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

4
votes
1answer
350 views

“Coordinate functions” on the structure-sheaf definition of a smooth manifold

I've been reading Bredon's Topology and Geometry recently; what an excellent book! He defines smooth manifolds in two distinct ways and then shows they are in fact equivalent. The "non-standard" ...
1
vote
0answers
57 views

The winding number of $f$ about $0$

Let $f: S^1 \to \mathbb{R}^2 - \{0\}$ be a smooth map. Define the winding number of $f$ about 0 and prove that it equals $\frac{1}{2\pi}\int_{S^1}f^*(d\theta).$ According to the definition, we ...
1
vote
2answers
153 views

If $M$ is diffeomorphic to $N$, then $\mathbf{H}_{DR}^p(M)$ is isomorphic to $\mathbf{H}_{DR}^p(N)$.

I thought I got this, but no..... Given $\mathbf{H}_{DR}^2(S^3)$ is trivial but $\mathbf{H}_{DR}^2(T^3)$ is not, how can I show $S^3$ and $T^3$ are not diffeomorphic? I am also wondering about the ...
1
vote
0answers
122 views

form exact $\Leftrightarrow$ pull-back exact

Is the form exact $\Leftrightarrow$ pull-back exact? Since $$f^*\omega = \omega \circ df,$$ which seems irrelavant. Because the composition with $df$ does not change $\omega$ is exact or not. The ...
2
votes
1answer
280 views

How do I prove that a subset of a manifold is not a submanifold?

I know of ways to prove that a given subset of a smooth manifold is a smooth submanifold, but what if I have some subset which I suspect is not a smooth submanifold? What are some approaches to ...
0
votes
0answers
23 views

The orientation preserving of folliation.

A foliation $\mathcal{F}$ on a compact, oriented manifold $X$ is a decomposition into $1-1$ immersed oriented manifolds $Y_\alpha$ (not necessarily compact) that is locally given (preserving all ...
3
votes
1answer
116 views

Let $M$ be a smooth simply-connected compact manifold of dimension $n$. Is there an immersion of $M$ into the $n$-torus $T^n = S^1 × . . . × S^1?$

I do not know how to do the following qualifying exam problem. Any helped is nice. Let $M$ be a smooth simply-connected compact manifold of dimension $n$. Is there an immersion of $M$ into the ...
1
vote
1answer
65 views

$S^3$ and $T^3$ are not diffeomorphic.

Let $f : T^2 \to S^3$ be the smooth map of a 2-torus into $S^3$, therefore $$\int_{T^2}f^*\omega = 0.$$ There is a closed $2$-form $\beta$ on $T^3 = S^1 \times S^1 \times S^1$ and a map $g : ...
1
vote
1answer
132 views

Is volume form equal to the top-dimensional form with coefficient $1$?

Wikipedia says a volume form on a differentiable manifold is a nowhere-vanishing top-dimensionial form. And Guillemin and Pollack says $f: V \to U$ is a diffeomorphim of two open sets in ...
5
votes
1answer
308 views

Functionally structured spaces and manifolds

The question requires some definitions that I have listed below for your convenience. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups. On a ...
2
votes
1answer
261 views

Show that the set $M$ is not an Embedded submanifold

How can I prove that $M=\{(x,y)\in \mathbb{R}^2\ ; y=|x|\}$ is not an embedded smooth submanifold of $\mathbb{R}^2$?
4
votes
1answer
490 views

Example of commuting vector fields generating globally noncommuting flows

Recently, I discovered that a theorem from my differential geometry lecture is false due to too big generality - it stated that for vector fields $X,Y$ we have the equivalence: Incorrect! $[X,Y] = ...
5
votes
1answer
265 views

1- forms on a torus

I think this is a very simple question but I'm not really confident in mathematics (even if I like it very much) Let's fix a cube $[0,1]^3$ in $R^3$ and identify opposite sides, so as to construct a ...
4
votes
3answers
211 views

Is it possible to make the set $M:=\{(x,y)\in\mathbb{R}^2:y=|x|\}$ into a differentiable manifold?

