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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Bundle that is isomorphic to the bundle of Whitney sum

I am involved with one question that a friend of mine asked. If a vector bundle $(E,\pi,M)$ has two sub-bundles, $(F,\xi,M)$, $(L,\psi,M)$, and $\pi^{-1}(x) \cong \xi^{-1}(x) \oplus \psi^{-1}(x),$ ...
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50 views

Degree theory and Invariance of domain

We'll use the Proposition (F) to show that: (Invariance of domain) Let $f: M \to N$ be a proper smooth mapping of two oriented, boundaryless, smooth manifolds of dimension $m$; furthermore, $N$ is ...
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80 views

compactification of $\Bbb R^n$

I want to show that $n$ dimensional real projection space is compactification of $\Bbb R^n$, how can I do that? I can show PR^n is compact, and R^n is localy compact, but I have problem to show R^n ...
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24 views

Coset spaces of a lie group and fiber bundles

I am reading Steenrod. He writes: Another example of a bundle is a Lie group $B$ operating as a transitive group of transformations on manifold $X$. The projection is defined by selecting a point ...
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28 views

Topologies of partially exponentiated lie algebras, especially in regard to $SU(2)$

Consider the fundamental respresentation of $\mathfrak{su}(2)$ given in terms of the Pauli matrices as $\mathfrak{su}(2) = \langle ...
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Normal bundle of sub-manifold is a manifold

This is exercise 2.3.12 of Guillemin and Pollack: Let $Z$ be a sub-manifold of $Y$, where $Y \subset \mathbb{R}^M$. Define $N(Z;Y)=\{(z,v):z\in Z, v\in T_z(Y), v \perp T_z(Z)\}$. Prove that $N(Z;Y)$ ...
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How to show that is a submanifold or How to derivate the determinant function?

I am trying to show that the space of $2\times 2$ matrix with rank equals $1$ is a submanifold of $\mathbb{R}^4 - \{0\}$ whoose the dimension equals $3$. To do this, I have defined $\det : ...
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On computing the Differential of a Smooth Map

In class, we proved Jacobi's formula for the differential of the determinant using the following formula for the differential of a smooth map $F$ between manifolds $M$ and $N$ $$D_A F(B) = ...
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Covering spaces as fiber bundles

I am reading Steenrod. I think there is something wrong or sloppy in his definition of a covering space as a fiber bundle. He writes: A fibre bundle consists of: (i) A topological space $B$ (ii) a ...
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26 views

The tangent space of the moduli space of connection?

I'm reading one of Floer's paper. (An Instanton-Invariant for 3-Manifold). Let $M$ be a $3$-manifold. A principal $SU_2$-bundle P over $M$ must be trivial. Fixed a trivialization $P \cong M \times ...
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covariant derivative of a helicoid

Given a helicoid $S$ parametrized by $x(u,v)=(v\cos(u),v\sin(u),u)$, a point $p=(1,0,0)$ on the helicoid, a tangent vector $v=(2,1,1)$ on $T_pS$ and a tangent vector field ...
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When the boundary of a manifold is orientable?

I am not sure whether the boundary of some manifold is definitely a manifold, but let's assume it is anyway. Then in what case the boundary is an orientable manifold. Maybe when the manifold can be ...
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25 views

Construct manifold from vector fields and point in $\mathbb{R}^n$

Suppose one is given a set of vector fields $X_{\alpha} \in \mathbb{R}^n$. Is it generally possible to construct a manifold embedded in $\mathbb{R}^n$ going through a particular point and whose ...
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39 views

Is there $f: U \to \mathbb{R}^{n}$ injective such that…

Let $f: U \to \mathbb{R}^{n}$ $C^{1}$ injective where $U$ is a open in $\mathbb{R}^{n}$ (so $f$ is open by invariance domain theorem). a) Is there exist $f$ such that dim $ker(df_{x}) >$ dim ...
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Show that if $M^n$ is a smooth manifold and $A,B$ are closed disjoint sets on $M$ then there is a smooth function $ 0\le f \le 1$

Show that if $M^n$ is smooth a manifold and $A,B$ are closed disjoint sets on $M$ then there is a smooth function $ 0\le f \le 1$ with $f(A) = 0$ and $f(B) = 1.$ What I am trying is: Note that ...
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83 views

Doubt in definition of symmetric continuous function and norm in Kupka's paper

In this article "Counterexample to the Morse-Sard theorem in the case of infinite-dimensional manifolds" of I. Kupka has the following passage: For $H=l^{2}$ "Let $H^{*}$ be the dual of $H$. A base ...
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66 views

Proof that this is a smooth manifold

Let $F: \mathbb{R}^2 \rightarrow \mathbb{R}^4$ be defined by $F(u,v) = (u+v, uv, u-v, v^3)$ and let $M = F(\mathbb{R}^2)$. Prove that $M$ is a smooth manifold. The proof that my TA posted online was ...
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36 views

Is an isometric and bijective mapping between two metric spaces complete?

