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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If the linearization of a section is surjective on a slice, is the image under a submersion also a smooth manifold?

Let $V \rightarrow M \times N_1 $ be a vector bundle, where $M$ and $N_1$ are smooth manifolds and $s: M \times N_1 \rightarrow V$ a smooth section such that if $s(p,q) =0$ then $$ \nabla ...
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64 views

Torus/Moebius Band homeomorphism

Is a fattened Moebius Spiral Band homeomorphic to a Torus? ( due to the same Euler Characteristic $\chi$ ?) Are both non-orientable? Following (3D printable plastic) Torus has a square section ...
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Boundary of $\mathbb{R}^4$ and fundamental group of $\mathbb{R}^4/\mathbb{R}^2$

a) To what I know boundary makes no sense in open sets. Does it make any sense to talk about the boundary of $\mathbb{R}^4$? In physics they consider it when discussing wether the universe has a ...
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32 views

What is $C_n(X)$?

This article says the following: Let $X$ be a triangulated space and let $C_n(X)$ be a real vector space with $n$-simplices $[x_0,x_1,x_2,\dots,x_n]$. Each different combination of $x_i's$ forms a ...
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Definition of a Manifold from Guillemin Pollack

I have been studying differential topology from Guillemin and Pollack (GP). Unlike many other books that define differentiable manifolds using maximal atlases GP starts by saying $ X \subset R^{N}$ ...
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Equivalent definitions of Euler characteristic for closed manifolds

It is well-known that the Euler characteristic of a closed manifold $M^n$, which can be defined as $\chi(M)=\sum_{k=0}^n (-1)^k \operatorname{dim}H^k(M)$, equals the intersection number ...
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Show that $R^n -X$ has at most 2 connected component without using the Jordan-Brouwer theorem.

Here is what I got but my professor say it's wrong Let $X$ be a compact, connected hyperspace in $R^n$, then $ R^n-X$ consist of 2 open sets $D_0$ – the outside component and $D_1$ the inside ...
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42 views

Basis for a line bundle

I have a basic confusion. The following is drawn from Huybrechts' Complex Geometry text: "Let $L$ be a holomorphic line bundle on a complex manifold $X$ and suppose that $s_0,\dots, s_n \in H^0(X, ...
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66 views

Is $\mathbb{H}P^\infty$ an H-space or not?

$\mathbb{R}P^\infty$ is H-space. $\mathbb{C}P^\infty$ is H-space. Is $\mathbb{H}P^\infty$ an H-space or not?
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40 views

Prove directly Morse lemma for real line $\mathbb{R}$

The exercise 9 section 7 chapter 1 in Guillemin & Pollak state the next Prove directly Morse lemma for real line $\mathbb{R}$.(Hint:Use this elementary calculus lemma: for any function on ...
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Lie group quotient structure

Let $G$ be a Lie group and $H$ a normal finite subgroup. Let $\pi : G \to G/H$ be the quotient surjection. How would one show that $G/H$ is a Lie group?
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12 views

Subsets of a manifold.

I read that every open subset $A$ of a manifold $M$ is a submanifold (it is a manifold with the induced topology by $M$). If I understand correctly, the argument is that, for an element $x \in A$, one ...
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1answer
49 views

Extenting a homeomorphism on subsurface to the entire surface

Suppose you have surface and a subsurface. The complement of subsurface is union of open discs and once punctured open discs. Can all homeomorphisms of the subsurface be extended to homeomorphisms of ...
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2answers
17 views

Preimage of 0 for a differentiable function.

If a subset $N$ of a manifold $M$ can be written as $f^{-1}(\{0\})$ being $f:M \longrightarrow \mathbb{R}$ a differentiable function, can I conclude that $N$ is a submanifold of $M$?
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How to prove that a certain set is a submanifold.

Let $P^{n-1}(\mathbb{R})$ the real projective space of dimension $n-1$. Consider the set $$B=\{(x,y)\in\mathbb{R}^n \times P^{n-1}(\mathbb{R}) / x=(x_1,..,x_N), y=[y_1;..;y_N], x_iy_j=x_jy_i ...
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69 views

Only one fixed point for $f:\bar{\mathbb{D}}\rightarrow\bar{\mathbb{D}}$ on the boundary.

We know for Brouwer theorem that $f$ (continuous bijective function) have a fixed point. My questions are: 1) Is there a function with only one fixed point $x_0\in Int(\bar{\mathbb{D}}) $ (open ...
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22 views

1-forms as a direct sum

Let us define the spaces $C_0, C_1$ and $C_2$ of differential $0,1,2$ forms respectively on the sphere $S^2.$ Is it true that $C_1$ is the direct sum of $d(C_0) \oplus \delta^* (C_2)$? I think this ...
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Geometrical Explanation of Borsuk Theorem

Assume $K$, $L$ are $n$-pseudomanifold, and $K$ is compact. Let $f$ be a simplicial map between $K$ and $L$. We denote $n$-simplexes of $K$ and $L$ by $S_n(K)$, $S_n(L)$. Define ...
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2answers
41 views

Smooth maps homotopic to the inclusion - updated.

