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.

learn more… | top users | synonyms

1
vote
1answer
287 views

Tubular neighbourhood theorem

in your opinion is it possible to get the existence of a tubular neighborhood for a manifold M even if it not embeds smoothly (but only topologically) in some R^N? Thank you!
4
votes
1answer
54 views

The Hopf invarient with coefficients other than Z.

So generally one defines the Hopf invariant of a map $f: S^{2n-1} \to S^n$ as the coefficient $H(f)$ in $\alpha^2 = H(f) \beta$ where $\langle \alpha \rangle = H^n(C_f)$ and $\langle \beta \rangle = ...
1
vote
1answer
240 views

Surgery on trivial knots

I know a theorem that any closed orientable 3 manifold can be obtained from the sphere $S^3$ by surgery along a framed knot. I think I read or heard somewhere that as a surgery link, we can take ...
1
vote
1answer
248 views

Finding Reeb Vector Field Associated with a Contact Form

I would greatly appreciate it if you could help me with the following: I'm curious as to how to find the Reeb field $R_w$ associated to a specific contact form $w$; does one actually find $R_w$ as ...
6
votes
1answer
180 views

First Pontryagin class on real Grassmannian manifold?

I wonder if real Grassmannian manifold $SO(p+q)/SO(p) \times SO(q)$ have nontrivial first Pontryagin class? I only have physics background and know really little about characteristic class theory.
9
votes
2answers
604 views

Distinguishing the Cylinder from a “full-twist” Möbius strip

Playing around with the definition of a fiber bundle, I found that while a Möbius strip (with its usual "half-twist") is a nontrivial fiber bundle, it seems that a Möbius strip with a "full-twist" is ...
2
votes
1answer
96 views

Stokes' Theorem for the upper half space.

How can I prove Stokes' Theorem $\int_M dω = \int_{∂M} ω$ where $M = \mathbb{H}^ n$, the upper half space.
3
votes
1answer
224 views

Why is the priemage of a regular value of $F(x,v)=df_x(v)$ finite?

This comes up as part of a larger issue of showing a compact $n$-dimensional manifold can be immersed in $\mathbb{R}^{2n-1}$ except at finitely many points. Suppose $X$ is a compact, $n$-dimensional ...
0
votes
1answer
41 views

Prove a$T_0$ topological group is $T_1$

How to prove that a $T_0$ topological group is $T_1$. I am a beginner in topological group. Also I want some good reference.
2
votes
1answer
119 views

Compact manifolds can almost be immersed?

The Whitney Immersion theorem states that any $n$-dimensional manifold can be immersed in $\mathbb{R}^{2n}$. However, I seem to remember that if $X$ is a compact $n$-dim manifold, then $X$ can be ...
0
votes
3answers
104 views

What dimensions are possible for contours of smooth non-constant $\mathbb R^n\to\mathbb R$ functions?

While for $n=2$ it is pretty clear that the contours of a non-constant $f:\mathbb R^n\to\mathbb R$ are either extrema (and therefore points) or (the union of) 1-dimensional isolines, for $n=3$ I am ...
0
votes
1answer
48 views

Does the structure group of S^n homotopic to O(n+1)?

It is easy to show that Diff(S^1) is homotopic to O(2), but in the case of n bigger than 1 things become really complicated, I cannot see the conclusion directly.
1
vote
0answers
108 views

Why isn't the graph of $y=|x|$ a smooth manifold? [duplicate]

Consider the graph of $y=|x|$ from $-1<x<1$. Equip it with a single chart, the projection onto the $x$-axis. Is it now a smooth manifold? It seems like it shouldn't be smooth, but perhaps with ...
4
votes
4answers
565 views

What is the codimension of matrices of rank $r$ as a manifold?

I'm reading through G&P's Differential Topology book, but I hit a wall at the end of section 4. There is a result stating The set $X=\{A\in M_{m\times n}(\mathbb{R}):\mathrm{rk}(A)=r\}$ is a ...
0
votes
1answer
46 views

Rotating about the z-axis defines a smooth function on $S^{2}$

I want to show that rotating about the z-axis defines a smooth function on $S^{2}$. To do this I used the function: $f(x,y,z)=(x\cos(\theta)-y\sin(\theta),x\sin(\theta)+y\cos(\theta),z)$ where ...
1
vote
0answers
98 views

Moduli space of stable principal $SL(2, {\mathbb C})$-bundles on a compact Riemann surface.

Let $C$ be a compact Riemann surface with genus $2$. How can I describe the moduli space of stable principal $SL(2, {\mathbb C})$-bundles on $C$?
1
vote
0answers
91 views

What's wrong with my argument to compute the vector bundles on $S^1$?