If I have the two metric spaces $(X,d_x)$ and $(Y,d_y)$ with the mapping $f : X \to Y$ that is both an isometry and bijection between X and Y. How do I show that $(Y,d_y)$ is complete iff $(X,d_x)$ ...
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30 views

The value of the integral of the curvature of a complex line bundle

I am trying to show that if $L \to M$ is a complex line bundle endowed with a connection $\nabla$, $F$ is the curvature form, and $S \in M$ a closed surface, then $\int \limits _S F \in 2 \pi \textrm ...
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Non-orientable manifolds and mod 2 homology

I was reading the wonderful book "The wild world of 4-manifolds" by Alexandru Scorpan and I found the following sentence: "We are able to orient $\mathfrak{M}$ (else we only get modulo 2 ...
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Representable homology classes on smooth manifolds

Let $X$ be a closed (compact without boundary) smooth manifold. We can consider its singular homology $H_*(X,\mathbb{Z})$. Let $H_{k}(X,\mathbb{Z})$ be the $k$-th singular homology group of $X$ and ...
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Poincaré-Hopf Index Theorem - Intuitive explanation

I heard about an interesting theorem. Poincaré-Hopf Index Theorem : If $\vec{v}$ is a smooth vector filed on the compact, oriented manifold $X$ with only finitely many zeros, then the global ...
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How to calculate Fourier coefficient of $f\in C^{\infty} (\mathrm{T^3})$?

I was trying to calculate the $k$-th fourier coefficient $c_k$ of some smooth functions on $T^3$, say $k=(m,n,p)\in \mathbb{Z}^3$. In a write-up I found online, it has the following definition: $$c_k ...
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1answer
33 views

Visualizing sections of nontrivial vector bundles

My question is simply: how does one think about sections of nontrivial vector bundles on a smooth manifold, for example? The canonical example I think of is a vector field, i.e. a section of the ...
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1answer
31 views

Differential of the multiplication and inverse maps on a Lie group

I'm trying to solve the following two problems: Let $G$ be a Lie group with multiplication map $\mu\colon G\times G\to G$, and let $\ell_a\colon G\to G$ and $r_b\colon G\to G$ be left and right ...
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1answer
33 views

Homotopy classes of manifolds of dimension $2n$ which are $(n-1)$ connected

In his essay "Classification of (n − 1)-connected 2n-dimensional manifolds and the discovery of exotic spheres" Milnor states (in passing) "The manifold can be built up (up to homotopy type) by taking ...
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1answer
98 views

Potential proof for the Slice-Ribbon conjecture (may be wrong).

Let $f:(D^2,S^1)\to(D^4,S^3)$ be a smooth embedding (so called a slice disk), and we set $M:=f(D^2)$. Then, is the restriction map $C^{\infty}(D^4)\to C^{\infty}(M)$ open map with relative to $C^2$ ...
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Is $\overline{u^{-1}((-\infty,a))}$ a $C^k$-manifold with boundary for a $C^k$ function $u$?

Let $u\in C^k(\mathbf{R}^d)$, $M:=\overline{u^{-1}((-\infty,a))}=\overline{\{x\in\mathbf{R}^d \mid u(x)<a\}}$, where $a$ is a constant such that $M\neq\emptyset$. Is $M$ a $C^k$-manifold with ...
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Simple singularities of a vector field

Let $M^n \subset \mathbb{R}^{n+1}$ a compact and orientable hypersurface of even dimension, with Gauss map $\gamma : M \to S^n$, that is, $\gamma(p)$ is normal to $M$ at $p \in M$. Let $a, b \in S^n$ ...
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Existence of incompressible surface in a non-orientable manifold.

Let $M$ be a compact $P^2$ -irreducible 3-manifold. If $M$ is non-orientable, then there is a compact surface $F$ properly embedded in $M$ such that $F$ is two-sided, non-separating and ...
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Chart-free definition of manifold

There is some way to avoid charts in the definition of a (topological, or smooth, or any other) manifold? The choice of a cover by charts are not really important for the manifold; many manifolds that ...
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1answer
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Degree of map between surfaces of genus $g>1$ is $1$, $0$ or $-1$

Let $M$ be an orientable surface of genus $g>1$, I can assume compact. Let $f$ be a continuous map from $M$ to $M$. I want to prove that the degree of $f$ is $1$, $0$ or $-1$. For a surface of ...
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81 views

I cannot understand an explanation why 2-sphere is simply connected.

