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Questions tagged [differential-topology]

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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$TS^2$ coordinate expression and coordinate change.

I want to express coordinate the (co)tangent bundle of sphere. I think $TS^2$ (or $T^*S^2$) $=\{(x_1,x_2,x_3,v_1, v_2 , v_3 | x_1^2+x_2^2+x_3^2=1, <(x_1,x_2,x_3),(v_1,v_2,v_3)>=0 \}$ and using ...
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A difficulty in understanding a part of solution of Q.1.3.10 in Allan Pollack and Guillemin.

"Generalization of the inverse function theorem.let $f:X \rightarrow Y$ be a smooth map that is 1-1 on a compact submanifold Z of X. Suppose that for all $x \in Z$, $$df_{x}: T_{x}(X) \rightarrow T_{f(...
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A difficulty in understanding a part of the solution of Q.10 section 1.3 in Allan Pollack and Guillemin(3).

Q.10 section 1.3 in Allan Pollack and Guillemin is the following: "Generalization of the inverse function theorem.let $f:X \rightarrow Y$ be a smooth map that is 1-1 on a compact submanifold Z of X. ...
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First generalization of the inverse function theorem Q.10 section 1.3 in Allan Pollack and Guillemin(2).

Part of Q.10 section 1.3 in Allan Pollack and Guillemin is the following: "Generalization of the inverse function theorem.let $f:X \rightarrow Y$ be a smooth map that is 1-1 on a compact submanifold ...
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21 views

First generalization of the inverse function theorem Q.10 section 1.3 in Allan Pollack and Guillemin(1).

Part of Q.10 section 1.3 in Allan Pollack and Guillemin is the following: "Generalization of the inverse function theorem.let $f:X \rightarrow Y$ be a smooth map that is 1-1 on a compact submanifold ...
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Clarification about the index of singular points and curves on the sphere.

I am having some problems understanding the concept of index of a singular point of a vector field on a Manifold (in particular on a 2-dimensional sphere) and some of its properties. And hope that you ...
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Geometric interpretation of torsion homology classes

Suppose I have a homology class $x \in H_1(M)$ which is torsion of order $k$ say. Suppose furthermore that $M$ has Dimension big enough, such that every element of $H_1$ and $H_2$ can be relalized as ...
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A vector field is differentiable if and only if the map $X: M \to TM$ is differentiable

Let $X$ be a vector field defined on a manifold $M$. Then $X$ is differentiable if and only if the application $\psi:M\rightarrow TM$ such that $\psi(p)=(p,X_p)$ is differentiable. I have some ...
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One-degree map between manifolds with boundary

Let $F:M\rightarrow N$ be a map between orientable compact connected $n$-manifolds such that $F^{-1}(\partial N)=\partial M$. The degree of $F$, $deg(F)$, is given by the equation $$F_{\#}([M])=deg(F)[...
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How to think about exotic differentiable structures in manifolds?

I apologize in advance for the vagueness of this question. It is known that there exist differential manifolds that are homeomorphic but not diffeomorphic to spheres (Milnor), and likewise there are ...
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Which open subsets of $\mathbb{R}^n$ are homeomorphic to $\mathbb{R}^n$ itself?

The motivation of this question is related to the fact that in various differencial geometry books I have seen three different criteria for chart maps. These were: If $(U,\varphi)$ is a local chart, ...
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Smooth curve at the boundary - inside a closed shape

I'm unsure whether this fits in topology or differential geometry. I have a basic intuitive idea, and I'm trying to find either the appropriate definition or theorem relating to this idea within ...
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86 views

If a composition of functions is smooth and one of them is smooth, then the other is smooth

Show that a map $\xi$ between smooth manifolds $M$ and $N$ is smooth if and only if $f ◦\xi$ is a smooth function on $M$ whenever $f$ is a smooth function on $N$. One implication is clear because I ...
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$\mathbb{HP}^2$, exotic 7-spheres, and Bott manifolds

I am looking for some explanation how $\mathbb{HP}^2$, exotic 7-spheres, and Bott manifolds are related? And how the construction of a Bott manifold is related to $\mathbb{HP}^2$ ...
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How to show that vector field is continuous?

In the book the definition of a a vector field over $U$(open)$ \subseteq S^n$ is given by a continuous map $s: U \to T(U)$ such that $p_U \circ s=id_U$ where $p_U$ is the base point projection from $T(...
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Linking number and cup product

Let $S^p$ and $S^q$ be disjoint spheres in $\mathbb{R}^n$ with $n=p+q+1$ and let $X= \mathbb{R}^n- (S^p\cup S^q)$. By Alexander duality, their fundamental classes yield cohomology classes in $\tilde{H}...
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Clarification on definition of smooth map between smooth manifolds

Let $M$ and $N$ be $n$-dimensional smooth manifolds. A map $F: M \to N$ is smooth if for each $p \in M$ there exists smooth charts $(U, \varphi)$ containing $p$ and $(V, \psi)$ containing $F(p)$ such ...
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Basis for the Whitney topology

In the book of Differential Topology by Hirsch, the Whitney topology is defined. (See below) I tried to prove the given collection $\{N^r(f;\Phi;\Psi;K;\varepsilon) \}$ is a basis. My proof starts ...
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A difficulty in understanding why there is a problem in the function constructed for Q.1.1.8(c) in Alan Pollack and Guillemin

I am asking about part(c) of this question: The solution is given below: But I could not understand in the solution of (c) why the implicit function $x \rightarrow r$ is not smooth at 0 and where we ...
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How do I show that this set is dense in the function space?