1) Is it possible that be given a suitable smooth atlas to the set $M:=\{(x,y)\in\mathbb{R}^2: y=|x|\}$ so that $M$ to be a differentiable manifold? Why? How? 2) How can I prove that M is not an ...
4
votes
1answer
38 views

Embedding of the cotangent of the n sphere in R^2n

It is true that $T^* S^1$ embeds in $\mathbb{R}^2$ as the cotangent of $S^1$ is trivial and $S^1 \times \mathbb{R}$ is diffeomorphic to the punctured plane. Is it true in general that $T^* S^n$ embeds ...
2
votes
1answer
152 views

Integration formula with wedge product.

On $\mathbb{R}^3$, consider a compactly supported $2$-form $$\omega = f_1 \, dx_2 \wedge dx_3 + f_2 \, dx_3 \wedge dx_1 + f_3 \, dx_1 \wedge dx_2.$$ Choose for $S$ the parametrization $h: ...
0
votes
0answers
36 views

Written any surface as the graph of a function locally.

I see an interesting statement: Locally any surface may be written as the graph of a function, although one must sometimes write one as a function of the others. So, does this infer such case, ...
7
votes
0answers
145 views

1-form with positive integral over a path

Given any smooth path $\gamma$ on a manifold $M$, when can we construct a 1-form $\omega \in \Omega^1(M)$ so that $\int_{\gamma} \omega > 0$? It is easy to see that such $\omega$ exists if the path ...
1
vote
1answer
46 views

Extending a 2-frame field - manifolds with boundary

It is known that every smooth manifold can be homotoped to a cell complex. In particular this is true for manifolds with boundary. My question: Under the homotopy to a cell complex, is the boundary ...
0
votes
1answer
26 views

Examples of non-parametrizable sets?

Encountered the term parametrizable for the first time: The support of $\omega$ is contained inside a single parametrizable open subset $W$ of $X$. So I am just curious, what kind of sets are ...
5
votes
1answer
154 views

Compensation of the anticommutativity of wedge product.

In Guillemin and Pollack, Differential Topology Page 166, The automatic appearance of the compensating factor $\det (df)$ is a mechanical consequence of the anticommuntative behavior of $1$-forms: ...
0
votes
1answer
42 views

$f^*\left(\sum_I a_I \, dx_I\right) = \sum_I (f^*a_I) \, df_I$

By linearity of $f^*$, $$f^*\left(\sum_I a_I \, dx_I\right) = \sum_I f^* (a_I \, dx_I)$$ And if I want $df_I$, I would have to use the formula $f^*dx_i = df_i$. So $f^*$ disappears when introduced ...
4
votes
1answer
29 views

Commutativity of $Y^j$.

$U \subset \mathbb{R}^k$ and $V \subset \mathbb{R}^l$ be open subsets. Let $f: V \to U$ to smooth. Use $x_1, \dots, x_k$ for the standard coordinate functions on $\mathbb{R}^k$ and $y_1, \dots, y_l$ ...
2
votes
0answers
76 views

Vector field from Lamination.

Let $S$ be a smooth closed (i.e. compact without boundary) surface. A geodesic lamination on $S$ is a nonempty closed subset of $S$ which is a disjoint union of geodesics. Suppose $\alpha$ is a ...
2
votes
1answer
52 views

$ (f^*dx_i)(\frac{\partial}{\partial y^j}) = dx_i(f_*(\frac{\partial}{\partial y^j}))$

$U \subset \mathbb{R}^k$ and $V \subset \mathbb{R}^l$ be open subsets. Let $f: V \to U$ to smooth. I am wondering how can I show $$ (f^*dx_i)(\frac{\partial}{\partial y^j}) = ...
10
votes
1answer
388 views

Folliation and non-vanishing vector field.

