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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2answers
34 views

Is the restriction map continuous and surjective?

Consider $C^{\infty}(U, \mathbb{R}^{k})$ and $C^{\infty}(V, \mathbb{R}^{k})$, where $V\subset U $, with $U$ and $V$ open subsets of $\mathbb{R}^n$. Is it true that the restriction \begin{align} R:C^{...
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1answer
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Can we define differentiation without using a norm?

Since all norms on $\mathbb R^n$ are equivalent, the following question makes sense: Can we define the notion of "differentiability" of a map $\mathbb R^n \to \mathbb R$ without refering to a norm ...
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Hyperplane intersects submanifold transversally

Let $M\subseteq \mathbb{R}^p$ an n-dimensional submanifold. Show that there is a hyperplane in $\mathbb{R}^p$ that intersects $M$ transversally. My ideas: I shall use the theorem of Sard but I don't ...
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Simply connected neighborhood of a compact set contained in an open set

To continue this line of thought, say we have a compact, simply connected subset $K$ of a manifold $M$. Can a simply connected neighborhood of $K$ be found? I suppose that if $K$ is a submanifold, ...
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Does 2-dimensional Gauss-Bonnet theorem applicable in higher dimensions?

I want to know Does 2-dimensional Gauss-Bonnet theorem applicable (any topological or geometrical obstruction) in higher dimensions? My idea is that one can consider 2-dimensional embedded sub-...
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Integral of a differential form on a torus

Can someone check if my work here is on the right track: Compute $$\int_{S^{1} \times S^{1}} f^{*} \mathbb{\omega}$$ where $f\colon S^{1} \times S^{1} \rightarrow \mathbb{R}^{4}$ is a smooth map ...
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1answer
38 views

De Rham cohomology of $S^1$ with compact supports (Bott/Tu)

This is a question about Example 2.9, in Bott/Tu - Differential Forms in Algebraic Topology. Consider the decomposition of $S^1=U\cup V$ by two open sets, as in the figure above. Then both $U$ and $V$...
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1answer
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Showing that the differential is an immersion

If $f: X \rightarrow Y$ is an immersion of smooth manifolds, then show that $df: TX \rightarrow TY$ is also an immersion. The definition of immersion(when dim$X <$ dim$Y$) that I have is that for ...
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3answers
25 views

Examples of no-zero-measure meagre set

I know cantor set and rational numbers in $\mathbb{R}$ are meagre. But they are all zero measure. So is there any meagre set that is non-zero measure?
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On every metric space $(T, d)$ metric $d:T^2 \to \mathbb{R}$ is continuous function? [duplicate]

On every metric space $(T, d)$ metric $d:T^2 \to \mathbb{R}$ is continuous function? How to prove it?
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1answer
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The vertex of a tetrahedron is homeomorphic to a disk. [closed]

I want to prove that the tethahedron is a 2-manifold, but I have difficulties when it comes to deal with the vertices. It's natural that they are homeomorphic to a disk, but I find it hard to prove.
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Understanding Hessian on Manifold (without Riemannian Geometry)

I've been going through notes on Morse theory and Handlebody theory and I've been having some trouble with the definition of the Hessian provided. The notes are on pages 3-4 here http://people.math....
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Showing that all fixed points of a homotopy are isolated - possible using fixed point index?

I have a function $F(X,y)$, with $F: [0,1]^n \times [0,\infty)\rightarrow[0,1]^n$ i.e. essentially a self-map on a compact subset of $R^n$ with one parameter $y$. $F$ ist real-analytic. I know that ...
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1answer
40 views

Connection of product spaces with constraints [closed]

I have a question. If I have the following topological space $\mathbb{R}^2\times\mathbb{S}^1\times\mathbb{S}^1$ that corresponds to the following coordinate vector in $\mathbb{R}^4$, $x= [x_1,x_2,x_3,...
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1answer
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Continuous map from the plane $\mathbb{R}^2$ to a polygon with $n$ vertices.

Let $\mathbb{P}$ be the set of all points contained in a polygon defined by $n$ vertices in the plane $p_i=(x_i, y_i)$, $i=0,\ldots,n-1$. Does there exist a continuous bijective map $f:\mathbb{R}\to\...
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2answers
103 views

Relation/Difference between moduli spaces and classifying spaces.

From what I have read so far, a classifying space is a representing object of some (co)representable functor. For example, the $n^\text{th}$ Eilenberg–MacLane space is the classifying space for the $...
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Prove that the boundary orientation of $S^k = \partial B^{k+1}$ is the same as its preimage orientation

I would like to verify if my approach to this problem is the correct one or not. This problem is from "Differential topology" by Victor Guillemin and Allan Pollack . More specifically is the problem 3....
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Principle curvature from torsion over curvature ratio?

If we have a curve $\gamma$ that is paramterized over its arc length, lets say for example $\gamma ~[a,b] \rightarrow \mathbb{R}^{3}$, can we obtain the principle curvature of its tangent surface ...
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Reference request: complete, rigorous proof of compactification of moduli spaces of flow lines in Morse homology?

