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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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An intutive proof of 'replacing two-caps by a handle'

I am trying to understand a statement given in Polchinski Vol.1 - a torus with cross-cap can be obtained either as (g,b,c) = (0,0,3) or as (1,0,1), trading two cross-caps for a handle. Here, g is ...
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Non-existence of local inverse for smooth map $S^3 \rightarrow \mathbb{R}^3$.

I am working on a problem which asks me to show that for any smooth map $f : S^3 \rightarrow \mathbb{R}^3$, there must exist at least one point $p \in \mathbb{R}^3$ where there fails to exist a local ...
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Curve shortening flow and strong maximum principle

I am in particular uncertain about how the strong maximum principle is used in the argument below. Could someone please clarify and add more detailed explanations. Thanks So assume we have a regular ...
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What's the geometrical meaning of immersion?

Does that just mean to different tangent vectors, their images are different tangent vectors?
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Does a proper map have to be continuous?

In Pollack's differential topology, the proper map is defined by the preimage of every compact set is compact. Here it doesn't require the map to be continuous. However, in his following claim, to a ...
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Is pointwise multiplication by a smooth non zero function a diffeomorphism

Say $f: \mathbb R \to \mathbb R$ is nowhere zero (like e.g. the constant map 1). Is the map $x \mapsto x f(x)$ a diffeomorphism? It seems to me that the answer is no because the derivative of a ...
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The first Kirby move and $\mathbb{C}P^2$

A surgery on a link in $S^3$ can be regared as 2-hadnle attachements to $D^4$ (resulting 4-manifold $W$ whose boundary is the result of the surgery). I would like to know how the first Kirby move ...
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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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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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Vector Bundles over Spheres

I would like to understand how to construct a vector bundle over the n sphere give a map of its equatorial $(n-1)$-sphere into the general linear group $GL_n(\mathbb{R})$. My thought is that one ...
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Pullback of $1$-form in coordinates

Let $$\theta(p) = \sum_{i=1}^n f_i(p) \, dx_i$$ be a $1$-form in local coordinates. then we define $F^*(\omega(p))(X_1,\ldots,X_n) = \omega(F(p))(DF(p)(X_1),\ldots,DF(p)(X_n))$ as the pullback of a ...
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Differentiation calculation

$L(E)$ espace fonction continuous and linear $$\begin{array}{llll} \psi:& L(E)\times E&\longrightarrow& E\\ &(u,x)&\longrightarrow &u(x) \end{array}$$ proved the application ...
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Lemma 2 - p.204 (Differential topology - Guillemin and Pollack)

Let $A$ be a compact subset of $\mathbb{R^n}$. Suppose that $A \cap V_c$ is contained in an open set $U$ in $V_c$. Then for any suitably small interval $I$ about $c$ in $\mathbb{R}$, $A \cap V_I$ is ...
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Orientability of Surfaces and the Fundamental Group

Let $(M,g)$ be a compact riemannian 3-manifold and $\Sigma \subset M$ an embedded compact surface homeomorphic to the projective plane. Consider the application $i_\#:\pi_1(\Sigma)\to \pi_1(M)$ given ...
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Why do we require differential manifolds to be Hausdorff? [duplicate]

Among the requirements for a differential manifold $M$ is that it be connected and Hausdorff. What fails if a manifold is not Hausdorff?
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$G$-structure defined by a tensor

Let $M$ be an $n$ dimensional manifold with its bundle of linear frames $\pi:L(M)\to M$. Suppose $T_0$ is a tensor on $\mathbb R^n$ and $u\in L(M)$. We may view $u$ as a linear map $u:\mathbb R^n\to ...
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Lemma 1 p.205 (Differential topology - Guillemin and Pollack)

Lemma 1 : Let $S_1, ..., S_N$ be the covering of the closed intervalle $[a,b]$ in $\mathbb{R^1}$. Then there exists another cover $S'_1, ..., S'_M$ such that each $S'_j$ is contained in some $S_i$, ...
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Show that the natural copy of $\mathbb{R^{n-1}}$ inside $\mathbb{R^{n}}$ - namely, $\{(x_1, x_2,…, x_{n-1},0)\}$ - has measure zero

Question P.202 (Differential Topology - Guillemin, Pollack) : Show that the natural copy of $\mathbb{R^{n-1}}$ inside $\mathbb{R^{n}}$ - namely, $\{(x_1, x_2,..., x_{n-1},0)\}$ - has measure zero. ...
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Determine when $T(S^n \times S^k)$ is a trivial tangent bundle.

My partial answer is that , to make $T(S^n \times S^k)$ trivial, a neccessary condition is that $n$ or $k$ is odd. My argument is as follows: Suppose $T(S^n \times S^k)$ is trivial, that is, $S^n ...
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Confused (disoriented?) by questions about orientation

I believe I have a reasonable basic understanding of orientation. Yet, i'm finding myself utterly confused when facing a specific question. Here are several examples: Exhibit an ordered basis ...
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Huisken's distance comparison principle and type II singularities.

I've been reading Huisken's paper on his distance comparison principle and he remarked that in particular his theorem rules out the formation of type II singularities. These are singularities where in ...
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Curvature shortening flow for immersions

I was wondering if for an immersed curve in the plane, is it true that if the singularity points are evolved appropriately, then the curve becomes more embedded. And if so, would it eventually become ...
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Is the Zariski topology equipped with Eisenstein's metric an analytic submanifold?

Using $M=(C(\mathbb{R}),T_z)$ with the norm $(x,y) \to \log(\partial_x+\partial_y)$, we can easily define a derivative using distributions. I was wondering: Does this make $M$ an analytic manifold ...
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Curvature shortening flow of embedded curves

QUESTION: I'm not sure how they proved part c in particular. Note that theorem 2.1 refers to Huiskan's distance comparison principle for evolving curves. I don't see why a separating boundary curve ...