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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Symplectic matrix

I want to show that if $\lambda$ is a real eigenvalue of a symplectic matrix $A$ then its char poly is of the form $det(A-\mu id) = (\lambda-\mu)(\frac{1}{\lambda}-\mu) det(\hat{A}- \mu id) $ where ...
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Preimage of a submanifold is a submanifold - Transversality

It is well known that if a smooth Map $f : M \to N$ between two smooth manifolds (finite dimensional) is transversal to a submanifold $L \subset N, L \pitchfork f$, than $f^{-1}(N)$ is a submanifold ...
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How to calculate homotopy invariant winding number?

Consider a map $f:S^1\to U(1)=S^1$, since we know $\pi_1(S^1)=\mathbb{Z}$, which measures how many times the map "wind" around the circle. Given some explicit form of the function $f(\phi$), where ...
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Bijective local isometry to global isometry

Suppose that I have a bijective local isometry $f: X \rightarrow Y$ where $X$ and $Y$ are length spaces. Can I show that $f$ is a global isometry? My thought is to consider a path $\gamma$ from $x$ to ...
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15 views

Tangent Spaces of Distinct Points are Disjoint?

I'm reading Tu's "An Introduction to Manifolds", and he defines the tangent bundle on $M$ as the disjoint union $TM:=\bigcup_{p\in M}\{p\}\times T_pM$, but he remarks that for $p\neq q$, we already ...
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22 views

Smooth approximation to a continuous curve

Let $\gamma: [0,1] \rightarrow M$ be a continuous curve in a smooth manifold $M$. Is there a standard way to approximate $\gamma$ by a smooth curve? My thought was to look at every point $p$ where ...
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Is it possible to define submanifold like this

Wikipedia offers the following definition for an (embedded) submanifold: An embedded submanifold (also called a regular submanifold), is an immersed submanifold for which the inclusion map is a ...
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13 views

Image is not a manifold when considered as a subset: how is this possible?

Wikipedia offers two definitions of a submanifold: One is that it is the image of an immersion. But I can't make sense of the remark that " in general this image will not be a submanifold as a ...
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78 views

Perturb a piecewise-linear path to make it $C^\infty$

I'm trying to prove that any two points on a path connected smooth manifold can be joined by a smooth path. It becomes easy if I can prove the following: Given a curve $\gamma :\mathbb{R} \to ...
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23 views

Show that the outward unit normal is smooth vector field

Exercise 2.1.8 of Guillemin and Pollack asks us to prove that if $X^m \subset \mathbb{R}^n$ is an embedded submanifold with boundary, then the outward unit normal to $\partial X$ is a smooth function ...
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Generalization of Inverse Function Theorem to noncompact submanifolds

In Guillemin and Pollack's Differential Topology, they give as an exercise (#1.8.14) to prove the following generalization of the Inverse Function Theorem: Use a partition-of-unity technique to ...
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36 views

Vector field left-invariant then also its respective flow?

I was wondering whether left invariance of a vector field $X$ to a respective Lie group $G$ (so $dL(a)(x)(X(x))= X(ax))$ is transfered to the respective flow defined by $\frac{d}{dt} \phi^{t}(x) = ...
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Definition of a length metric.

Let $(X, d)$ be a metric space. Define the induced intrinsic metric $\widehat{d}$ as follows \begin{equation} \widehat{d}(x,y) = \inf_{\gamma} \sup_{t_0, \ldots, t_N} \sum_{i=1}^{N}d(\gamma(t_{i-1}, ...
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1answer
35 views

Pullback of a normal bundle

Consider $\Sigma$ a compact surface embedded into a compact 3-manifold, such that $\Sigma$ is diffeomorphic to $\mathbb{R}\mathbb{P}^2$ (real projective plane) and $\varphi:\mathbb{S}^2 \to \Sigma$ is ...
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Closed form on any submanifold closed?

Let $M$ be a manifold and $\omega$ a closed differential form, so i.e. $d \omega =0.$ If I now consider a submanifold $N$ of $M$. Does this mean that $\omega$ is still closed on $N$? This statement ...
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Sending unitary group to reals, second differential at critical points.