The result I'm looking for can be stated as follows (taken from Hutchings' notes): Here the moduli spaces are referring to the spaces of flow lines of the negative gradient flow induced by the Morse ...
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1answer
28 views

Degree of a function maps $\Omega \subset \mathbb{R}^n \to \mathbb{R}^m$ for $m < n$

I am doing self-reading on some degree theory using "Topological Degree Theory and Applications" by Donal O’Regan, Yeol Je Cho, and Yu-Qing Chen; I am totally new to this field and not quite familiar ...
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1answer
30 views

Every contractible manifold is simply connected

To clarify the definitions here: We call a differentiable manifold contractible if the identity is homotopic to some constant function on that manifold. Furthermore we call a differentiable manifold $...
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0answers
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Proof that Morse complex is a complex using coherent orientation

I'm reading the book Morse Homology by M. Schwarz, which aims to develop Morse homology in strict analogy with Floer homology. For orientation matters, the book follows the paper A. Floer and H. ...
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1answer
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Guillemin and Pollack Exercise 4.4.5

A closed curve $\gamma$ in a manifold $X$ is defined to be a smooth map $\gamma: S^1 \to X$, and I am tasked with finding an explicit formula for the line integral $\oint_{\gamma}\omega = \int_{S^1} \...
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0answers
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Well-definedness of the Einstein-Hilbert action

I am corrently working on mathematical general relativity and stumbled over the following question: The (vacuum) Einstein Hilbert action is defined to be $$\mathcal{S}_{\mathrm{EH}}(g):=\int_{\...
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Convergence in $C^{\infty}_{\text{loc}}$ on a manifold

Usually on a domain in $\Omega \subset \mathbb R^n$, convergence in $C^{\infty}_{\text{loc}}(\Omega)$ just means that for each compact subset $K$ of $\Omega$ we have $C^{\infty}(K)$ convergence. ...
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1answer
45 views

Weakly irreducible manifold

Let $M^3$ be a compact, connected and orientable manifold with boundary. I will say that $M$ is weakly irreducible if every smoothly embedded $2$-sphere $S \subset \operatorname{int}(M)$ separates $M$,...
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1answer
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Prove the set of continuous functions on $S^1$ is isomorphic to the set of continuous periodic functions with period $2\pi$

Let $S^1 = \{ (x,y)\in \mathbb{R}^2 | x^2+y^2=1$. Let $z:\mathbb{R} \to S^1$ given by $z=(cos(\theta),sin(\theta))$. Define the map $z^* : C^0 (S^1) \to C^0 (\mathbb{R})$ by $z^* (f)= f \circ z$. I'm ...
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Differential operetor For Poincare duality

Let $M$ be smooth manifold, dim$M=n\ \ $ and $\ \ [M] \ $ the fundamental class of $M$. Let's consider $$I : H_{dR}^{n-k}(M)\longrightarrow H_{k}(M)$$ What's this boundary operator equal $\ \ \partial ...
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1answer
95 views

What is the “full” topological requirements for a (classical) spacetime?

The model for (classical) spacetimes are, fundamentally, topological Manifolds. I know that this isn't the complete structure (because in fact we need to realize what is lorentz manifolds and so on...)...
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46 views

Smooth simple representatives of elements of the fundamental group

I am trying to define a closed simple curve (that is, a smooth curve) on a manifold $M$ such that it belongs to a specific class of $\pi_1(M)$. How can I be sure that such a curve exists? The ...
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1answer
27 views

Under surjective submersion, pushforward of a $k$-form is smooth

Let $\pi:P\to M$ be a principal $G$-bundle with some connection $\omega\in\Omega^1(P,\mathfrak g)$. Let $\psi\in \Omega^k(P)$ such that $\psi_{p\cdot g}\big(r_{g*}(v_1),...,r_{g*}(v_k)\big)=\psi_p(v_1,...
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1answer
16 views

Can we say that the moduli of differentiable structures of a topological manifold is discrete?

Differentiabl structure is defined by tangent bundle and any bundle determines a homotopy class of a map from the manifold to the Grassmanian manifold of the planes whose dimension is the rank of the ...
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1answer
93 views
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Heinz integral formula in the degree theory

The following problems can be found in chapter 8 of the PDE book by Evans. Here $U$ is an open, bounded set in $\mathbb{R}^n$ with smooth boundary. After problem 4, I was asked to solve problem 5. To ...
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1answer
72 views

Canonical identification of cohomology of nearby fibers in a fiber bundle

Let's say we have a map $f : X \to B$ of complex manifolds, with $B$ simply connected, such that $f$ is diffeomorphic (as a map of smooth manifolds) to the projection $F \times B \to B$. For $b \in B$,...
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1answer
47 views

Sign of $df_x$ is locally constant

This question is about the book Topology from the Differentiable Viewpoint of Milnor. Let $M$ and $N$ be oriented $n$-manifolds without boundary, and assume $M$ is compact and $N$ is connected. Let $...
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Embedding Iso(M) into M x $(TM)^n$ where M is a connected Riemannian manifold

