Questions tagged [differential-topology]
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.
1
vote
1answer
18 views
$TS^2$ coordinate expression and coordinate change.
I want to express coordinate the (co)tangent bundle of sphere. I think
$TS^2$ (or $T^*S^2$) $=\{(x_1,x_2,x_3,v_1, v_2 , v_3 | x_1^2+x_2^2+x_3^2=1, <(x_1,x_2,x_3),(v_1,v_2,v_3)>=0 \}$
and using ...
1
vote
1answer
36 views
A difficulty in understanding a part of solution of Q.1.3.10 in Allan Pollack and Guillemin.
"Generalization of the inverse function theorem.let $f:X \rightarrow Y$ be a smooth map that is 1-1 on a compact submanifold Z of X. Suppose that for all $x \in Z$, $$df_{x}: T_{x}(X) \rightarrow T_{f(...
0
votes
1answer
24 views
A difficulty in understanding a part of the solution of Q.10 section 1.3 in Allan Pollack and Guillemin(3).
Q.10 section 1.3 in Allan Pollack and Guillemin is the following:
"Generalization of the inverse function theorem.let $f:X \rightarrow Y$ be a smooth map that is 1-1 on a compact submanifold Z of X. ...
0
votes
1answer
24 views
First generalization of the inverse function theorem Q.10 section 1.3 in Allan Pollack and Guillemin(2).
Part of Q.10 section 1.3 in Allan Pollack and Guillemin is the following:
"Generalization of the inverse function theorem.let $f:X \rightarrow Y$ be a smooth map that is 1-1 on a compact submanifold ...
1
vote
1answer
21 views
First generalization of the inverse function theorem Q.10 section 1.3 in Allan Pollack and Guillemin(1).
Part of Q.10 section 1.3 in Allan Pollack and Guillemin is the following:
"Generalization of the inverse function theorem.let $f:X \rightarrow Y$ be a smooth map that is 1-1 on a compact submanifold ...
0
votes
1answer
21 views
Clarification about the index of singular points and curves on the sphere.
I am having some problems understanding the concept of index of a singular point of a vector field on a Manifold (in particular on a 2-dimensional sphere) and some of its properties. And hope that you ...
7
votes
0answers
79 views
Geometric interpretation of torsion homology classes
Suppose I have a homology class $x \in H_1(M)$ which is torsion of order $k$ say. Suppose furthermore that $M$ has Dimension big enough, such that every element of $H_1$ and $H_2$ can be relalized as ...
2
votes
1answer
71 views
A vector field is differentiable if and only if the map $X: M \to TM$ is differentiable
Let $X$ be a vector field defined on a manifold $M$. Then $X$ is differentiable if and only if the application $\psi:M\rightarrow TM$ such that $\psi(p)=(p,X_p)$ is differentiable. I have some ...
2
votes
0answers
27 views
One-degree map between manifolds with boundary
Let $F:M\rightarrow N$ be a map between orientable compact connected $n$-manifolds such that $F^{-1}(\partial N)=\partial M$. The degree of $F$, $deg(F)$, is given by the equation $$F_{\#}([M])=deg(F)[...
4
votes
0answers
36 views
How to think about exotic differentiable structures in manifolds?
I apologize in advance for the vagueness of this question.
It is known that there exist differential manifolds that are homeomorphic but not diffeomorphic to spheres (Milnor), and likewise there are ...
1
vote
1answer
45 views
Which open subsets of $\mathbb{R}^n$ are homeomorphic to $\mathbb{R}^n$ itself?
The motivation of this question is related to the fact that in various differencial geometry books I have seen three different criteria for chart maps. These were:
If $(U,\varphi)$ is a local chart, ...
0
votes
1answer
33 views
Smooth curve at the boundary - inside a closed shape
I'm unsure whether this fits in topology or differential geometry.
I have a basic intuitive idea, and I'm trying to find either the appropriate definition or theorem relating to this idea within ...
3
votes
2answers
86 views
If a composition of functions is smooth and one of them is smooth, then the other is smooth
Show that a map $\xi$ between smooth manifolds $M$ and $N$ is
smooth if and only if $f ◦\xi$ is a smooth function on $M$ whenever $f$ is a smooth function on $N$.
One implication is clear because I ...