I am studying Elementary Differential Geometry written by Barrett O'Neill. In page 188, Chapter 4.7, there is an explanation why 2-sphere is simply connected. The following is from the text : ...
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Trouble with definition of signature of a compact manifold

The signature of a manifold, as I understand it, is defined as follows: Given a connected, compact, and oriented manifold $M$ of dimension $4n$, we may define a quadratic form on the cohomology group ...
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On the map $E: O\subset TM \to M \times M$ by $E(v)=(\pi(v), \exp(v) )$

Let $\exp$ be the exponential map on the Riemannian manifold M and $O$ is its domain in $TM$. Consider the map $E: O\subset TM \to M \times M$ by $E(v)=(\pi(v), \exp(v) )$, where $\pi$ is the ...
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If $M$ is a connected manifold, does $M\setminus\{p\}$ have finitely many components?

Let $M$ be a connected manifold and $p\in M$. Is it true that $M\setminus\{p\}$ has only finitely many connected components? (We can also suppose $M$ is compact if that helps.) I think this ...
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Being morse function for a determinant map on M(n)

Show that the determinant map on M(n) is Morse function if n=2. I know that f to be a morse function, all critical points for f must be nondegenerate. But i dont know how i calculate the ...
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Smooth vs topological orientation

I am confused about the different notions for a closed smooth manifold to be oriented. In my mind there are several equivalent ones: 1) Coherent pointwise orientation of the tangent spaces. 2) ...
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Using Poincare duality to show a closed manifold is a homology sphere

Suppose that $M$ is an orientable, compact, $(n-2)$-connected, $(2n-3)$-dimensional smooth manifold, where $n$ is a natural number. I want to show that $M$ is a homology sphere if and only if the ...
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Finding a basis for the cohomology vector space of 1-forms in the 2-torus, $H^1 (T^2)$

I would like help in understanding where I am going wrong here: If I consider the 2-torus $T^2 = S^1 \times S^1$ with an atlas $(\theta_1,\theta_2)$, I can define 2 closed 1-forms $\omega_1 = ...
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Hartman-Grobman Theorem - Necessary?

The Hartman-Grobman theorem states, in layman's terms, that a nonlinear system and the corresponding liniarized system behave similarly around a hyperbolic equilibrium point (in terms of vector fields ...
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Do the level sets of a constant rank map give a foliation of the domain?

I'm working through Smooth Manifolds by Lee and came to problem 19-8, where you're asked to prove that the level sets of a submersion form a foliation of the domain. However when I try to solve this ...
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A new definition of a focal point

Let $X$ be a manifold lying in $\mathbb{R^n}$, with $dim(X) = n-1$. Define $h: N(X) \to \mathbb{R^n}$ by $h(x, v) = x + v$, where $$N(X) = \{(x, v) \in X \times \mathbb{R^n}: v \perp T_x(X)\}$$ is the ...
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Immersions are open maps

Let $M,N$ be manifolds with $\dim M = \dim N$. If $f:M\to N$ is an immersion then $f$ is open. I thought that I have solved it, but then I thought there could be a mistake: Let $p\in M$. As $f$ ...
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Is $T(M)$ necessarily equivalent to the normal bundle of $M$ in $\textbf{R}^{2n}$, if $M$ is totally real?

Consider real submanifolds, $M^n \subset \textbf{C}^n$. ($\textbf{C}^n \equiv (\textbf{R}^{2n}, J)$, where $J(x, y) = (-y, x)$). $M^n$ is totally real if $J(T_pM) \cap T_pM = 0$, for all $p \in M$. ...
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Embedding of $2$-torus minus one point into $\mathbb{R}^2$ as an open set?

Let $M^2$ be the $2$-torus minus one point. Is there an embedding of $M^2$ into $\mathbb{R}^2$ as an open set?
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Can $S^k \times S^n$ be embedded into $\textbf{R}^{k + n + 1}$? [duplicate]

As the question title suggests, can $S^k \times S^n$ be embedded into $\textbf{R}^{k + n + 1}$, i.e. in codimension $1$?
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Immersion of $M^n$ into $\mathbb{R}^n$, is $M^n$ orientable? Compact? [closed]

Say we have an immersion of $M^n$ into $\mathbb{R}^n$ (same dimension). I have two questions. Is $M^n$ orientable? Is $M^n$ compact? Thanks in advance!
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1answer
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Show the parametrized torus is a 2-dimensional smooth submanifold of$\mathbb{R}^3$ [duplicate]

How can I show that the parametrized torus $T=\{(x,y,z)\in \mathbb{R}^3 : (\sqrt{x^2 +y^2}-a)^2 +z^2 =b^2 \}$ is a 2-dimensional smooth submanifold of $\mathbb{R}^3$ ? I was thinking of using the ...
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2answers
77 views

Let $f : A\subset \mathbb{R}^{n+1} \to \mathbb{R}$, what does mean that $f$ is a submersion?

I am trying to answer the following question: Let $M_a := \{ (x^1,\ldots,x^n,x^{n+1}) \in \mathbb{R}^{n+1} : (x^1)^2 + \cdots +(x^n)^2 - (x^{n+1})^2 = a\}$. For which values of $a$, $M_a$ is a ...