This is a specific continuation of the question of embedding the group of isometries = Iso(M) of a riemannian manifold in $M$ x $(TM)^n$ Having proved the naturality of the exponential map, that is, ...
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How to prove the smooth manifold in $R^3$

Let T be smooth manifold in $\mathbb{R}^3$ such that , $(x^2+y^2+r^2-z^2-1)^2-4(x^2+y^2)(r^2-z^2)=0;\text{ with } 0<r<1$ Now my Question how to prove that $(x,y,z)\in T$ if and only if $r^2=(\...
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1answer
70 views

Cartesian product of embeddings is embedding

Let $M_1,M_2,N_1$ and $N_2$ be differentiable manifolds and $f_1:M_1\to N_1$, $f:M_2\to N_2$ two embeddings which in my literature is defined as an injective, proper immersion and a proper function is ...
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2answers
41 views

Homeomorphism induces differentiable structure

Let $M$ be a differentiable manifold and $f:M\to N$ a homeomorphism. I want to show that there is exactly one differential structure on $N$ that makes $f$ a diffeomorphism. I have to show that there ...
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1answer
33 views

Constructing a vector field with no zeros

Let, $\phi: X \rightarrow X$ be a diffeomorphism of a smooth compact manifold $X$ with no boundary. Let, $X_{\phi}$ be the quotient manifold $(X \times[0,1])/ \sim$ and $(x,1) \sim (\phi(x),0)$. How ...
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1answer
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$\mathscr{C}^{k-1}$-manifold structure on the cotangent bundle

Let $X$ be a $\mathscr{C}^k$-manifold of dimension $n$ and atlas $(U_\alpha,\varphi_\alpha)$ (i.e. its atlas is such that the transitions maps $\varphi_\beta\circ\varphi_\alpha^{-1}$ are $\mathscr{C}^...
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0answers
31 views

Why is the horizontal plane field of $\mathbb{H}^2\times\mathbb{R}$ integrable?

In Peter Scott's "The geometry of 3-manifolds" he shows that $\widetilde{SL(2,\mathbb{R})}$ is distinct from $\mathbb{H}^2\times\mathbb{R}$ by showing that the horizontal plane field induced from $T\...
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0answers
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Connected sum and h-cobordant: if $M_1 ~ M_1'$ then $M_1\#M2 ~ M_1'\#M2$

I'm trying to understand the proof of lemma 2.2 in Kervaire and Milnor's paper Groups of Homotopy spheres. I do not understand how to show that $M_1\# M_2$ is a deformation retract of the manifold $W$...
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1answer
14 views

If $F: M\rightarrow N$ is smooth then its coordinate representation w.r.t any smooth charts $(U,\phi),~ (V,\psi)$ s.t. $F(U)\subseteq V$ is smooth

I want to prove that if $F: M\rightarrow N$ is smooth, where $M,N$ are smooth manifolds, then its coordinate representation w.r.t any smooth charts $(U,\phi),~ (V,\psi)$ such that $F(U)\subseteq V$ is ...
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1answer
51 views

Cone of complex and isomorphism

let $M$ be a smooth manifold with boundary $\partial M= X\times F$, where $X$ and $F$ are smooth manifolds, $F$ compact. The embedding $i : \partial M\longrightarrow M$ induces the restriction $$i^{*...
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1answer
70 views

Does every path in $SO(2k)$ from $1$ to $-1$ pass through the space of complex structures?

Recall the space of (normalised) complex structures $\mathcal{J}_{2k} : = \{J \in SO(2k) \mid J^2 = -1\}$ on $\mathbb{R}^{2k}$. I am curious to know if every path from $1$ to $-1$ in $SO(2k)$ must ...
5
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1answer
102 views

Why integrate on cubes that's not injective?

Again, this is a conceptual(soft) problem I had while reading Spivak's calculus on manifold. There, to develop the theory of integration, Spivak chose to integrate k-forms on singular cubes. However, ...
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0answers
26 views

How to show explicitly that $\{(x, y)| x^3 = y^2\}$ is not a smooth manifold? [duplicate]

Lately I've been watching a course in differential topology and the instructor gave the set $\{(x, y)| x^3 = y^2\}$ as a subset of $\mathbb{R}^2$ which is not a smooth manifold with no explanation, ...
2
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1answer
68 views

General dimension hairy ball theorem and division algebras

Question: Can someone please give a clear explanation, or point to a clear visual, that explains how the existence (or non-existence) of a non-vanishing continuous $n$-vector field on an $n$-sphere ...
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1answer
26 views

Stochastic process, why this is that?

Let $W_t$ be a process such that: $dW_t = (r+\pi* (\mu-r)) ) * W_t * dt + \pi * \sigma * W_t * d_{z_t}$ where $r, \pi, \mu, \sigma $ are constant, $Z_t$ is the Geometric Brownian motion. Applying ...

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