3
votes
0answers
37 views
$\mathbb{HP}^2$, exotic 7-spheres, and Bott manifolds
I am looking for some explanation how
$\mathbb{HP}^2$,
exotic 7-spheres, and
Bott manifolds
are related? And how the construction of a Bott manifold is related to $\mathbb{HP}^2$ ...
1
vote
0answers
38 views
How to show that vector field is continuous?
In the book the definition of a a vector field over $U$(open)$ \subseteq S^n$ is given by a continuous map $s: U \to T(U)$ such that $p_U \circ s=id_U$ where $p_U$ is the base point projection from $T(...
7
votes
1answer
80 views
Linking number and cup product
Let $S^p$ and $S^q$ be disjoint spheres in $\mathbb{R}^n$ with $n=p+q+1$ and let $X= \mathbb{R}^n- (S^p\cup S^q)$. By Alexander duality, their fundamental classes yield cohomology classes in $\tilde{H}...
2
votes
2answers
37 views
Clarification on definition of smooth map between smooth manifolds
Let $M$ and $N$ be $n$-dimensional smooth manifolds. A map $F: M \to N$ is smooth if for each $p \in M$ there exists smooth charts $(U, \varphi)$ containing $p$ and $(V, \psi)$ containing $F(p)$ such ...
0
votes
0answers
21 views
Basis for the Whitney topology
In the book of Differential Topology by Hirsch, the Whitney topology is defined. (See below)
I tried to prove the given collection $\{N^r(f;\Phi;\Psi;K;\varepsilon) \}$
is a basis. My proof starts ...
1
vote
0answers
28 views
A difficulty in understanding why there is a problem in the function constructed for Q.1.1.8(c) in Alan Pollack and Guillemin
I am asking about part(c) of this question:
The solution is given below:
But I could not understand in the solution of (c) why the implicit function $x \rightarrow r$ is not smooth at 0 and where we ...
2
votes
2answers
53 views
How do I show that this set is dense in the function space?
Consider the smooth manifold $\mathbb{S}^3$ embedded in $\mathbb{R}^4$, note that $$\widetilde{T}:= \frac{1}{\sqrt{2}}\mathbb{T}^2 = \left\{(x_1,x_2,x_3,x_4) \in \mathbb{R}^4; \ x_1^2 + x_2^2 = x_3^2 ...
0
votes
1answer
25 views
problem 4 in Alan Pollak and Guillmin differential topology.
Let $a > 0$, and set $B_a = \{x \in \mathbb{R}^n : |x|^2 < a \}$. Let $\phi : B_a \to \mathbb{R}^n$ be given by $\phi(x) = \frac{ax}{\sqrt{a^2 - |x|^2}}$. Prove that $\phi$ is a diffeomorphism ...
0
votes
0answers
46 views
Equivalence of definitions of a vector bundle
Let $n\in\mathbb N$, let $E,B$ be topological spaces and let $p:E\to B$ be a continuous map. For every $b\in B$, let $p^{-1}(b)$ be equipped with the structure of an $n$-dimensional real vector space. ...
1
vote
0answers
25 views
Pontryagin square: lifting to an integral cocycle, and an element $H^4 (B^2 \mathbb{Z}_r, \mathbb{Z}_{2r})=$Hom$(\mathbb{Z}_{2r},\mathbb{Z}_{2r})$?
Let $M$ be a simplicial complex and $Π$ be a finite abelian group. In
the simplest case $Π = \mathbb{Z}_r$ finite group with $r$ even, the Pontryagin square is a cohomological operation which maps an ...
3
votes
0answers
21 views
Sum formulas for Pontrjagin square and Postnikov square
Inspire by this, I wonder
Pontrjagin square: There is a geometric interpretation of $\mathfrak{P}_2$, due to Morita.
Assume $q=2k$, so that the Pontrjagin square is a map
$$\mathfrak{P}_2 \colon H^{...
1
vote
0answers
30 views
$\mathcal{C}^1$-topology of a submanifold with boundary
Let $M \subset \mathbb{R}^n$ be a compact connected manifold without boundary embedded in $\mathbb{R}^n$, then we can define the $\mathcal{C}^1$-topology of the functions $\mathcal{F}(M)= \{f: M\to \...
