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

6
votes
3answers
252 views

Prove where exp: Skew($3\times 3$) $\rightarrow SO(3)$ is local homeomorphism

The matrix exponential on skew-symmetric $3\times3$ matrices onto $SO(3)$ is not local homeomorphism everywhere. I have been instructed that one problem is with the spheres of radius $2n\pi$ ...
0
votes
1answer
29 views

Does 2 manifolds can be “isotoped away”?

Let $M,N\subset P$ be two manifolds such that dim($M$) + dim($N$) < dim($P$), suppose that $M$ is compact and $N$ is closed, is it true that there exists an isotopy $F$ of $M$ such that $F(M,1)\cap ...
2
votes
1answer
54 views

Topological property of some manifold

I am provided with a smooth map $g : N \rightarrow N^\prime $ between differentiable manifolds. $N$ is assumed to be compact and connected. Moreover, the differential $dg_x: T_x N \rightarrow ...
0
votes
1answer
53 views

Show $\lambda$ is smooth

Let D be the closed unit disc in $\mathbb R^2$ and $S^1= \partial D$. Let $f$, $g$: $S^1 \rightarrow \mathbb R^3$ be smooth embeddings s.t. $f(S^1) \cap g(S^1) = \emptyset$. Define $\lambda: S^1 ...
1
vote
1answer
287 views

Confused on Guillemin and Pollack's proof of the $\epsilon$-Neighborhood theorem.

On pg. 71 of Guillemin and Pollack they prove the $\epsilon$-Neighborhood theorem. Here $Y$ is a compact boundaryless manifold in $\mathbb{R}^M$. They say Proof: Let $h:N(Y)\to\mathbb{R}^M$ be ...
1
vote
0answers
60 views

Restriction of a line bundle to a a two-cycle

I am reading a paper on Chiral Differential Operators http://arxiv.org/pdf/hep-th/0604179v3.pdf and it says on page 23 that a line bundle over a manifold $C$ can be characterized completely by its ...
3
votes
2answers
278 views

Hairy ball theorem : a counter example ?

Recall the hairy ball theorem : any continuous vector field on a sphere $S^n$ of even dimension must vanish at least once. Now consider the Riemann sphere $\mathbb{C} \cup \infty \simeq S^2$. Let ...
5
votes
1answer
65 views

When can a connection be lifted?

Let $P \rightarrow X$ be a principal $G$-bundle, and $P' \rightarrow X'$ be a principal $G'$-bundle. Let $(f',f'')$ be a morphism from $P'$ to $P$, i.e., a pair of maps $f': P' \rightarrow P$ and ...
0
votes
1answer
44 views

Assumptions required for an implicitely defined surface/manifold to have a specified dimension

What are some normal assumptions made on implicitly defined manifolds? More specifically, by implicitly defined manifold, I mean the definition of a surface such as $g(x,y)=x^2+y^2-1=0$ for the ...
3
votes
0answers
88 views

Does a “typical” reproducing kernel on a manifold generate an infinite-dimensional RKHS?

Let $M$ be a finite dimensional manifold and $k:M\times M\rightarrow\mathbb{R}$ a smooth reproducing kernel. Is the set of all such $k$ with the property that the reproducing kernel Hilbert space ...
1
vote
1answer
243 views

Question about a lemma from Milnor's Topology from the Differentiable Viewpoint

I've been reading John W. Milnor's Topology from the Differentiable Viewpoint for some time and currently I'm stuck at a little lemma. I would appreciate if someone can clarify it to me. The details ...
8
votes
1answer
739 views

Preparing for “differential forms in algebraic topology”?

I'd very much like to read "differential forms in algebraic topology". Apart from background in calculus and linear algbra I've thoroughly went through the first 5 chapters of Munkres. I'm thinking ...
0
votes
1answer
150 views

Submersion implies every point is in the image of a local section

I want to show the following: Let $\pi:M\to N$ be a submersion. Then, every point of $M$ is in the image of a smooth local section of $\pi$. Since $\pi$ is a submersion, it is also an immersion ...
2
votes
1answer
129 views

Transversality of Vector Fields Defined in terms of Diff. Forms and Open Books.

