mathbb{R}\to TM$, $k$-form $w:M\to T^{k,0}{M}$ and the isotopy $\rho:M\times \mathbb{R} \to M$ generated by $v$, ...
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1answer
61 views

Inner product on the space of sections

Let $L\to M$ be a real line bundle over a manifold $M$, and let us denote by $\Gamma(L)$ its space of sections. I am trying to find a product in $\Gamma(L)$ to make it into an algebra. The naive ...
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2answers
50 views

Definition of critical point

Let $f:M→N$ be a smooth function between two smooth manifolds. Then $p\in M$ is a critical point if $df_p$ is not surjective. I feel very confused about this definition, even in the case where ...
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39 views

Estimating vector fields on the product of compact manifolds

Let $M,N\subset \mathbb{R^n}$ be compact embedded manifolds, $X_1,...X_i$ vector fields on $M\times N$ and $\delta\colon M\times N\rightarrow (0,\infty)$ a continuous function. Are there vector ...
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0answers
52 views

Are there hypersurfaces with connected complement in a Banach space?

In $\mathbb{R}^n$ it is well-known that a smooth hypersurface $M$ (closed as a subset of $\mathbb{R}^n$) is the zero locus of a global smooth function (whose gradient is nonzero on $M$); from this one ...
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1answer
52 views

Integral over vector field

Can someone help me with this one: Let $X$ be a vector field on $R^{3}$ such that $X(x)=x$, for each $x\in S^{2}$. Calculate $$\int\limits_{\overline{B^3(1)}} \left(\frac{\partial X_1}{\partial x_1} ...
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1answer
135 views

Visualization of Lens Spaces

I am trying to visualize lens spaces geometrically. While I am aware of the fact that most manifolds which cannot be embedded in $\mathbb{R}^3$ are hard to visualize because of the obvious ...