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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Is every open subset of a manifold homeomorphic to some Euclidean space?

Let $M^n$ be a connected topological manifold. Is every proper open subset of $M$ homeomorphic to some open set in $\mathbb{R}^n$?
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Is $(\#^k \Bbb{RP}^2) \times I$ an $\mathbb{RP}^2$-irreducible 3-manifold?

Consider $S$ a surface homeomorphic to a connected sum of $n$ projective planes, $n \geq 2$. Can there be a two sided projective plane embedded in $[-\epsilon,\epsilon]\times S$?
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why the Poincaré Duality morphism induces a morphism from cohomology to dual cohomology

I am studying the de rham theorem and Poincaré Duality from http://www.few.vu.nl/~vdvorst/DeRham.pdf and I have a question about the Poincaré map \begin{align*} \mathcal{PD} : \Omega^p(M) ...
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Distances in geodesic triangles

Let $a, b \in \mathbb{R}^2$ be two points in the plane and let $\Pi$ be their perpendicular bisector (see left figure). Let $c \in \Pi$ be any point and consider the triangle $\triangle abc$. Suppose ...
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The motivation of differential forms

The motivation of differential form, I think, is created to deal with the integral on manifold. But in most textbooks, differential forms is introduced by some other knowledges about covectors, tensor ...
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how to evaluate homotopy group of this specific structure

I am a Ph.D. student of physics and now I have some problems regarding the evaluation of homotopy group of a specific structure. In a paper, a specific topological structure is defined. The structure ...
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49 views

Possible error in Guillemin and Pollack RE de Rahm cohomology?

The context is Guillemin and Pollack, Chapter 4.6, Cohomology with Forms. Let $U$ be an open subset of $\mathbf{R}^k$ and let $\omega$ be a $p$-form on $\mathbf{R} \times U$, represented as $$ ...
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Exists homeomorphism which carries each fiber isomorphically to itself, composition… make rigorous.

See here for a question I asked. Let $\mu$ and $\mu'$ be two different Euclidean metrics on the same vector bundle $\xi$. How do I see that there exists a homeomorphism $f: E(\xi) \to E(\xi)$ ...
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The real points of $\operatorname{mSpec}( C(\mathbb{R}))$?

Known fact: If $X$ is a compact, Hausdorff space, then $X$ is homeomorphic to the max spectrum of $C(X)$ with the Zariski topology. This fails for non-compact spaces, as for instance there may be too ...
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is the vector space of n- forms of an n-manifold equal to the vector space of compactly supported n-forms?

Let $\Omega^{n}(M)$ be the real vector space of smooth n-forms of an n-manifold $M$. It is a real vector space of dimension 1. $\Omega^{n}_c(M)$ is the real vector space of compactly supported smooth ...
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62 views

First order PDE system on a complete Riemannian manifold

Let $X_1, X_2$ be two orthogonal everywhere non-zero vector fields on a complete Riemannian manifold $M$. Can one always solve the solve the system $$X \phi = 1, Y\phi = 0,$$ $$X\psi = 0, Y\psi = 1 ...
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23 views

Is there a minus thickening operator on a metric space?

Let $S$ be a metric space and $A$ a subset. For some $\varepsilon>0$ define the $\varepsilon$-thickening of $A$ as $$A^{\varepsilon} = \left\{p \in S \mid \exists q \in A \;\;\text{with}\;\; ...
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116 views

Tangent vectors in $\mathbb{R}^n$

I am confused with the idea of tangent vector or tangent space. First of all, I learned that there is an isomorphism from $ \mathbb{R}_a^n$ onto $T_a( \mathbb{R} ^n)$ from John M.Lee' book ...
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98 views

Morse functions and connected sum

My question is closely related to this post but it is slightly different. Let $M_1$ and $M_2$ be two smooth closed $n$-manifolds such that there is a Morse function $f_i:M_i\rightarrow \mathbb R$ for ...
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69 views

Hints for an exercise on Morse theory

Exercise: Let $M$ be a $3$-dimensional smooth manifold with boundary $\partial M$ which is a surface of genus $g$. Moreover let $f:M\longrightarrow [0,1]$ be a Morse function with the following ...
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26 views

Does the Morse homology depend on the orientation?

Before asking my question I need to define some objects. I will follow the book "M. Audin, M.Damian - Morse theory and Floer homology", but the terminology is quite standard: Let $M$ be a smooth ...
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How to prove this null space can not be completely contained in $R^{m-1}\times0$?

A capture from "topology from differentiable viewpoint" by Milnor How to prove this statement in the proof of lemma 4 in the picture: the forth line from the bottom. "but the hypothesis that ...
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Is there a reason why Harmonic functions are defined on open sets?

