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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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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56 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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1answer
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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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46 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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33 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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20 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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34 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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21 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
38 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
48 views

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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22 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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24 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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1answer
43 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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26 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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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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30 views

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
49 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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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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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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123 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 ...
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1answer
71 views

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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1answer
54 views

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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133 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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60 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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1answer
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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 ...
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1answer
52 views

Several questions concerning Alexander's Theorem

I'm reading Hatcher's proof of Alexander's theorem in his 3 manifolds notes. The statement is the following: Let $\Sigma \subset \mathbb{R}^3$ be an embedded $2$-sphere; then $\Sigma$ bounds a ...
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orientability of a vector bundle plus trivial bundle

I'm trying to understand the following: if a vector bundle $E$ over $X$ is not orientable, then neither is $E \oplus \underline{\mathbb{R}}$, where $\underline{\mathbb{R}} = X \times \mathbb{R}$ is ...
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Understanding Euler density

I know the definition of Euler density in terms of antisymetrized contractions of products of the Riemann curvature tensor, ie Euler density is the $\mathcal{R}^n$ in these formulae: And I know ...
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1answer
66 views

Intuitive description of what a topological space is?! [duplicate]

I have read several texts that give a very technical description of what a topological space is, but I can't find any notes that really give an intuitive description of what it is and I'm really ...
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31 views

Zeros of vector field extended from field lines

Suppose I have a finite number of 1-dimensional curves embedded in a 3-dimensional space. What can I say topologically about vector fields chosen so that these closed curves are field lines? For ...
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35 views

Smooth manifolds and cartesian products

So assume you have two smooth maps $f:M\rightarrow N$ and $g:P\rightarrow Q$. If we define the map $f\times g:M\times P\rightarrow N\times Q$ to be such that $(f\times g)(x,y)=(f(x),g(y))$. Does it ...
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Properties of immersions and embeddings

Definition 1: $f \in C^{\infty}(M,N)$ is called immersion iff $\forall x \in M: \operatorname{Ker} d_x f = 0$. Definition 2: $f \in C^{\infty}(M,N)$ is called embedding iff it is immersion and ...
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Smooth structure after surgery

I'm having some trouble understanding how surgery again produces a smooth manifold. My understanding of surgery is something like this: start with a smooth manifold $M$ of dimension $m$ and, suppose ...
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Are the following sets a submanifold?

I have got stuck in showing whether this particular subsets are submanifolds(smooth): Let $f:\mathbb{R}\to\mathbb{R}$ be the function $f(x)=|x|$. Then the graph of the function f is smooth ...
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1answer
39 views

Embedding, local diffeomorphism, and local immersion theorem.

Suppose $f: M \to N$ is smooth and an immersion, i.e $df_p : T_p(M) \to T_p(N)$ is one-to-one. Since $f$ is an immersion, we have the following theorem, $\textbf{Local Immersion Theorem:}$ Suppose ...
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Applying Lie bracket to a smooth function.

Let $\textbf{X}$ and $\textbf{Y}$ be vector fields on an $n$-dimensional manifold, $M$. Let $f^{-1} : M \to \mathbb{R}^n$. We can represent $$\textbf{X} = a^1 \frac{\partial}{\partial x^1} + \dots + ...
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Surfaces, vector fields, and the Lie bracket.

$\textbf{Theorem:}$ Suppose that $X_1, \dots, X_k$ are vector field on a manifold $M$ and at a point $p \in M$, we have that $X_1(p), \dots, X_k(p) \in T_p(M)$ are linearly independent, then the Lie ...
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27 views

Atiyah-Bott fixed point formula; signs

In classical paper by Atiyah-Singer on page 16 (or 560) stated formula $(3.1)$. It should give classical Lefschetz fixed-point formula if the operator is $d + d^* : \Omega^{even} \rightarrow ...
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