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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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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0answers
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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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0answers
13 views

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
32 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
29 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
27 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
85 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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1answer
40 views

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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0answers
23 views

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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1answer
76 views

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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0answers
21 views

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
31 views

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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1answer
79 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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2answers
47 views

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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2answers
57 views

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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1answer
53 views

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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1answer
12 views

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 ...
2
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0answers
58 views

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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0answers
41 views

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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1answer
25 views

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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0answers
40 views

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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1answer
29 views

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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1answer
47 views

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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1answer
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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
45 views

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
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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 ...
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1answer
35 views

If $dF_p$ is nonsingular, then $F(p)\in$ Int$N$

Here is the problem 4-2 in John Lee's introduction to smooth manifolds: Suppose $M$ is a smooth manifold (without boundary), $N$ is a smooth manifold with boundary, and $F: M \to N$ is ...
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1answer
19 views

Properties related to connectedness of manifold

Suppose a manifold $M$ is connected. (Here I assume that a manifold is a Hausdorff, second countable space and each point $x\in M$ has a neighbourhood homeomorphic to $\mathbb{R}^{n}$, where $n$ can ...
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Induced orientation on manifold

Suppose $M$ is a naturally oriented $n$-manifold embedded in $\mathbb{R}^n$ with non-empty boundary. Give $\partial M$ the induced orientation. The induced orientation defines a unit normal vector ...
2
votes
1answer
42 views

What does Hochschild (co)homology mean

What does hochschild (co)homology mean. Is it a statement of a topological invariant? What is a way of picturing what it is doing. I am only asking this question because my experience with other ...
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0answers
23 views

Trajectories of vector fields on compact manifolds

Suppose that $X$ is a smooth vector field on a smooth manifold $M$. The trajectories of $X$ are curves $p(t)$ in $M$ which satisfy $d{p(t)}/{dt} = X(p(t))$. It's well known that $p(t)$ exists ...
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0answers
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Diffeomorphism $F((z,v), \lambda) = (z, v + \lambda z)$

Show that $F: TS^n \times \mathbb{R} \to S^n \times \mathbb{R}^{n+1}$ given by $F((z,v), \lambda) = (z, v + \lambda z)$ is a diffeomorphism. Here we interpret tangent vectors to a submanifold as being ...
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1answer
35 views

Mathematical definition of the word “generic” as in “generic” singularity or “generic” map?

I've been trying to work out what generic means but I'm not making much progress. You can find an example of the usage of the word generic for example here: "School on Generic Singularities in ...
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0answers
21 views

Differentiability of the norm in connexion with duality map

Let $(X,\|\cdot\|)$ be a Banach space and let $J$ be the duality mapping defined for all $x\in X$ by: $J(x)=\{x^∗∈X^∗\mid ⟨x^∗,x⟩=\|x\|^2=\|x^∗\|^2\}$, where $X^∗$ is the dual space of $X$. I'm ...
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0answers
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proper submersion

I have the following question: Let $X,W$ be smooth manifolds with $W\subset X\times \mathbb{R}\times \mathbb{R}^n$ and the projection $p_{1}:W\rightarrow X$ a surjective submersion. Let ...
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Uniqueness of a PDE solution

Suppose $F:U \to \mathbb{R}^n$ is a $\mathcal{C}^1$ vector field on an open set $U \subseteq \mathbb{R}^n$. Let $\lambda \in \mathbb{C}$. Consider the partial differential equation $$F(x) \cdot ...
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1answer
39 views

Frobenius theorem for differential forms

I have to check the next version of the theorem Frobenius: Let $M$ a smooth manifold and $\{\omega^1,\ldots,\omega^k\}\subset\Omega^{1}(U)$ $l.i.$ on $U\subset M$ and $P(x)=\{ v\in T_x M\vert ...
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0answers
36 views

Gaussian curvature of $S^3$

It is easy to see that Gauss curvature of $S^2$ is $1/R^2$. How can we find the Gaussian curvature of $S^3$? What about $S^n$ in general?
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0answers
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Degree of Gauss map coincides with Euler characteristic

Let $M^n \subset \mathbb{R}^{n+1}$ be a compact hypersurface, oriented with the smooth normal vector field $N(X) \perp T_xM$. Let $G: M^n \to S^n$ be the corresponding Gauss map. Does it follow that ...
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1answer
32 views

Diffeomorphisms between smooth manifolds with boundary

In the professor Lee's introduction to smooth manifolds 2nd edition, the notion of diffeomorphism is defined for smooth manifolds with or without boundary. However, I saw some propositions that seems ...
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1answer
34 views

Definition of Integral Morse Homology

I am reading through "Morse Homology and Floer Homology" by Audin and Damian and I am confused about the definition of the differential in integral Morse homology. Let $V$ be a compact manifold, ...
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1answer
57 views

Showing that $\{(x,|x|):x\in\mathbb{R}\}$ is not the image of an immersion [duplicate]

Show that $A=\{(x,|x|):x\in\mathbb{R}\}$ is not the image of an immersion $f:\mathbb{R}\to\mathbb{R}^2$. Here are the definitions I know: Let $M$ be a differentiable manifold. A tangent vector ...
3
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1answer
137 views

Differentiable function under specific topological constraints

Can you give an example of$\:\:\emptyset\neq D\large⊂$$\:\mathbb{R}$ and a differentiable function $f$ : $D → \mathbb{R}$ such that $\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:D ⊂$$Acc(D)$, ...
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1answer
46 views

Isometric spheres in euclidean space

I would like to prove that the sphere of radius $R>0$, $S^2(R)\subset \mathbb{R}^3$, with the induced metric is isometric to the sphere with radius $1$, $S^2\subset \mathbb{R}^3$, furnished with ...
4
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1answer
38 views

Riemannian metric given in polar coordinates

the Riemannian metric of the euclidean plane is given in polar coordinates as \begin{align*} ds^2=dr^2+r^2d\theta^2. \end{align*} Consider more generally, \begin{align*} ds^2=dr^2+\psi(r)^2d\theta^2, ...
0
votes
1answer
39 views

Show there exists a unique map $g$ such that $g \circ f_{2} = h$

I was wondering if somebody could give me some help on this question. Any hints etc. would be greatly appreciated. Let $0 < b < a$. Define a smooth map $h: \mathbb{R}^2 \rightarrow ...