Whenever I see a definition of a harmonic function, it's always defined as follows A function $f : U \to \Bbb{R}$ is called harmonic (where $U$ is an open subset of $\Bbb{R}^n$) iff it is twice ...
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33 views

Canonical Bundle Isomorphism $T_v(E)\cong \pi^* E$

Given a smooth vector bundle $\pi:E\to M$, the vertical bundle $T_v(E)$ is by definition $\ker T\pi$, which is a subbundle of $T(E)$. It is asserted in this answer that there is a canonical ...
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Cell-structure for Grassmann manifolds, is restriction homomorphism an isomorphism for $p < k$? [closed]

Is the restriction homomorphism$$i^*: H^p(G_n(\mathbb{R}^\infty)) \to H^p(G_n(\mathbb{R}^{n+k}))$$an isomorphism for $p < k$? Here, any coefficient group may be used.
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Given two points on a manifold, is there a chart containing both?

Consider a connected manifold $M$ and two distinct points $p,q$ on the manifold. Is it true that there exists a chart containing both? How can one prove this result? OBS: I've seen an answer on the ...
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61 views

Stably trivial bundle is trivial

I have a smooth embedding $f:S^2\to \mathbb{R}^4$ and would like to show that the normal bundle $\nu\to K$ of the image $K:=f(S^2)$ is trivial. I have already shown that it is stably trivial, i.e. $ ...
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33 views

Example of a $C^\infty$ function which cannot be approximated by local diffeomorphisms

I'm trying to understand the proof of Whitney's theorem characterizing generic maps from $\mathbb R^2$ to $\mathbb R^2$ as in chapter 8 of Brocker and Lander's Differentiable Germs and Catastrophes. ...
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156 views

Vector bundle $\gamma^1$ over infinite real projective space doesn't have finite type? [closed]

Using Steifel-Whitney classes, how do I see that the vector bundle $\gamma^1$ over $\textbf{RP}^\infty$ does not have finite type?
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41 views

curved space vs linear space with curved basis

What is actually the difference between a curved space and an euclidean space represented in curvilinear basis?
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29 views

The $f^{-1}(y)$ is locally constant where $ y$ ranges through regular values

Let $f: M \longrightarrow N$ be a smooth map between two manifolds of the same dimension, with M compact, and a regular value $y \in N$. Then the number of points in $f^{-1}(y)$ is locally constant as ...
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1answer
31 views

why the space of smooth mappings has the homotopy type of a CW complex?

let $N$ be a smooth compact manifold. I want to know why $C^\infty (N,\mathbb{R})$ has the homotopy type of a CW complex? I know that if Y has the homotopy type of a CW complex and X is a finite CW ...
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49 views

Sections of a sheaf

In algebraic geometry, specifically Hartshorne, for a given sheaf $\mathscr{F}$, we say that the sections of the sheaf over some open set $U$ are merely elements $s \in \mathscr{F} (U)$. However, I ...
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35 views

Let $\Omega$ be a star-shaped open set of $\mathbb{R}^3$. Under which conditions is $\Omega$ analytically diffeomorphic to $\mathbb{R}^3$?

In this post A star shaped open set in $\mathbb{R}^n$ is diffeomorphic to $\mathbb{R}^n$ I found a proof that $\Omega$ is always diffeomorphic to $\mathbb{R}^3$. In which cases can such a ...
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Map to exterior power gives rise to smooth embedding of Grassmannian in projective space?

How do I see that the map$$(x_1, \dots, x_n) \mapsto x_1 \wedge \dots \wedge x_n$$from $V_n(\mathbb{R}^m)$ to the exterior power $\wedge^n(\mathbb{R}^m)$ gives rise to a smooth embedding of ...
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21 views

Frobenius Condition for Singular Integrable Distributions

A smooth "singular" distribution $D\subseteq TM$ on an $n$-dimensional manifold $M$ is integrable if it is tangent to immersed submanifolds $N_\alpha$ that are disjoint and cover $M$. If dim$D=k$ ...
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129 views

What kind of object is the push forward of a vector field?

I was actually not sure about asking this question since I think I know what the answer is, but here it goes: Let $M$ and $N$ be two smooth manifolds and $\mathbf{X}$ a vector field defined on $M$. ...
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Question about Extension Lemma for Smooth Functions: Lee, Smooth Manifolds, Lemma 2.26

I may be overlooking something very simple, but I am reading the proof of Lemma 2.26 in Lee's Smooth Manifolds 2nd ed., and am a bit confused by the following statement: For each $p\in A$, choose a ...
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Grassmannian, symmetric, idempotent matrices of trace $n$?