Consider the smooth manifold $\mathbb{S}^3$ embedded in $\mathbb{R}^4$, note that $$\widetilde{T}:= \frac{1}{\sqrt{2}}\mathbb{T}^2 = \left\{(x_1,x_2,x_3,x_4) \in \mathbb{R}^4; \ x_1^2 + x_2^2 = x_3^2 ...
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problem 4 in Alan Pollak and Guillmin differential topology.

Let $a > 0$, and set $B_a = \{x \in \mathbb{R}^n : |x|^2 < a \}$. Let $\phi : B_a \to \mathbb{R}^n$ be given by $\phi(x) = \frac{ax}{\sqrt{a^2 - |x|^2}}$. Prove that $\phi$ is a diffeomorphism ...
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Equivalence of definitions of a vector bundle

Let $n\in\mathbb N$, let $E,B$ be topological spaces and let $p:E\to B$ be a continuous map. For every $b\in B$, let $p^{-1}(b)$ be equipped with the structure of an $n$-dimensional real vector space. ...
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Pontryagin square: lifting to an integral cocycle, and an element $H^4 (B^2 \mathbb{Z}_r, \mathbb{Z}_{2r})=$Hom$(\mathbb{Z}_{2r},\mathbb{Z}_{2r})$?

Let $M$ be a simplicial complex and $Π$ be a finite abelian group. In the simplest case $Π = \mathbb{Z}_r$ finite group with $r$ even, the Pontryagin square is a cohomological operation which maps an ...
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Sum formulas for Pontrjagin square and Postnikov square

Inspire by this, I wonder Pontrjagin square: There is a geometric interpretation of $\mathfrak{P}_2$, due to Morita. Assume $q=2k$, so that the Pontrjagin square is a map $$\mathfrak{P}_2 \colon H^{...
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$\mathcal{C}^1$-topology of a submanifold with boundary

Let $M \subset \mathbb{R}^n$ be a compact connected manifold without boundary embedded in $\mathbb{R}^n$, then we can define the $\mathcal{C}^1$-topology of the functions $\mathcal{F}(M)= \{f: M\to \...
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Coorientation of contact structures

When reading about contact geometry one quickly encounters the notion of a cooriented contact structure/form. But I do not seem to be able to find a definition of "coorientation". In some places they ...
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Critical values and real-analytic mappings between manifolds

Let $M^m,N^n$ be smooth manifolds and $f \in C^\infty(M, N)$. Then, for any regular value $y \in N$ we have that $f^{-1}(y)$ is a smooth submanifold of $M$ of dimension $m-n$. Moreover, by Sard's ...
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Help understanding proof of orbit manifold

I attach some photos. I am reading the proof of (16.10.3). Here is the statement. The authorclaims (16.10.3.1) is a consequence of another lemma. See: Obviously, to understand how it is clear ...
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1answer
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Degree of a map and Suspension

I want to verify the following "geometric" proof that if we have a continuous $f:S^n \rightarrow S^n$ then $\textit{deg}f=degSf$ , $Sf$ denoting the suspended map from $S^{n+1}$ to $S^{n+1}$. It is ...
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Topology and smooth structure on tangent bundle

My lecture notes on differential geometry read the following (without proof): For $M$ a manifold, let $TM = \bigcup_{p \in M} T_p M$ be the (disjoint) union of all its tangent spaces. Then, there ...
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Neighbourhood of a point where $(\theta_1)_{t_1} \circ (\theta_2)_{t_2} \circ \cdots \circ (\theta_k)_{t_k}$ is defined

This is actually a detail in Lee’s smooth manifold 2nd ed p.234, 2nd paragraph in the Theorem 9.46. What i found is possibly a correction for the book. Forgive me if it’s not even close. Let $M$ be a ...
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Manifolds with varying (local or global) dimension

What subject should one look into to understand manifolds with varying dimensions throughout it's structure. For example, Imagine a 2-sphere with a line going through it defining some type of ...
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Prove the theorem of level sets as manifolds.

THEOREM. Let $f_1,..f_r \in C^\infty (\mathbb R^n$ and $X=\{ x\in\mathbb R^n | f_i(x)=0, 1\le i\le r\}.$ If $$Df_1(x)=(\frac{\partial f_1}{\partial x_1}(x),...,\frac{\partial f_1}{\partial ...
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1answer
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One-degree map between orientable compact manifolds

Let $F:M\rightarrow N$ be a map between orientable compact connected $n$-manifolds such that $F^{-1}(\partial N)=\partial M$. The degree of $F$, $deg(F)$, is given by the equation $$F_{\#}([M])=deg(F)[...
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0answers
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Preimage by homotopy family of smooth functions is invariant under homeomorphism

Suppose $f_t:M^n\rightarrow \mathbb{R}^n$ is a family of smooth functions depending smoothly on $t\in[0,1]$, where $M$ is a manifold. I would like to know when the preimage of $0$ (assume not empty) ...
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Can a basis of a tangent space be mapped to a basis of another tangent space if the map between the spaces is a homeomorphism and vice versa?