Let$$X = U(n) = \{B \in \text{GL}_n(\mathbb{C}): BB^* = I\}$$be the group of $n \times n$ unitary matrices over $\mathbb{C}$, viewed as a real manifold. Fix a diagonal $n \times n$ matrix $A$ with ...
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Embedding of a topological manifold

We know the celebrated 'Whitney embedding theorem' for smooth manifold that says any n-dimensional manifold can be smoothly embedded in $\ \mathbb R^{2n} \ $. Now my question is: Is there similar ...
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113 views

Index of zero of a gradient vector field at a critical point

Let $M$ be a Riemannian manifold with a Morse function $f: M \to \mathbb{R}$. The zeroes of the gradient vector field of $f$ are the critical points of $f$. How do you show that a critical point ...
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Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure?

$\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an orientation-reversing diffeomorphism (complex conjugation!), so this is diffeomorphic to the blowup of ...
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Is the connected sum of complex manifolds also complex?

Let $M$ and $N$ be real manifolds of dimension $n$ which happen to admit complex structures (so that necessarily $n=2k$ and both are orientable). Then does their connected sum $M\# N$ also admit a ...
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1answer
88 views

Questions on CW-complexes

I am trying to proof the following two statements. If $X_1 \subset \dots \subset X_i \subset \dots$ is a infinite sequence of CW-complexes, then $X = \bigcup X_i$ is a CW-complex and each $X_i$ is a ...
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62 views

Application of Morse Inequalities

I am an undergraduate student interested in morse theory. I understand, that the morse inequalities provide an lower bound for the number of critical points morse functions on a manifold can take. One ...
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77 views

Diffeomorphism group of a manifold is never a Lie group?

Let $M$ be a smooth manifold. I heard there is a way to introduce a topology and a structure of infinite dimensional manifold (something like a Banach or a Frechet manifold) on $\text{diff}(M)$ ...
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Characterization of graphs of maps between smooth manifolds

Theorem 6.52 in Lee's Introduction to Smooth Manifolds, 2nd ed., says Suppose $M$ and $N$ are smoothe manifolds and $S \subset M \times N$ is an immersed submanifold. Let $\pi_M$ and $\pi_N$ ...
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51 views

Proof of Brouwer fixed point theorem using Stokes's theorem

$\omega$ is the volume form on the boundary B -ball $f\colon B \to \partial B$ $$ 0 < \int_{\partial B}\omega = \int_{\partial B} f^*(\omega) = \int_{B} df^*(\omega) = \int_B f^*(d\omega) = ...
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1answer
37 views

Proving a given set is a submanifold

Let $S \subseteq \mathbb R^n$. I have been faced with showing that $S$ is a submanifold and I have some ideas but I want to get the complete picture. (Main) Question 1: What methods are there to ...
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21 views

C^1 mapping of a non-metric topological space - does this make sense?

Is there a way to define a derivative on a mapping between general topological spaces without invoking a metric?
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48 views

Are these “Stiefel-Whitney numbers” invariants of the (smooth)topology of the total space?

Let $E$ be a smooth real vector bundle over a smooth compact $n$-dimensional manifold $M$. It is relatively easy to see that the Stiefel-Whitney classes $w_i(E)$ are not diffeomorphism invariants of ...
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1answer
36 views

Locally flat manifold from Frobenius, differential forms approach

It is a known fact that a Manifold $M$ is locally flat if and only if the Riemann curvature tensor (Levi-Civita connection) vanishes. To show the if part is easy, but I am trying to follow the details ...
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Pulling back vector fields

I want to find conditions under which one can pull-back vector fields (if it is at all possible). Let $F:M \to N$ be a smooth surjective map between two $C^{\infty}$ manifolds of the same dimension. ...
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38 views

Orientability and volume form

I was wondering if there is an easy argument that manifolds that are orientable always have a volume form and vice versa? So I am not looking for a full proof of this, but rather a good argument how ...
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97 views

Prerequisites for differential topology

I want to self study differential topology. I know the basics of point set topology. I need some multivariate calculus and linear algebra, what is a good and short set of books which covers what I ...
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1answer
35 views

Can a single point in a manifold be seen as a sub manifold?

In Pollack's differential topology, in Transversality, p.28, it reduced the study of the submanifold $Z$ to the simpler case, where $Z$ is a single point. But by the definition of manifold, it seems ...
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Every differentiable structure is smoothable to a smooth structure.