I have read the solution in Vector bundles of rank $k$ with base $S_{1}$, and I am sure that answer is correct. But I have another approach and I don't know where did I make a mistake. So as circle is ...
2
votes
0answers
130 views

Maximal submanifold

I wonder if the notion of "maximal submanifold" exists or is relevant? I'm surprised because I found pretty much nothing about it on the web (after a quick search). The definition, which seems ...
1
vote
2answers
198 views

Is there any standard N-sphere that has non-trivial first Pontryagin class?

I am wondering if there is a standard $N$-sphere that has non-trivial first Pontryagin class on its tangent bundle $TS^n$and frame bundle $FS^n$? I know that only $S^4$ has non-trivial $H^4(S^n, R)$ ...
4
votes
1answer
131 views

The degree of every smooth map $\mathbb{R}^n \to \mathbb{R}^n$ is one…

Let $\varphi : M^n \to N^n$ be a proper smooth map between two connected smooth manifolds. Then $\varphi$ induces a linear map $\varphi^* : H_c^n(N) \simeq \mathbb{R} \to H_c^n(M) \simeq \mathbb{R}$ ...
2
votes
2answers
133 views

Extending diffeomorphism to disk

I am trying to prove the following If $f:S^1 \to S^1$ is a diffeomorphism it can be extended to a diffeomorphism $F: D^2 \to D^2$. But I can't seem to prove it. I proved it for homeomorphisms using ...
0
votes
1answer
46 views

does composition of maps is smooth and one map is smooth imply the other is also smooth?

If $f\circ g$ is smooth and $f$ is smooth, does it follow that $g$ is smooth? Note that I cannot simply take the inverse of $f$. Do I have to use implicit function theorem?
1
vote
1answer
358 views

Diffeomorphism and determinant of Jacobian

I don't remember where I read it and if I remember it correctly but does the following hold true? If $M,N$ are two (smooth?) surfaces and $f: M \to N$ is a homeomorphism such that $det(J_f)$ (the ...
3
votes
0answers
144 views

Products of homeomorphisms

I was wondering if there is a theorem like "If $f_i:X_i\to Y_i$ are homeomorphisms then $\prod_i f_i : \prod_i X_i \to \prod_i Y_i$ is a homeomorphism" for $I$ finite. What about $I = \mathbb N$? ...
3
votes
1answer
80 views

Locally Euclidean but not topological manifold

I'm having trouble solving one part of one of the initial exercises of the classic Boothby book "An Introduction to Differentiable Manifolds and Riemannian Geometry" (exercise I.3.1). To be more ...
1
vote
1answer
326 views

The derivative of the inclusion map is the inclusion map of tangent spaces.

Let $X$ and $Y$ be smooth manifolds, let $i:X\to Y$ be the inclusion map, prove $di_x$ is the inclusion map from $T_x(X)$ to $T_x(Y)$. I know this is pretty basic, but can someone show me how to do ...
0
votes
1answer
58 views

Why is a vector space equal to its tangent space for any point?

I'm self-studying Guillemin and Pollack, but I'm stuck on Problem 3 of section 2. It says that if $V$ is a vector subspace of $\mathbb{R}^N$, then $T_x(V)=V$ if $x\in V$. If $x\in V$, then since $V$ ...
5
votes
0answers
65 views

The space of paths

Let $ (M,g) $ be a compact Riemannian manifold, and let the path space $ \Omega $ of $ M $ be the set of equivalence class of smooth maps $ \gamma : [0,1] \rightarrow M $ (equivalent under ...
4
votes
1answer
169 views

Defining a quotient manifold with gluing

I'm trying to find conditions on the gluing map between two manifolds so that the quotient space will be a smooth manifold, and the inclusion map will be a diffeomorphism. Specifically, Suppose ...
1
vote
0answers
20 views

Functions from $[E_8]$ to $E_8$

Let $f: [E_8] \to [E_8]$ be a function between 4-manifolds with intersection form $E_8$. What we know (due to Rocklin) is that $[E_8]$ can't have any smooth structure. Questions: Is it true for all ...
1
vote
0answers
31 views

Transforming the measure in $CP^1$ mapping from Riemann sphere to $\mathbb{C}^2$-plane

I would like to know how the measure changes in $CP^1$ mapping from Riemann sphere (2-sphere) to $\mathbb{C}^2$-plane. Let a point on the 2-sphere is given by the vector ...
1
vote
0answers
91 views

Is this function just zero?

I asked a question like this a few minutes, but am about ready to strike the problem out as errata for the book. The problem defines a function $g(x)=f(x-a)g(b-x)$. The function has bound $|f|<1$, ...
4
votes
1answer
117 views

Does a germ of a smooth (i.e., $C^\infty$) function at a point of a manifold always extend to a global smooth function?

Obviously this doesn't hold if we replace "smooth" with something like "analytic" or "regular," which are the contexts I'm more familiar with. And obviously we can't extend a smooth function defined ...
3
votes
0answers
88 views

Is $X$ diffeomorphic to its diagonal?