2
votes
1answer
27 views
Coorientation of contact structures
When reading about contact geometry one quickly encounters the notion of a cooriented contact structure/form. But I do not seem to be able to find a definition of "coorientation".
In some places they ...
3
votes
0answers
18 views
Critical values and real-analytic mappings between manifolds
Let $M^m,N^n$ be smooth manifolds and $f \in C^\infty(M, N)$. Then, for any regular value $y \in N$ we have that $f^{-1}(y)$ is a smooth submanifold of $M$ of dimension $m-n$. Moreover, by Sard's ...
0
votes
0answers
12 views
Help understanding proof of orbit manifold
I attach some photos.
I am reading the proof of (16.10.3). Here is the statement.
The authorclaims (16.10.3.1) is a consequence of another lemma. See:
Obviously, to understand how it is clear ...
2
votes
1answer
34 views
Degree of a map and Suspension
I want to verify the following "geometric" proof that if we have a continuous $f:S^n \rightarrow S^n$ then $\textit{deg}f=degSf$ , $Sf$ denoting the suspended map from $S^{n+1}$ to $S^{n+1}$.
It is ...
4
votes
2answers
220 views
Topology and smooth structure on tangent bundle
My lecture notes on differential geometry read the following (without proof):
For $M$ a manifold, let $TM = \bigcup_{p \in M} T_p M$ be the (disjoint) union of all its tangent spaces. Then, there ...
1
vote
0answers
101 views
Neighbourhood of a point where $(\theta_1)_{t_1} \circ (\theta_2)_{t_2} \circ \cdots \circ (\theta_k)_{t_k}$ is defined
This is actually a detail in Lee’s smooth manifold 2nd ed p.234, 2nd paragraph in the Theorem 9.46. What i found is possibly a correction for the book. Forgive me if it’s not even close.
Let $M$ be a ...
0
votes
0answers
22 views
Manifolds with varying (local or global) dimension
What subject should one look into to understand manifolds with varying dimensions throughout it's structure.
For example, Imagine a 2-sphere with a line going through it defining some type of ...
3
votes
0answers
33 views
Prove the theorem of level sets as manifolds.
THEOREM.
Let $f_1,..f_r \in C^\infty (\mathbb R^n$ and $X=\{ x\in\mathbb R^n | f_i(x)=0, 1\le i\le r\}.$ If
$$Df_1(x)=(\frac{\partial f_1}{\partial x_1}(x),...,\frac{\partial f_1}{\partial ...
4
votes
1answer
39 views
One-degree map between orientable compact manifolds
Let $F:M\rightarrow N$ be a map between orientable compact connected $n$-manifolds such that $F^{-1}(\partial N)=\partial M$. The degree of $F$, $deg(F)$, is given by the equation $$F_{\#}([M])=deg(F)[...
1
vote
0answers
25 views
Preimage by homotopy family of smooth functions is invariant under homeomorphism
Suppose $f_t:M^n\rightarrow \mathbb{R}^n$ is a family of smooth functions depending smoothly on $t\in[0,1]$, where $M$ is a manifold. I would like to know when the preimage of $0$ (assume not empty) ...
3
votes
1answer
34 views
Can a basis of a tangent space be mapped to a basis of another tangent space if the map between the spaces is a homeomorphism and vice versa?
If I have an open subset $U$ of a n-dimensional $C^k-$manifold $M$ and a homeomorphism $f:U \to \Bbb R^n$ (Basically I am talking about a chart $(U,f)$) can I say that under this map a basis of $T_pU=...
3
votes
1answer
22 views
Making a homeomorphism a $\mathscr{C}^k$-map
I'm solving the following exercise and I just need a little push for the one step I'm failing to do:
Let $M$ be a $\mathscr{C}^k$-manifold, $N$ be a topological manifold, and $\alpha: M\to N$ be a ...
4
votes
1answer
101 views
An integral map from 3-torus $\mathbb{T}^3$ to 3-sphere $S^3$
Let $\phi_1, \phi_2, \phi_3, \phi_4 \in \mathbb{R}$ be real valued functions, such that
$$\phi_j(x,y,z):(x,y,z) \in \mathbb{T}^3 \to \phi_j(x,y,z) \in \mathbb{R}.$$
Here $\mathbb{T}^3$ is a 3-torus,...