The canonical foliation on $\mathbb{R}^k$ is its decomposition into parallel sheets $\{t\} \times \mathbb{R}^{k-1}$ (as oriented submanifolds). In general, a foliation $\mathcal{F}$ on a compact, ...
1
vote
3answers
136 views

Notation on the tangent space.

Consider $Y$ an element of the $n$-dimensional tangent space $T_yY$. The the canonical basis is $(\frac{\partial}{\partial y^1}, \cdots, \frac{\partial}{\partial y^n}).$ Then should I write $$Y = ...
1
vote
1answer
373 views

Thinking of giving up..

I got really stuck to the end of Guillemin and Pollack (in particular, here) and plan to give up. Give up Guillemin and Pollack, not math though. It seems John Milnor's classic little book topology ...
2
votes
1answer
83 views

Why can an orientation on $X$ be written as a sum of cycles on this open cover?

I'm reading a proof of the following theorem Let $X$ and $Y$ be compact connected oriented $n$-manifolds, and let $f:X\to Y$ be continuous. Let $y\in Y$, and assume that $f^{-1}(y)$ is finite. ...
5
votes
1answer
190 views

Topologist's Sine Curve not a regular submanifold of $\mathbb{R^2}$?

I am trying to work out the details of following example from page 101 of Tu's An Introduction to Manifolds: Example 9.3. Let $\Gamma$ be the graph of the function $f(x) = \sin(1/x)$ on the ...
2
votes
0answers
53 views

Problem reduced to analyzing solutions of a family of nonlinear systems of equations

This was posted on mathoverflow about two weeks ago and I got no response so I'm asking here in case anyone has any ideas. Original post is here. I was able to reduce a research problem relating to ...
1
vote
2answers
123 views

Winding number of $f$ is equal to $\frac{1}{2\pi}\int_{S^1}f^*(d\theta).$

Let $f: S^1 \to \mathbb{R}^2 - \{0\}$ be a smooth map. Define the winding number of $f$ about 0 and prove that it equals $\frac{1}{2\pi}\int_{S^1}f^*(d\theta).$ I have no clue, except for the ...
2
votes
1answer
299 views

The Generalized Stokes Theorem.

The Generalized Stokes Theorem. If $\omega$ is any smooth $(k-1)$ form on $X$, then $$\int_{\partial X} \omega = \int_X d\omega.$$ Let $C \subset \mathbb{R}^2$ be a (smooth) simple closed ...
2
votes
1answer
285 views

Change of Variable vs. Change of Coordinates.

Are they the same thing? So given an example, I could work out by change of coordinates, but how can I apply Change of Variable to replace this process? Change of Variable in $\mathbb{R^k}$. ...
0
votes
1answer
43 views

Abbreviation of volumn form

Change of Variable in $\mathbb{R}^k$. Assume that $f: V \to U$ is a diffeomorphism of open sets in $\mathbb{R^k}$ and $a$ is an integrable function on $U$. Then $$\int_U a dx_1 \cdots dx_k = ...
5
votes
2answers
165 views

Textbooks with exposition done mostly in proof outlines or exercises?

As the title indicates, I'm trying to find books where the exposition of the main course of thought is done entirely or mostly in outlines of proofs, or as exercises with or without hints. I'm trying ...
2
votes
1answer
199 views

Define pull-back on a manifold by pull-back on the linear space.

It appears to me that pull-back on a manifold If $f: X \to Y$ is a smooth map and $\omega$ is a $p$-form on $Y$, define a $p$-form $f^*\omega$ on $X$ as follows: $$f^*\omega(x) = ...
2
votes
2answers
115 views

Linearity of push forward $F_*$

How can I prove the linearity of $F_*$? What does $F_*$ eat? If $N$ is smooth manifolds and $F: M \to N$ is a smooth map, for each $p \in M$ we define a map $F_*: T_pM \to T_{F(p)}N$, called the ...
1
vote
1answer
129 views

Linearity of $f_*, f^*$.