All: Sorry for the length of the post, but I think it is necessary to set things up so that the post is understandable. I'm trying to understand how it is that the transversality (in this case , the ...
4
votes
1answer
117 views

Concrete non trivial computation of Morse homology

I am studying Morse homology and have found only examples on spheres and tori so far. Of course the homology of these manifolds is better understood by other more standard methods, so I am having ...
0
votes
0answers
53 views

Some questions on definition of Grassman manifolds

In W.M. Boothby, An introduction to differentiable manifolds and Riemannian geometry , Page 64. We use coordinate correspondences $\phi_j : U_j \to R^{k(n-k)}$ in order to define Grassman manifolds ...
1
vote
0answers
126 views

Splittings of 4 manifolds

I am wondering why a copy of $CP^2$ may be split off from $CP^2\mathbin\#9\overline{CP^2}$ to leave $\overline M_{E_8}\mathbin\#\overline{CP^2}$(«The wild world of 4-manifolds» by Alexandru Scorpan, ...
3
votes
0answers
105 views

Homological Interpretation of the intersection number

Let $M^r$ and $N^s$ be smooth submanifolds of the smooth manifold $V^{r+s}$ and $M,N,V$ compact, connected and orientable. The Thom-Isomorphism together with the Tubular Neighbourhood Theorem gives me ...
1
vote
0answers
73 views

$\Bbb{R}P^1$ bundle isomorphic to the Mobius bundle

This was asked several times on math.se but none of them was answered. I'm trying to construct an explicit isomorphism from $E = \{([x], v) : [x] ∈ \Bbb{R}P^1, v ∈ [x]\}$ to $T = [0, 1] × R/ ∼$ where ...
0
votes
0answers
234 views

Uniqueness of smooth manifold obtained from glueing two manifolds along their boundary

I have a question about a result in Lee's Introduction to Smooth Manifold(2nd Edition). Theorem 9.29 (Attaching Smooth Manifolds Along Their Boundaries). Let $M$ and $N$ be smooth $n$-manifolds with ...
6
votes
1answer
181 views

Question about image of proper smooth map of constant rank. (Undergraduate)

I have to find a proof of the following theorem Given a smooth function f between smooth manifolds X and Y that: has constant rank is proper the preimage of every point in f(X) is connected and ...
0
votes
2answers
113 views

differmorphism and homeomorphism for manifolds

For two abstract manifolds that are differmorphic, why are they always homeomorphic? Why does differentiability imply continuity for abstract manifolds? (for $R^n$ this is certainly clear)
0
votes
1answer
365 views

Reference request : Study of Differential topology post Milnor's book

I am just about to finish my study of Milnor's book 'Topology from the Differentiable Viewpoint' and I really love the subject. I would like to continue my study of Differential Topology and am ...
2
votes
0answers
171 views

Definition of linking number for disjoint submanifolds of the sphere- A problem from Milnor's book

In problem number 13 of Milnor's 'Topology from the Differentiable Viewpoint', the linking number for two compact boundary-less manifolds $M,N \subset \mathbb{R}^{k+1}$ of dimensions $m,n$ such that ...
1
vote
1answer
128 views

Real line bundle is smoothly isomorphic to Möbius bundle

I'm stuck on this question and tried to follow the partial answer of Neal. Erno's answer is fine too but it seems like I need to find the local trivializations of the Mobius bundle, which requires a ...
3
votes
2answers
156 views

zero of vector field with index 0

I'm currently studying vector fields on surfaces in the $\mathbb R^3$ and I currently I am doing some reading on the index of zeros of vector fields, which got me wondering: Is it possible to find a ...
1
vote
1answer
140 views

Is the classical derivative the weak derivative on any domain?

I was looking at the following definition of the weak derivative: Let $\Omega$ be a domain (ie an open connected subset of $\mathbb{R}^n$). Suppose $u,v \in L_{1,loc}(\Omega)$ and \begin{equation} ...
0
votes
1answer
129 views

open sets in a regular surface in $R^3$

I am reading Differential geometry of curves and surfaces by Do Carmo. Let S be a regular surface in $R^3$. I wonder How is open set of S defined? Is a subset of S open if and only if it is the ...
0
votes
1answer
102 views

Morse Sard Theorem

I'm reading the Morse-Sard's theorem in the book Differential Topology by Hirsch, and I wonder if anyone has a paper on the theorem with more details? I'm not proving it for manifolds, just open sets ...
3
votes
1answer
302 views

Is it possible to learn differential topology before analysis?

Currently I'm self studying for my own enjoyment topology and algebra (munkres and herstein). Since I start at the university next year everything I'm learning now is for my own enjoyment and I will ...
0
votes
1answer
47 views

zeroes of vector fields on surfaces

I know that for compact (smooth) surfaces in $\mathbb R^3$ the 2-torus is the only surface that has vector fields with no zeroes. What happens if we take the compactness of the surface away? Does this ...
1
vote
0answers
106 views

Confusing remark in Guillemin and Pollack

On page 23 of Guillemin and Pollack they say the following. Suppose $g_1,\dots,g_l$ are smooth, real-valued functions on a manifold $X$ of dimension $k\geq l$. Under what conditions is the set ...
1
vote
0answers
37 views

Cylinder as Fibre bundles

I have to show that the cylinder C is a fibre bundle over $S^1$ with fibre an open interval and I have to write a trivialization and the cocycles. I think that this is a trivial bundle, because I can ...
1
vote
0answers
31 views

Multiplication in homotopy groups and cobordism

Each homotopy class of a map of an n-sphere into the Thom space of the universal vector bundle determines a cobordism class of embedded smooth manifolds in Euclidean space. How do the cobordism ...
3
votes
2answers
216 views

How does one prove that the Klein bottle cannot be embedded in $R^3$?