How do I see that $G_n(\mathbb{R}^m)$ is diffeomorphic to the smooth manifold consisting of all $m \times m$ symmetric, idempotent matrices of trace $n$?
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Generalized Gauss map, giving rise to second fundamental form

I know that the tangent bundle of $G_n(\mathbb{R}^{n+k})$ is isomorphic to $\text{Hom}(\gamma^n(\mathbb{R}^{n+k}), \gamma^\perp)$, where $\gamma^\perp$ denotes the orthogonal complement of ...
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Closures of one-parameter subgroups of lie groups

I'm reading some basic facts of Lie groups. I meat with difficulties when I try to solve the following statement: "Prove the closure of a non-closed one-parameter subgroup of a Lie group is a torus." ...
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Can $\mathbb C P^4$ be smoothly embedded in $\mathbb R^{12}$?

In Bott and Tu's Differential Forms in Algebraic Topology, the authors show using Pontrjagin classes that $\mathbb CP^4$ cannot be smoothly embedded in $\mathbb R^k$ when $k\le 11$. The obvious ...
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Direct sum of $\mathbb{R}(B)$-modules consisting of all cross sections.

Let $B$ be a Tychonoff space and let $\mathbb{R}(B)$ denote the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$ let $S(\xi)$ denote the $\mathbb{R}(B)$-module ...
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Set consisting of all unoriented cobordism classes of smooth closed $n$-manifolds can be made into additive group? [closed]

How do I see that the set $\mathfrak{N}_n$ consisting of all unoriented cobordism classes of smooth closed $n$-manifolds can be made into an additive group?
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smooth vector field on the boundary of unit ball

Let C be the boundary of the unit ball in $\mathbb{R}^n$, and let $v$ be a smooth vector field on $C$. What does the condition $x\cdot v(x)>0$ for all $x$ in C mean?
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Fundamental groups of codimension 1 manifold complements

Let $M$ be a smooth manifold of dimension at most $3$ and $S \subset M$ a smoothly embedded compact connected codimension $1$ manifold, separating $M$ into two components, $M_1$ and $M_2$. I wonder ...
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$\mathbb{R}^1$-bundle $\xi$ possesses Euclidean metric iff $\xi$ represents an element of order $\le2$

The set of isomorphism classes of $1$-dimensional vector bundles over $B$ forms an abelian group with respect to the tensor product operation. How do I see that a given $\mathbb{R}^1$-bundle $\xi$ ...
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Any $\mathbb{R}$-linear mapping $X: C^\infty(M, \mathbb{R}) \to \mathbb{R}$ with $X(fg) = X(f)g(x) + f(x)X(g)$ given by $X(f) = Df_x(v)$?

Let $M$ be a smooth manifold, and let $C^\infty(M, \mathbb{R})$ denote the collection of smooth real valued functions on $M$. For $x \in M$, how do I see that any $\mathbb{R}$-linear mapping $X: ...
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Smooth manifold $M$ is completely determined by the ring $F$.

For any smooth manifold $M$, the collection $F = C^\infty(M, \mathbb{R})$ of smooth real valued functions on $M$ can be made into a ring, and every point $x \in M$ determines a ring homomorphism $F ...
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If $M$ is compact, every maximal ideal in $F$ arises in this way as a point of $M$?

For any smooth manifold $M$, the collection $F = C^\infty(M, \mathbb{R})$ of smooth real valued functions on $M$ can be made into a ring, and every point $x \in M$ determines a ring homomorphism $F ...
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42 views

Collection of smooth real valued functions on smooth manifold has ring structure.

For any smooth manifold $M$, how do I see that the collection $F = C^\infty(M, \mathbb{R})$ of smooth real valued functions on $M$ can be made into a ring, and that every point $x \in M$ determines a ...
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Tensor product for vector bundles is commutative, associative, and has an identity element?

How do I see that the tensor product operation for vector bundles over a fixed base space is commutative, associative, and has an identity element?
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How do I see that $\mathbb{RP}^4$ and $\mathbb{RP}^6$ do not admit fields of tangent $2$-planes? [duplicate]

A manifold $M$ is said to admit a field of tangent $k$-planes if its tangent bundle admits a subbundle of dimension $k$. How do I see that $\mathbb{RP}^4$ and $\mathbb{RP}^6$ do not admit fields of ...
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Subset $V$ of projective space is open iff $q^{-1}(V)$ is open?

Let $\mathbb{P}^n$ denote the set of all lines through the origin in the coordinate space $\mathbb{R}^{n+1}$. Define a function$$q: \mathbb{R}^{n+1} - \{0\} \to \mathbb{P}^n$$by $q(x) = \mathbb{R}x =$ ...
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Extending Morse-Smale pair from submanifolds?

The following proposition is extracted from Audin & Damian's Morse Theory and Floer Homology, Proposition 4.6.3: Let $(f,X)$ be a Morse-Smale pair on $V$ (a submanifold of $W$). Then there ...