2
votes
0answers
28 views
Proving that for a $C^k$ function, the measure of the set of singular values is $0$
Let $f:R^m\to R^n$ be a $C^k$ function, where $k>\max \{0,,m-n\}$. Then the lebesgue measure of the set of singular values is $0$.
I've been trying to prove this. I came up with the following ...
1
vote
0answers
27 views
de Rham cohomology of vector bundles: advantages of different definitions of compact vertical cohomology
I'm currently reviewing the Thom isomorphism and the de Rham cohomology of vector bundles over a compact manifold $M$.
I'm familiar with the case of topological disk bundles, where the Thom class ...
0
votes
1answer
42 views
Lower and Upper Bound for Ricci Curvature
As mentioned in first chapter of John M. Lee: Riemannian Geometry, one of our goal in differential geometry is connecting geometry and topology. For this reason it is natural to compare curvature ...
1
vote
0answers
47 views
Monodromy of the family of hypersurfaces on moduli space
Let $\bar{\mathfrak X}\to \mathbb P^N$ be the universal family of hypersurfaces in $\mathbb P^{n+1}$ of degree $d$ and $\mathfrak X \to U$ ($U\subset \mathbb P^N$) be the sub-family of smooth ...
1
vote
1answer
27 views
Is a map with invertible differential a diffeomorphism onto its image near a boundary point?
Let $M,N$ be smooth manifolds of the same dimension, and suppose $M$ has a non-empty boundary. Let $f:M \to N$ be a smooth map, and suppose that $df_p$ is invertible for some $p \in \partial M$.
...
0
votes
0answers
18 views
Dimension of isotropy group in homogenous space
Just sincerely ask a fundamental problem as the following:
We know
$G = GL(2,\mathbb{R})$ is a transitive group of dimension $4$.
The vector space $GL(2,\mathbb{R})$ acting on is $X=\mathbb{R}^2$...
4
votes
2answers
65 views
$p\mapsto c_p$ is continuous where $p\in S^1$ and $c\in \mathbb{R}$.
Let $f: S^1\to S^1$ be a diffeomorphism. Consider the derivative map $$ Df_p:T_p S^1\to T_{f(p)}S^1,\ (p,ip)\mapsto (f(p),ic_pf(p)). $$ we say $f$ is orientation preserving if given $p\in S^1,\ c_p>...
0
votes
0answers
62 views
$\min_{w}(w'(A+x_{t+1}x_{t+1}')w-2(b-yx_{t+1})'w+\sum_{s=1}^ty_s^2-y^2)-\min_{w}(w'Aw-2b'w+\sum_{s=1}^ty_s^2)$
Consider $A$ to be a symmetric positive definite matrix, vectors $w,x,b\in\mathbb{R}^n$ and scalar $y\in\mathbb{R}$. I am trying to solve $Q_1(w)-Q_2(w)$ where:
$$Q_1(w)=\min_{w}(w'(A+x_{t+1}x_{t+1}')...
1
vote
0answers
37 views
spin mapping class group of circles
The MCG of the circle is $\mathbb{Z}/2$, generated by an orientation-reversing diffeomorphism.
The circle has two spin structures, a periodic one and an anti-periodic one. Each has a nontrivial ...
4
votes
0answers
41 views
Every immersed submanifold can be deformed to have transverse self-intersection
Let $f : M^n \to \overline{M}^{n+k}$ be an immersion between smooth manifolds. Is it true that there exists a smooth map $F : M \times [0,1] \to \overline{M}$ such that the following conditions hold?
...
3
votes
1answer
59 views
$M$ is orientable iff $\bigwedge^n TM \setminus \{ \text{$0$-section} \}$ has $2$ connected components
Let $M$ be an manifold of dimension $n$. We say that $M$ is orientable if it has an $n$-form that doesn't varnishes in any point if $M$. My question is about the following claim:
$M$ is orientable ...
1
vote
0answers
79 views
Value changes after using a permutation in a wedge-product
We defined the wedge-product as follows: $ w \wedge \mu = \frac{(k + l)!}{k!l!} Alt (w \otimes \mu) $ (here you'll find additional information).
In the context of a proof we came to a point where we ...