How does one prove that the Klein bottle cannot be embedded in $R^3$? I'm talking about smooth embeddings.
5
votes
3answers
279 views

Application of Hodge decomposition

Hodge decomposition states any $p$ form can be decomposed into three orthogonal $L^2$ components: exact form, co-exact form and hamonic form. But actually we don't know how to decompose a general one. ...
1
vote
1answer
109 views

Locally Euclidean can be defined for whole of $\mathbb{R}^n$, or an open set or open ball of $\mathbb{R}^n$?

Topological manifolds are defined to be locally Euclidean (e.g. John Lee). That is, any point is in an open set that is homeomorphic to either $\mathbb{R}^n$, an open ball in $\mathbb{R}^n$ or an open ...
5
votes
5answers
437 views

Trivial tangent bundle of sphere with handles

I am wondering if there is a simple proof of this statement: A sphere with $g$ handles has trivial tangent bundle iff $g=1$ I know that it is a corollary of Poincaré-Hopf theorem, but it seems ...
2
votes
1answer
42 views

isometric imbedding of the projective plane

How to isometrically imbed the projective plane (identifying antipodal points of the unit sphere) in $\mathbb{R^5}$? My textbook indicates that there is an isometric imbedding of the projective ...
1
vote
0answers
76 views

Reference: forms invariant under Lie group action give the de Rham cohomology?

I'm looking for a reference for a proof of the following fact: Let $G$ be a compact, connected Lie group acting on a smooth manifold $M$. Then inclusion of the differential forms invariant under the ...
0
votes
1answer
130 views

Meaning of differentiability

Could anyone give an intuitive idea of the meaning of differentiability in general in any dimension and any space?
2
votes
1answer
115 views

Product of manifolds with boundary

If $M$ and $N$ are manifolds with boundaries and $\{(U_a,f_a)\}$ and $\{(V_a,g_a)\}$ are their respectives $C^r$ atlas, why $\{(U_a \times V_b,f_a \times g_b)\}$ isn't an $C^r$ atlas for $M \times ...
0
votes
2answers
70 views

Isometry of surfaces in $\mathbb{R}^3$

Let $F$ be an isometry of the Euclidean space $\mathbb{R}^3$. Hence $F$ is orthogonoal transform followed by translation by a constant vector. Let M be a surface of $\mathbb{R}^3$ that is connected, ...
2
votes
1answer
165 views

About Stokes' theorem

I noticed that in the proof of Stokes' theorem on manifolds, the condition that the form $\omega$ is compactly supported ensures that the sum is finite so that we can change the order of sum and ...
4
votes
1answer
199 views

differentiable maps between topological spaces without using the idea of manifolds

Is it possible to define differentiable maps between topological spaces without using the idea of manifolds?
1
vote
1answer
75 views

Moves for regular homotopies of immersions of $S^1$ in the plane

What is a set of moves to combinatorially describe regular homotopies of (smooth) immersions $S^1\to \mathbb R^2$?
2
votes
1answer
130 views

Orthonormal frame bundle orthogonal to a curve

Let $M$ be a $n$-dimensional smooth riemannian manifold and $\varphi\colon(-\varepsilon,\varepsilon)\rightarrow M$ an embedding. $\varphi$ will denote the image of $\varphi$, too. Consider the bundle ...
2
votes
3answers
340 views

How to prove there exists a solution? Guillemin Pollack

Prove there exists a complex number $z$ such that $$ z^7+\cos(|z|^2)(1+93z^4)=0. $$ (For heaven's sake don't try to compute it!)
6
votes
3answers
1k views

Top homology of an oriented, compact, connected smooth manifold with boundary

Let $M$ be an oriented, compact, connected $n$-dimensional smooth manifold with boundary. Then, is it true that $n$-th singular homology of M, that is $H_n(M)$, is vanish? I can't make counterexamples ...
1
vote
1answer
290 views

Is a curl-free vector field always a gradient?

I tried to prove this problem using the Helmholtz decomposition theorem, but it seems the two are entirely contradictory--thus leaving me with empty hands. Does anyone know how to proceed? Thanks ...