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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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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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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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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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 ...
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53 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 ...
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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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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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27 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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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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64 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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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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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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56 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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77 views

Good text on Differential manifolds?

I am new in field of topology.I am finding to read good self readable text on differential manifolds.
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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
54 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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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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45 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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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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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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44 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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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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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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23 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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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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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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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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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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22 views

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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38 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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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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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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60 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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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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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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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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51 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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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 ...
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proving the injectivity half of de Rham's Theorem when $p>1$ (if a $p$-form $\omega$ vanishes on all $p$-cycles, then $\omega$ is exact)

Let $M$ be a smooth manifold of dimension $n$, and let $\omega$ be a differential $p$-form on $M$. Then we have the following theorem: $\omega$ is exact if and only if $\oint_c\omega=0$ for all ...
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Cotangent space explicit definition

Given a tangent space $T_xM$, where $M$ is a differentiable manifold homeomorphic to $\mathbb{R}^n$, we have the cotangent space $T^{*}_xM$ defined as being the set of linear functionals $\eta: T_xM ...
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80 views

Torus diffeomorphic to $S^1\times S^1$.

This is an exercise from Guillemin/Pollack's Differential Topology. In a previous exercise, I'm asked to give a complete set of parametrizations of $S^1\times S^1$, which I've succeeded in (I think) ...
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homomorphism inducing Galois cover

We are given a homomorphism $\rho: \pi_1(\Sigma_g - \{p_1,...,p_k\}) \rightarrow G$ , where $\Sigma_g$ is genus g Riemann surface and $p_i$'s are points on it, G is a finite group. Then it is claimed ...
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Differential topology, fundamental theorem of algebra

I am reading Milnor's Topology from the Differentiable Viewpoint, in particular page 8-9 about applying regular values to prove the fundamental theorem of algebra. So he defines stereographic ...
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If $f\colon M^n\to N^n$ is proper, $N$ is connected, and $f_*$ preserves orientation at regular points, then $f$ is surjective?

I'm attempting Exercise 8.21 from Spivak's Differential Geometry. It is not for homework or anything. The problem states Let $f\colon M^n\to N^n$ be a proper map between oriented $n$-manifolds ...
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Why is the local representation of a connection a projection on $T_{\xi}E$?

In Klingenberg's Lectures on Closed Geodesics, he states a proposition that goes as follows: Proposition: A connection $K$ on $\pi: E \rightarrow M$ defines a splitting $TE=T_hE \oplus T_vE$ ...
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Calculation of Poincare dual to $\mathbb{CP}^n$ in $\mathbb{CP}^{n+1}$

I would like to go about finding an explicit representative of the Poincare dual to $\mathbb{CP}^n$ in $\mathbb{CP^{n+1}}$, I am following Bott & Tu and would like an explicit form to be wedged ...
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58 views

proving linear interpolation of Level Set

I tried to explain figure below in mathematics form. As you can see I have got triangle (v1, v2, v3). The signed shortest distance form red interface (level set value) is calculated for each vertex ...
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Transversality of a function to a sphere

I'm working through problem 6-9 in Lee Smooth Manifolds and I'm stuck. Let $F: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ defined by $(x,y) \mapsto (e^{y}\cos(x), e^{y}\sin(x), e^{-y})$. For which ...