If I have an open subset $U$ of a n-dimensional $C^k-$manifold $M$ and a homeomorphism $f:U \to \Bbb R^n$ (Basically I am talking about a chart $(U,f)$) can I say that under this map a basis of $T_pU=...
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1answer
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Making a homeomorphism a $\mathscr{C}^k$-map

I'm solving the following exercise and I just need a little push for the one step I'm failing to do: Let $M$ be a $\mathscr{C}^k$-manifold, $N$ be a topological manifold, and $\alpha: M\to N$ be a ...
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101 views

An integral map from 3-torus $\mathbb{T}^3$ to 3-sphere $S^3$

Let $\phi_1, \phi_2, \phi_3, \phi_4 \in \mathbb{R}$ be real valued functions, such that $$\phi_j(x,y,z):(x,y,z) \in \mathbb{T}^3 \to \phi_j(x,y,z) \in \mathbb{R}.$$ Here $\mathbb{T}^3$ is a 3-torus,...
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Proving that for a $C^k$ function, the measure of the set of singular values is $0$

Let $f:R^m\to R^n$ be a $C^k$ function, where $k>\max \{0,,m-n\}$. Then the lebesgue measure of the set of singular values is $0$. I've been trying to prove this. I came up with the following ...
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de Rham cohomology of vector bundles: advantages of different definitions of compact vertical cohomology

I'm currently reviewing the Thom isomorphism and the de Rham cohomology of vector bundles over a compact manifold $M$. I'm familiar with the case of topological disk bundles, where the Thom class ...
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1answer
42 views

Lower and Upper Bound for Ricci Curvature

As mentioned in first chapter of John M. Lee: Riemannian Geometry, one of our goal in differential geometry is connecting geometry and topology. For this reason it is natural to compare curvature ...
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Monodromy of the family of hypersurfaces on moduli space

Let $\bar{\mathfrak X}\to \mathbb P^N$ be the universal family of hypersurfaces in $\mathbb P^{n+1}$ of degree $d$ and $\mathfrak X \to U$ ($U\subset \mathbb P^N$) be the sub-family of smooth ...
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Is a map with invertible differential a diffeomorphism onto its image near a boundary point?

Let $M,N$ be smooth manifolds of the same dimension, and suppose $M$ has a non-empty boundary. Let $f:M \to N$ be a smooth map, and suppose that $df_p$ is invertible for some $p \in \partial M$. ...
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Dimension of isotropy group in homogenous space

Just sincerely ask a fundamental problem as the following: We know $G = GL(2,\mathbb{R})$ is a transitive group of dimension $4$. The vector space $GL(2,\mathbb{R})$ acting on is $X=\mathbb{R}^2$...
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$p\mapsto c_p$ is continuous where $p\in S^1$ and $c\in \mathbb{R}$.

Let $f: S^1\to S^1$ be a diffeomorphism. Consider the derivative map $$ Df_p:T_p S^1\to T_{f(p)}S^1,\ (p,ip)\mapsto (f(p),ic_pf(p)). $$ we say $f$ is orientation preserving if given $p\in S^1,\ c_p>...
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$\min_{w}(w'(A+x_{t+1}x_{t+1}')w-2(b-yx_{t+1})'w+\sum_{s=1}^ty_s^2-y^2)-\min_{w}(w'Aw-2b'w+\sum_{s=1}^ty_s^2)$

Consider $A$ to be a symmetric positive definite matrix, vectors $w,x,b\in\mathbb{R}^n$ and scalar $y\in\mathbb{R}$. I am trying to solve $Q_1(w)-Q_2(w)$ where: $$Q_1(w)=\min_{w}(w'(A+x_{t+1}x_{t+1}')...
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spin mapping class group of circles

The MCG of the circle is $\mathbb{Z}/2$, generated by an orientation-reversing diffeomorphism. The circle has two spin structures, a periodic one and an anti-periodic one. Each has a nontrivial ...
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Every immersed submanifold can be deformed to have transverse self-intersection

Let $f : M^n \to \overline{M}^{n+k}$ be an immersion between smooth manifolds. Is it true that there exists a smooth map $F : M \times [0,1] \to \overline{M}$ such that the following conditions hold? ...
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1answer
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$M$ is orientable iff $\bigwedge^n TM \setminus \{ \text{$0$-section} \}$ has $2$ connected components

Let $M$ be an manifold of dimension $n$. We say that $M$ is orientable if it has an $n$-form that doesn't varnishes in any point if $M$. My question is about the following claim: $M$ is orientable ...
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79 views

Value changes after using a permutation in a wedge-product

We defined the wedge-product as follows: $ w \wedge \mu = \frac{(k + l)!}{k!l!} Alt (w \otimes \mu) $ (here you'll find additional information). In the context of a proof we came to a point where we ...