I was studying differentiable manifolds and smooth manifolds. While reading on the Wikipedia website about them, I came across one statement that I have no idea why is it. This statement is Every ...
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1answer
44 views

Non-orientable submanifolds

Let $M$ be a $n$-manifold and let $S \subset M$ be a non-orientable $n$-dimensional submanifold possibly with boundary. Under what conditions can I conclude that $M$ is also non-orientable? Is ...
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31 views

Integrate symplectic two-form

I am supposed to show the following (maybe falsely stated) theorem Let $\Phi$ be a symplectic group action $\Phi: G \times M \rightarrow M$ on a manifold $M$. Assume that the symplectic form $\omega$ ...
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Equivalent condition for non-orientability of a manifold

I've just came across this question, which gives us a great tool for showing that smooth manifold is non-orientable. Namely Thm. If $M$ is a smooth manifold and there are two charts ...
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82 views

Is the sphere $S^2$ diffeomorphic to a quotient of the square?

If we take the square $[0,1]\times [0,1]$ and collapse the border, the resulting quotient space is homeomorphic to the sphere. The same holds if we take the square $[0,1]\times [0,1]$ with the ...
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1answer
37 views

Generalizations of Inverse Function Theorem

A beautiful exercise in Guilleman and Pollack asks us to show the following generalization of the Inverse Function Theorem: Suppose $f: M \to N$ is a map of smooth manifolds, and $Z$ is a ...
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1answer
232 views

Analytic approximations of the step function

Consider the Heaviside step function: $$H:\mathbb{R}\to \mathbb{R} $$ defined by $$H(x)=\begin{cases} 0 & \mbox{if } x<0 \\ 1 & \mbox{if } x\geq 0\end{cases}$$ Fix any $\delta>0$. Given ...
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closed form is exact in euclidean space

Question is to show that $d(f)=0$ for a $0$ form on $\mathbb{R}^n$ then $f$ is a constant function. See that $$0=df=\sum_i\frac{\partial f}{\partial x_i}dx_i$$ implies that $\frac{\partial ...
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324 views

Which continuous functions are polynomials?

Suppose $f \in C(\mathbb{R}^n)$, the space of continuous $\mathbb{R}$-valued functions on $\mathbb{R}^n$. Are there conditions on $f$ that guarantee it is the pullback of a polynomial under some ...
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Reference request: Pullbacks along submersions are submanifolds and the induced map is a submersion

I'm looking for an introductory book in differential topology in which there are proofs of the following facts: Let $X,X',Y$ be smooth manifolds, $X\rightarrow Y$ a smooth map and $X'\rightarrow Y$ ...
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1answer
29 views

Preimage Lemma for transverse map. Help with some passages

I'm on my way proving the Preimage Lemma for a transverse smooth map but I've encountered some problems with two passages: Let $f\colon M \to N$ be a smooth map transverse to a submanifold $L$ of ...
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186 views

Lie bracket; confusing proof from lecture

I am having some difficulties understanding this proof. Let $G$ be a closed matrixsubgroup of the general linear group. We have a right translation $Y(g):=dR_g(e) Y(e)$ on the Lie algebra $Y \in ...
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1answer
130 views

Parallelizability of Lie groups

I have tried a lot to prove this well-known result. The basic idea behind the proof is clear to me. But I'm stuck at showing either of the following conditions: The map $\ G \rightarrow TG \ $ given ...
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Is the restriction of a maximal atlas on an open submanifold maximal?

Let $M$ be a $n$-manifold, with some maximal atlas $A$, and let $V \subset M$ be an open set. The standard open-submanifold-atlas on $V$ is $A|_V$ defined as $$A|_V = \big\{ (U \cap V,x|_{U \cap V}) ...
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Is it (not) possible for two vector fields on the Klein bottle to be a basis?

Lefschetz fixed point theorem (for example) implies that a compact manifold with a nowhere vanishing vector field must have Euler characteristic zero. Is there a way to draw stronger algebraic ...
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1answer
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How does an atlas give a notion of whether a function is differentiable or not?

Defintion. An atlas is a set of pairs $\{(O_i,ψ_i)\}$ such that $\cup_iO_i=M$ and $\phi:O_i \rightarrow \mathbb{R}^n$, and most importantly, for $O_i \cap O_j \neq \emptyset $, $\tau_{ij}: \phi_i(O_i ...
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I have no idea what “smooth structure” is

I know what a manifold is: it's a topological space such that for every point there is an open set that looks like $\mathbb{R}^n$. But I do not know what a smooth manifold is, because I have no clue ...