The definition of $f^*$ is given to me as below. But what is $f_*$? How can I justify $f_*, f^*$ is linear? Definition. If $f: X \to Y$ is a smooth map and $\omega$ is a $p$-form on $Y$, define a ...
1
vote
1answer
57 views

Linearity of everything

May I ask for details about how can I prove "linearity of everything" for the following step? $(f^*dx_i)(Y) = \sum_{j = 1}^lY^j (f^*dx_i)(\frac{\partial}{\partial y^j}) = \sum_{j = ...
2
votes
1answer
73 views

$(f\circ h)^* \omega = h^*f^*\omega$ - Legit now?

Three pullback identities are required to prove on Guillemin and Pollack's Differential Topology on Page 164. I am not sure if I get it since it is the first time I play with them. I hope I got the ...
1
vote
1answer
48 views

$f^*(w_1 + w_2) = f^*w_1 + f^* w_2$

A few pullback identities are required to prove on Guillemin and Pollack's Differential Topology on Page 164. I am not sure if I get it since it is the first time I play with them. $w$ is an ...
8
votes
3answers
476 views

$f^*dx_i = \sum_{j=1}^l \frac{\partial f_i}{\partial y_j} dy_j = df_i$

Guillemin and Pollack's Differential Topology Page 164: $U \subset \mathbb{R}^k$ and $V \subset \mathbb{R}^l$ be open subsets. Let $f: V \to U$ to smooth. Use $x_1, \dots, x_k$ for the standard ...
0
votes
1answer
201 views

About zeros of vector fields in compact surfaces

I'm studying compact surfaces and in particular the relationship between zeros of vector fields defined on them and Euler characteristic of the surface herself. Let be $S$ a compact (smooth) surface ...
2
votes
2answers
213 views

$f^*(w \wedge \theta) = (f^*w) \wedge (f^* \theta)$

I am reading Guillemin and Pollack's Differential Topology. For the proof on Page 164, I was not able to get through the last step. $$f^*(w \wedge \theta) = (f^*w) \wedge (f^* \theta)$$ ...
2
votes
1answer
96 views

Pullback expanded form.

Definition. If $f: X \to Y$ is a smooth map and $\omega$ is a $p$-form on $Y$, define a $p$-form $f^*\omega$ on $X$ as follows: $$f^*\omega(x) = (df_x)^*\omega[f(x)].$$ According to Daniel ...
1
vote
2answers
66 views

$f: X \to Y$ is a smooth map and $\omega$ is a $p$-form on $Y$, what is $\omega[f(x)]$?

I am reading Guillemin and Pollack's Differential Topology Page 163: If $f: X \to Y$ is a smooth map and $\omega$ is a $p$-form on $Y$, define a $p$-form $f^*\omega$ on $X$ as follows: ...
0
votes
1answer
85 views

How to show the smooth map $f : T^2 \to S^3$ is an orientation-preserving diffeomorphism?

Let $f : T^2 \to S^3$ be the smooth map of a 2-torus into $S^3$. Let $\omega$ be a closed 2-form on $S^3$. Show that $$\int_{T^2}f^*\omega = 0.$$ So apparently, if I can use the theorem on ...
5
votes
2answers
460 views

Definition of pullback.

I am reading Guillemin and Pollack's Differential Topology Page 163: If $f: X \to Y$ is a smooth map and $\omega$ is a $p$-form on $Y$, define a $p$-form $f^*\omega$ on $X$ as follows: ...
1
vote
1answer
85 views

Fill in the hole for the proof for $f^*(w \wedge \theta) = (f^*w) \wedge (f^* \theta)$

My proof has a hole there, wonder if anyone can help me fill it in? $$f^*(w \wedge \theta) = (f^*w) \wedge (f^* \theta)$$ By definition, $$f^*w(x) = (df_x)^*w[f(x)].$$ So I understand ...