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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isotopy equivalence of maps

In the book Encyclopaedia of Mathematics, Vol. 6, Question: I do not understand the part Does this mean $F_1\circ f_0=f_1: X\to Y$ or as subsets of $Y$, $$ \{y\in F_1(f_0(X))\}=\{y\in f_1(X)\}? ...
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Can the twisting of mobius band be represented by a U (1) bundle?

With the usual embedding of a mobius band, the strip is twisted by an angle pi, smoothly, as it goes round.I think this can be represented intrinsically, independent of the embedding, by attaching a ...
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Homotopy of certain maps induced homotopies

Let $\psi$ be a homeomorphism and $\gamma :[0,1] \rightarrow \mathbb{R}^2$ a path. Now assumme additionally that $(\psi \circ \gamma)(t) \neq \gamma(t)$ everywhere. Then we can look at the map ...
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Geometry of images of maps $f: \mathbb R \to \mathbb C$?

I am having trouble seeing what a continuous map $f: \mathbb R \to \mathbb C$ might look like. If it was linear it would look like a line but it's not clear to me what happens if it's any map. I ...
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Rigorous definition of an embedded connected sum.

Can someone point me to a rigorous definition of a connected sum of two smooth embeddings? I know about the usual construction, the problem is that I can't find a proof that this construction is well ...
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Proving that a polilinear operator is differentiable

A Polilinear map operator is $P:X^1 \times ... \times X^n \to Y$ such that the foolowing applies: $\lambda, \mu \in R$ $$ P( \lambda x_1^1 + \mu x_2 ^1, x^2,...,x^n)= \lambda P(x_1^1,x^2,...x^n)+ \mu ...
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$D^m\cup_h D^m$, joining $D^m \amalg D^m$ along the boundary $\partial D^m$

Given an orientation-preserving diffeomorphism $h: \partial D^m \to \partial D^m$, we can glue two copies of the closed unit disk $D^m$ along the boundary by identifying $x \sim h(x)$ to form the ...
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Concrete non trivial computation of Morse homology

I am studying Morse homology and have found only examples on spheres and tori so far. Of course the homology of these manifolds is better understood by other more standard methods, so I am having ...
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Is Heisenberg group Euclidean?

I'm reading an article speaking about Heisenberg group $\mathbb H^n$ and some of its properties. Now, I have some questions to ask, hoping to be clear enought. Reading the introduction I've ...
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A star shaped open set in $\mathbb{R}^n$ is diffeomorphic to $\mathbb{R}^n$

After googling this fact (which is used in Bott and Tu to show the existence of good covers of manifolds) I've gotten the impression this is somewhat difficult to prove. But I also came across this ...
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Easy classical physics made mathematically rigorous!

Consider the following: We are given a symplectic manifold $M$. Now, we define a Hamilton function $H : M \rightarrow \mathbb{R}.$ Additionally, we want that $H^{-1}(x)=:M_x$ is a submanifold. We can ...
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Immersion, embedding and category theory

The map between smooth manifolds $f:M \to N$ is called an immersion if the tangent map $Tf$ is injective for each $x \in M$. There is a corresponding notion of submersion. About embeddings I met two ...
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Doesn't $x^3+2y^3+3z^3=0$ give a surface in $R^3$?

In my last exam on Advanced Calculus (following Spivak's Calculus on Manifolds), I couldn't solve the following question. True or false: the set $S$ in $R^3$ given by $x^3+2y^3+3z^3=0$ is a ...
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Fixed points of diffeomorphisms: eigenvalues of the pushforward

I want to answer this question: Let $M$ be a smooth manifold, and let $f: M \rightarrow M$ be a diffeomorphism. Let $\mathrm{Fix}_f$ be the fixed points of $f$, and suppose that $x \in \mathrm{Fix}_f$ ...
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What exactly is a Kähler Manifold?