First problem - (my original question before the editing) Prove or disprove the following: Let $A$, $B$ be differentiable manifolds such that $A \subseteq B$, and $s: A \to B$ a smooth map. Then $s ...
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Smooth homotopy

Let $M,N$ be manifolds. Suppose that $f_0, f_1:M\stackrel{C^\infty}\to N$ are homotopic, i.e. there exists a continuous mapping $f:M\times[0,1]\to N$ s.t. $f(x,0)=f_0(x)$, $f(x,1)=f_1(x)$. Then is ...
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29 views

Embedding counterexample

Lee writes on page 156 of Introduction to Smooth Manifolds: A smooth embedding is a map that is both a topological embedding and an immersion, not just a topological embedding that happens to be ...
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42 views

How to orient a manifold in the Euclidean space?

I learned that the orientation of a smooth manifold is a smooth choice of an orientation for bases of tangent space. Also I sometimes read that an embedded manifold in $\mathbb{R^3}$ inherites an ...
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Iteratives of an invariant manifold

According to the stable manifold theorem, I want to calculate the first three Picard iteratives of the invariant manifold given by the system of differential equations: $ \dot x= -x + y^2$ $ ...
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Ref. Request Lefschetz Fibrations, Restriction of Base

All. Let $ M^4 \rightarrow S^2 $ be a Lefschetz fibration over $S^2$, where $M^4$ is a compact, oriented 4-manifold. I am still weak in this topic, and I would appreciate references to properties ...
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36 views

Prove that a map $f:S^1→S^1$ extends to the whole ball $B=\{|z|≤1\}$ if and only if $deg(f)=0$

Prove that a map $f:S^1→S^1$ extends to the whole ball $B=\{|z|≤1\}$ if and only if $deg(f)=0$ again I can only do one direction => $f:S^1→S^1$ is smooth, and $S^1 = \partial B$. Assume that ...
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124 views

Why is the pole generally outside the contour loop when its outside the contour loop in 2D?

The following contour integral is path dependent with the following results \begin{align} \oint_C\dfrac{dz}{z} = \begin{Bmatrix} 2\pi i && \text{when $z=0$ is inside C} \\ 0 && ...
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1answer
40 views

Prove that the Möbius band is not orientable.

Prove that the Möbius band is not orientable. I know that in the Möbius band the central circle is orientable. If I let $Y$ be the Möbius band and $Z$ be a compact submanifold of $Y$ with ...
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22 views

Suppose that $Z$ is a hyperspace in the oriented manifold $Y$, as sub manifold of codimension $1$. Prove that following statement are equivalent

Suppose that $Z$ is a hyperspace in the oriented manifold $Y$, as sub manifold of codimension $1$. Prove that following statement are equivalent a) $Z$ is orientable b) There exists a smooth field ...
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115 views

Coordinate-free definition of integration of differential forms?

Let $\omega$ be an $n$-form on an oriented $n$-manifold $M$. To integrate $\omega$, we choose an atlas $(O_\alpha, (x^1_\alpha,\dots, x^n_\alpha))_\alpha$ for $M$ and a partition of unity ...
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147 views

Extension of a map $g:\overline{B_1^n}\to \mathbb{R}^2$

Let $B_r^n\subset \mathbb{R}^n$ ($n\geq 6$) be the open ball with radius $r$ and let $g:B_2^n\to \mathbb{R}^2$ be an analytic map. How to define a continuous map $h:\mathbb{R}^n\to \mathbb{R}^2$ such ...
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Any real analytic Frobenius theorem used in the proof of integrable almost complex manifolds to arise from complex manifolds?

The reference book is S.S.Chern's Complex Manifolds Without Potential Theory. It's used in integrability condition for an almost complex structure to arise from a complex structure. It's claimed that ...
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40 views

Smooth structure on $M\cup_f N$?

Let $M$ and $N$ be two smooth manifolds with $$\textrm{dim}(M)=\textrm{dim}(N)=n.$$ Let $U\subseteq M$ and $V\subseteq M$ be two open sets and $f:U\longrightarrow V$ a smooth diffeomorphism. Consider ...
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Prove that the index of a vector field is well-defined.