I missed geometry in school, I'm trying to fill in the gaps by reading Pollack's Differential Topology. Am I doing this right? This is #16 in the first section. Show the diagonal $\Delta$ of $X\times ...
1
vote
1answer
90 views

Lifting problems existence

Let $g:\mathbb{R}^m \longrightarrow \mathbb{R}^m,g\in C^1(\mathbb{R}^m)$ such that: $\|g'(x)(v)\|\geq\|v\|,\forall v\in \mathbb{R}^m,\forall x \in \mathbb{R}^m$ show that any rectilinear path ...
2
votes
0answers
59 views

Isotopy between two open disks on a surface

So I have a (compact) surface $\Sigma$ and two open disks on the surface call them $A$ and $A'$ such that the intersection contains a simple curve $P$. What I want to do is construct an isotopy ...
1
vote
0answers
124 views

derivative along a curve with respect to a given vector field

This is question 7 of Do carmo's differenital geometry of curves of surfaces 3.4. Define the derivative $w(f)$ of a differentiable function $f:U \subset S\rightarrow R$ relative to a vector field $w$ ...
2
votes
1answer
68 views

Diffeomorphism on $\mathbb C$

Let $P= a_{0} z^{n} +a_{1} z^{n-1}+ \cdots +a_{n}$, with $ a_{0} \neq 0,z \in \mathbb C$. I don't know why $P$ fails to be a local diffeomorphism only at the zeros of the derivative polynomial ...
3
votes
1answer
152 views

What is nonhomogeneous linear mapping?

In Milnor's Topology from the differentiable viewpoint, page 3, he said: One thinks of the nonhomogeneous linear mapping from the tangent hyperplane at $x$ to the tangent hyperplane at $y$ which ...
2
votes
1answer
113 views

Are there Kirby diagrams for manifolds with boundaries?

There are Kirby diagrams for 3- and 4-manifolds which consist of framed links corresponding to 1- and 2-handles attached to a single 0-handle. Any such diagram will give a unique closed manifold since ...
3
votes
0answers
112 views

Does Clifford algebra depend on the topology of manifold?

We know the greatest feature of Clifford algebra is coordinate-free. One can do vector operations without knowing the representation of vectors. And due to its very characteristc, Clifford or ...
3
votes
0answers
132 views

(Topological quantum field theory) identifying objects of cobordism category

I am beginning the study of Topological quantum field theory(TQFT) and I am confused with the basic notions. Before writing down the question, to check if I understood the definition correctly, I ...
8
votes
1answer
162 views

Can the diagonal of a manifold be expressed as the zero set of a section of a vector bundle?

Let $\mathbb{C} \mathbb{P}^2$ be the two dimensional complex projective space and $$M:= \mathbb{C} \mathbb{P}^2 \times \mathbb{C} \mathbb{P}^2.$$ Let $ \gamma \rightarrow \mathbb{C} \mathbb{P}^2 $ be ...
3
votes
2answers
178 views

Generalization of the hairy ball theorem.

The hairy ball theorem of states that there is no nonvanishing continuous tangent vector field on even dimensional n-spheres. Can the hairy ball theorem be strengthened to say that there is no ...
12
votes
2answers
210 views

If a smooth manifold X is covered by an odd sphere, then X is orientable.

In solving some old qualifying exam questions, I've been thoroughly stumped. If a smooth manifold $X$ is covered by an odd dimensional sphere, then $X$ is orientable. I see this question has ...
8
votes
1answer
552 views

Is a connected sum of manifolds uniquely defined?

It is a standard excercise in differential geometry to prove that a connected sum $M\#N$ of two smooth manifolds $M,N$ of the same dimension is uniquely defined (under some assumptions regarding ...
2
votes
0answers
74 views

Smoothness does not depend on the choice of atlases

Here is a part of a lecture note: I need some help to solve the exercise. I want to show that if $\psi\circ f\circ\phi^{-1}$ is differentiable and $\alpha, \psi$ and $\beta,\phi$ are in the same ...
1
vote
1answer
520 views

Understanding the definition and meaning of cotangent space

I would like to understand definition and also meaning of cotangent space. Let's first of all define tangent space: generally we know that equation of tangent line of function $f(x)$ at point $x_0$ ...
2
votes
0answers
57 views

Signature form of $S^2 \times S^2$

Let $M=S^2 \times S^2$ be the product of two copies of the $2$-sphere. We have that $dim(M)=4$. So we can define the intersection form $$ I_{S^2 \times S^2} := H^2(M, \mathbb{Z}) \times H^2(M, ...
3
votes
0answers
80 views

Does the boundary of a handle decomposition obtain a handle decomposition?

Let $M$ be the $4$-manifold $D^4\cup2\text{-handle}\cup\ldots\cup2\text{-handle}$, where the attachment of the handles is specfied by an oriented framed link $L=L_1\cup\ldots\cup L_n\subseteq S^3$. By ...