Please scroll down to the bold subheaded section called Exact questions if you are too bored to read through the whole thing. I am a physics undergrad, trying to carry out some research work on ...
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Defining a metric in the tangent spaces $ T_xM $

I'm working with a metric $D$ over the manifold of grassman $G_n(\mathbb{R}^{d})$ and have difficulties to extend $D$. Let me explain: If $M$ is a submanifold of dimension $n$ in $\mathbb{R}^{d}$ ...
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Picture behind $SO(3)/SO(2)\simeq S^2$

Is there some kind of intuitive/waving hand argument to explain that $$SO(3)/SO(2) \simeq S^2 \; ?$$
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Tangent space of quotient space

Let $\pi : M \rightarrow M/G$ be the canonical projection, where $M$ is a manifold and $M/G$ is a quotient manifold. Now, what can we say about $d \pi (p) : T_pM \rightarrow T_p(M/G)$? From my ...
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Adjoint representation is Lie algebra homomorphism

Let $T_g:=L_g R_{g^{-1}}: G \rightarrow G$ be the standard automorphism of a Lie algebra, then $Ad_g:=DT_g(e): \mathfrak{g} \rightarrow \mathfrak{g} $is called the adjoint representation. Now, I want ...
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Lie group and stabilizer quotient

Let $G$ be a Lie group and $$G_x:=\{g \in G; Ad^*_g(x)=x\}$$ the stabilizer, where $Ad_g^*$ is the adjoint of the adjoint representation. Now, I was wondering why $G/G_x$ has a manifold structure. ...
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Does the Morse-Bott index of a critical point depend on the choice of metric?

By the Morse lemma there exists a coordinate chart $(x_1,...,x_n)$ in the neighbourhood of a critical point $p$ of a Morse function $f : M^n \to \mathbb{R}$ such that \begin{equation*} f(x) = f(p) - ...
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Orbits form a manifold?

A prominent example are the coadjoint orbits $O_x = \{Ad_u^*(x);u \in G\}$ where $x \in \mathfrak{g}$ and $G$ a Lie group with Adjoint map $Ad.$ Could anybody give me an easy argument why $O_x$ is a ...
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Condition (C) of Palais-Smale

In Klingenberg's Notes, he makes the following definition: $\Lambda M$ will be said to satisfy the condition (C) of Palais-Smale if: Given a sequence $\{c_m\}$ on $\Lambda M$ satisfying: ...
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Diffeomorphism between $\Bbb{R}^{4}$ and the cube

I'm looking for an explicit diffeomorphism between the four-dimensional euclidean space $\Bbb{R}^{4}$ and the four-dimensional open cube. I wonder whether there is a simple looking map, with simple ...
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How can a Kirby diagram fail to determine a handle decomposition?

I've read that a handle decomposition for 4-manifold determines a unique smooth structure, and I've also read that every 4-manifold admits a Kirby diagram. So when does a Kirby diagram fail to ...
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45 views

Smoothness of a map between $\Bbb {RP}^n$ and $\Bbb{RP}^k$

I would like to prove the following statement. If $P: \mathbb{R}^n \backslash \{0\} \to \mathbb{R}^k \backslash \{0\}$ is smooth and homogeneous of degree $d$, then the map $\widetilde P: ...
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Tangent space of coadjoint orbit

Let $\xi \in T_xOx$ be a tangent vector at $x \in O_x :=\{\mathrm{ Ad} _{u}^*(x); u \in G\}$ for $x \in g^*.$ ($g$ is the Lie-Algebra) Then I read that this $\xi$ can be represented as the velocity ...
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Exponential map only for matrix Lie algebras?

Recently, I stumbled over some proofs in Lie algebra theory and noticed that they often use the notion of an exponential map $e^{t \zeta}$ for $\zeta \in \mathfrak{g}$ such that $e^{t \zeta} \in G$ ...
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Understanding tangent space basis

Consider our manifold to be $\mathbb{R}^n$ with the Euclidean metric. In several texts that I've been reading, $\{\partial/\partial x_i\}$ evaluated at $p\in U \subset \mathbb{R}^n$ is given as the ...
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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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Killing form - strange definition

I was just reading about Killing forms. In my opinion, the definition of these forms is quite strange. I mean why would one define $B(X,Y) = \mathrm{tr} (\mathrm{ad} (X) \circ \mathrm{ad} (Y))$? I ...
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Lie algebra of affine linear maps

Let $G$ be the Lie group of affine transformations, $$\{x \mapsto Ax+b,A \in GL(n), b \in \mathbb{R}^n\}.$$ We can represent these maps as matrices $$\begin{pmatrix} A & b \\ 0 & 1 ...
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Moser's Twist Theorem for maps with reflection

Suppose I have a simple two dimensional integrable twist map, such as $x_{1}=x_{0}+y_{0}, \quad y_{1}=y_{0}$. Suppose that I perturb it in such a way that Moser's Twist Theorem is satisfied. What ...
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Lie algebra affine transformations [duplicate]