Consider first an open set $U \subset \Bbb{R^m}$ and a smooth vector field $v : U\to\Bbb{R^m}$ with an isolated zero at the point $z \in U$. The function $\overline{v}(x) = v(x)/\|v(x)\|$ maps a small ...
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56 views

For which k,n the k-covector is decomposable (14-2 from Lee)

This is homework so no answers please The problem is: Find for which k, n, a k-alternating map $\omega$ can be written as $\omega=\omega_{1}\wedge...\wedge \omega_{k}$ were $\omega_{i}$ are ...
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Submersion surjective on the complex projective space $\mathbb{C}P^1$.

If $S^3=\{ (z_1,z_2)\in\mathbb{C}^2\mid \vert z_1\vert^2+\vert z_2\vert^2=1\}$ and $\pi:S³\rightarrow\mathbb{C}P^1$ for $(z_1,z_2)\mapsto [(z_1,z_2)]$ since $[(z_1,z_2)]=\{ ...
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Definition of the hessian as a bilinear functional on the tangent space

In Milnor's Morse Theory, the Hessian of a smooth function $f : M \to \mathbb R$ defined on a manifold $M$ at a critical point $p$ is the bilinear functional on $T_p M$ defined as follows: ...
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Surgery or Partial Whitney Trick in McDuff and Salamon proof of Gromov squeezing

In the proof of Gromov squeezing in McDuff and Salamon's epic J-Holomorphic Curves and Symplectic Topology, the authors use a surgery argument without any explicit construction. In the following, ...
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Degree of Map between Pseudomanifold

There are two different ways to define a degree of map. Let $M$, $N$ be smooth, and $M$ is compact, $N$ is connected. If $f\in C(M,N)$, we define $\deg f$ by smooth approximation. [Milnor/Topology ...
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Orientability of integrable plane fields

Let $\xi \subset \text{T}M$ be a integrable plane field on a smooth 3-manifold (i.e. the tangent field of a foliation). Is it true that $\mathcal{F}$ is coorientable?
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Realizing a Contact Structure on S^1 x S^2 via an Open Book Decomposition

I am trying to learn about Contact Geometry and Open Book Decompositions. I went through the example of the Hopf Fibration for $S^3$ and how you can see a contact structure. I am now trying to do ...
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Prove that $f$ map a neighborhood of $Z$ diffeomorphically on to a neighborhood of $f(Z)$( more detail)

Supposed that the derivative of $f:X\to Y$ is an isomorphism whenever $x$ lies in the sub-manifold $Z \subset X$, and assume that $f$ maps $Z$ diffeomorphically onto a $f(Z)$ . Prove that $f$ map a ...
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1answer
40 views

Something about Degree of Map

I have been told that degree of map can be defined by the combinatorial method instead of the differential one. Assume $M$, $N$ is orientiable pseduomanifold, and $M$ is compact, $N$ is connected. Let ...
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Is a diffeomorphism's image automatically open?

Sorry if this question is trivial, I am new to smooth manifold theory. Let $\varphi : I \times \mathcal S^{n-1} \to X$ be a diffeomorphism. $I=(0,1)$, $\mathcal S^{n-1}$ is the unit sphere in ...
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1answer
52 views

Tubular neighborhood of $X^k$ compact submanifold with normal bundle $\perp X$ trivial

For $X^k\subset M^n$ compact submanifold with $\perp X$ trivial and set $S^k$ the $k$-sphere. Then there is a function $f:M^n\rightarrow S^k$ such that $X$ is the preimage for a regular value. My ...
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A function from a smooth manifold with boundary to $[0,\infty)$

Suppose $M$ is a smooth manifold with boundary, show that there exists a smooth function $f: M \rightarrow [0, \infty)$ such that $\partial M = f^{-1}(0)$. My attempt is that given a chart ...
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15 views

jacobian of the transition functions

Can you check if the $(21)^{th}$ entry of my Jacobian matrix is correct. Consider the cotangent bundle $\pi : T^*M \rightarrow M$. Compute the Jacobian for the transition functions on the overlaps ...
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Counterexpample for solution for D.E. of second order

Set $M$ a $n$-manifold with $1\leq n$. Show that not every curve in $M$ is the solution for a differential equation of second order. A curve on $M$ is a differentiable fuction ...
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Visualizing sums of bundles

So I was wondering about what the whitney sums of various line bundles would look like in general, since it is possible to visualize such sums. I know that the sum of two mobius bundles is just the ...
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Connected sum of orientable manifolds

I was reading through Lee's Smooth Manifolds on the part regarding orientations and I was wondering if the connected sum preserves the orientability of manifolds. Intuitively it seems to be true, but ...
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1answer
32 views

Arbitrary Smooth structure

Is it possible to give a smooth structure to any objects? Say two lines intersecting at a point. It seems there is a smooth structure though at the intersecting point it is not locally euclidean if ...