Tagged Questions

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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2
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1answer
21 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, ...
4
votes
1answer
31 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?
1
vote
0answers
22 views

Cones over a topological space and homotopy!

Let $X$ be a topological space and let $CX$ be the cone over $X$. We identify $X$ with the subspace $X\times \{0\}$ of the cone through the immersion $x \mapsto [(x,0)]$ for $x \in X$. Let $f:X \to Y$ ...
1
vote
1answer
19 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 ...
1
vote
0answers
40 views

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?
0
votes
1answer
10 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 ...
2
votes
1answer
38 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 ...
2
votes
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 $M$?
1
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0answers
18 views

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 ...
0
votes
1answer
51 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 ...
0
votes
0answers
8 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 ...
1
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0answers
20 views

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 ...
0
votes
2answers
37 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 ...
1
vote
2answers
48 views

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 ...
1
vote
0answers
26 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 ...
0
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2answers
37 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 ...
0
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0answers
18 views

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$ $ ...
0
votes
0answers
21 views

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 ...
0
votes
1answer
32 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 ...
0
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0answers
19 views

Prove that 2 maps of the circle $S^1$ into itself are homotopic if and only if they have the same degree.

Prove that 2 maps of the circle $S^1$ into itself are homotopic if and only if they have the same degree. Let $g_0,g_1: S^1 \to S^1$. => assume that $g_0$ and $g_1$ are homotopic, then for all ...
4
votes
2answers
121 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 && ...
1
vote
1answer
33 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 ...
1
vote
1answer
19 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 ...
3
votes
1answer
100 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 ...
0
votes
0answers
79 views
+50

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 ...
0
votes
1answer
18 views

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 ...
0
votes
1answer
37 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 ...
0
votes
1answer
31 views

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 ...
0
votes
1answer
48 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 ...
1
vote
2answers
37 views

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)]=\{ ...
2
votes
3answers
65 views

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: ...
3
votes
0answers
29 views

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, ...
1
vote
0answers
28 views

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 ...
0
votes
0answers
24 views

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?
0
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0answers
17 views

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 ...
1
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0answers
17 views

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 ...
0
votes
1answer
38 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 ...
3
votes
1answer
45 views

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 ...
-1
votes
0answers
32 views

definition of domain with boundary of class c2

I googled about domain with boundary of class c2 but I didn't find its exact definition. It seems every time that the given definition is missing some part. Can ...
0
votes
1answer
35 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 ...
2
votes
1answer
29 views

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 ...
0
votes
0answers
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 ...
1
vote
0answers
34 views

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 ...
0
votes
0answers
15 views

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 ...
1
vote
0answers
34 views

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 ...
2
votes
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 ...
1
vote
1answer
23 views

holomorphic function and simple zeros

How can I prove this? If $f$ is a holomorphic function in a domain $U$ and $f'(z)\neq0$ for all $z\in U$ then every zero of $f$ are simple and positive. Definition: $q\in U$ is a simple ...
5
votes
1answer
89 views

Does the compact manifold $f=0$ resist small perturbations?

Suppose we have a compact manifold of the form $\left\{f=0\right\}$ where $f:\mathbb R^n\to\mathbb R$ is a smooth Morse function. I am interested in showing that the manifold is topologically ...
0
votes
0answers
19 views

Prove that the index of a vector field is well-defined?

How can I prove that the index of a smooth vector field is well-defined? All I know that it is locally constant.
1
vote
0answers
39 views

Closed 1-forms implies exact 1-forms

I have two problems, the first one I think I´ve prove it, 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 ...