Let $G$ be the Lie group of affine transformations $$\{x \mapsto Ax+b,A \in GL(n), b \in \mathbb{R}^n\}.$$ Then we can represent these maps as matrices $\begin{pmatrix} A & b \\ 0 & 1 ...
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Adjoint and coadjoint orbits

I just read that for the Lie algebras $\mathfrak{gl}(N),\mathfrak{sl}(N),\mathfrak{so}(N),\mathfrak{sp}(2N)$ the adjoint and coadjoint orbits coincide. Now, the adjoint orbits are $O_{\xi} = ...
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Closed 1-forms implies exact 1-forms

I have two problems, the first one I think I've proved, but I have problems on the second one. Let $\omega$ a closed $1$-form defined on a open $U\subset \mathbb{R^2}$ and let $\gamma:[0,1]\rightarrow ...
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Do analytic functions on open subsets of $\mathbb{C}$ with an analytic square root form a sheaf? [duplicate]

I'm trying to learn algebraic geometry and am trying to think about what kinds of things are presheafs but not sheafs. One exercise I had was to show that bounded holomorphic functions on open ...
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1answer
36 views

Smooth maps preserve dimension

I stumbled over a useful consequence, that is apparently wrong for only continuous maps. Imagine $A \subset \mathbb{R}^{n-1}$ is a compact set and $F : \mathbb{R}^{n-1} \rightarrow S^{n}$ a smooth ...
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64 views

Differential forms- riddle me this.

I stumbled over this answer on math.stackexchange Let $x$ be a point in $\rm M$. Then because $\omega$ is non degenerate at $x$, the antisymmetric matrix $\omega_x$ has full rank on the tangent ...
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Exhibit a smooth map $f : \mathbb{R} \to \mathbb{R}$ whose set of critical values is dense.

Question 1.7.5 (Differential Topology - Guillemin and Pollack) Exhibit a smooth map $f : \mathbb{R} \to \mathbb{R}$ whose set of critical values is dense. From [Exercise 1.1.18], there is a function ...
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1answer
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Proving that Lie groups are parallellizable

Let $G$ be a Lie group. There is a diffeomorphism $$G \times T_e G \to TG$$ mapping $(g, [\gamma]) \mapsto [g \cdot \gamma]$. The inverse map then gives rise to the following isomorphism of bundles: ...
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isotopy equivalence between manifolds

The definition below is from Encyclopaedia of Mathematics: Volume 6. Question: For any $n\geq 1$, is the $n$-dimensional closed cube $$[0,1]^n=[0,1]\times [0,1]\times\cdots \times[0,1]$$ isotopy ...
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Cut out characteristic Submanifold N ($w_1(M)=w_1$(Normalbundle of N in M)). Remainder M-N is orientable? Orientation Character or CW-Structure?

So I try to understand the following (which is taken from Dold, "Structure of the cobordism ring", Page 3/274, in the paragraph "1. La suite exacte de Wall."): https://eudml.org/doc/109581 ): Giving a ...
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Basic examples topological manifolds with boundary

I've just started to study differential geometry and I've some problems with the first definitions. We have defined a topological manifold with boundary of dimension n as a topological space $M$ such ...
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Exterior derivative

Let $$\omega = \frac{1}{2} \sum_{i,j} \omega_{i,j} dx_i \wedge dx_j$$ be an antisymmetric form, so i.e. $\omega_{i,j} = - \omega_{j,i}.$ Now, in some lecture notes I found that $$d \omega ...
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Dimension of the $0^{\text{th}}$ de Rham cohomology group of $U$

Let $U\subset\mathbb{R}^n$ be an open set. I proved that $$ H^0_{DR}(U):=\frac{\text{closed forms}}{\text{exact forms}}=\{f\in C^{\infty}(U):\,f\,\,\text{is locally constant} \} $$ I have to show ...
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Symplectic geometry spectrum

I have a question about a step in the proof of the following theorem from symplectic geometry: The theorem is: Given any ellipsoid $E:=\{ w \in \mathbb{R}^{2n}; \sum_{i,j =1}^{2n} a_{ij}w_iw_j \le ...
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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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Finite universal covering induces injective maps on cohomology

I am trying to prove the following: Suppose $M$ is a smooth, connected manifold with finite fundamental group and $f : \widetilde{M} \rightarrow M$ is its (smooth) universal cover. Show that $f^* : ...
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Intersection form on $S^2 \tilde \times S^2$

The intersection form on the nontrivial $S^2$-bundle over $S^2$, denoted $S^2 \tilde \times S^2$, can be written as $\left[\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix}\right]$ with ...