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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When does $\pi_1(\Sigma)$ inject into $\pi_1(S^3 \setminus \Sigma)$?

Here's a fun fact from knot theory: $\quad$ If $\, \Sigma$ is a minimal-genus Seifert surface for a knot $K$, then $i_*:\pi_1(S^3 \setminus \Sigma) \to \pi_1(S^3 \setminus K)$ is injective, where ...
4
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1answer
29 views

Is there a one to one correspondence between Jones' polynomials and knots?

I know Jones' polynomial is a knot invariant. By using knot invariant like p-coloration one can only say whether two knots are different but not whether they are the same. So it is like injective ...
6
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3answers
476 views

Why is this map called a fold?

Consider the map $\varphi : \mathbb R^2 \to \mathbb R^2$ defined by $(x,y) \mapsto (x,y^2)$. Apparently this map is called a fold as the $(x,y)$-plane is folded over and creased along the axis ...
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0answers
25 views

Zips and Zippers

I'm currently reading Differential Manifolds by Antoni Kosinski, and the concept of a zip--defined as half of a zipper--is mentioned early on, of course with the intent of connecting manifolds. This ...
0
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0answers
19 views

Proving that Lie groups are parallellizable

Let $G$ be a Lie group. There is a diffeomorphism $$G \times T_e G \to TG$$ mapping $(g, [\gamma]) \mapsto [g \cdot \gamma]$. The inverse map then gives rise to the following isomorphism of bundles: ...
2
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2answers
49 views

Counting singularities

It is well known that a smooth vector field on a 2-sphere must vanish twice. What is the general technique for counting singularities of a smooth map between manifolds? For example, how many ...
3
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0answers
99 views

What do Set-Theoretic (General) Topologists study? [on hold]

I was reading in Elementary Topology by O Viro, O Ivanov, V Kharlamov, and N Netsvetaev and it caught my attention the following quotes by the authors: "...As a research field (refering to General ...
4
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0answers
23 views

G-P Exercise 4.8.2, proof verification.

Let $\gamma$ be a smooth closed curve in $\mathbb{R}^2 - \{0\}$ and $\omega$ any closed $1$-form on $\mathbb{R}^2 - \{0\}$. Prove that$$\oint_\gamma \omega = W(\gamma, 0) \int_{S^1} \omega,$$where ...
5
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1answer
63 views

Can a $1d$ space never be curved?

I was wondering about this: Wikipedia article I refer to (here I refer to the first part: metric) This wikipedia article claims that this hyperbolic space model has constant curvature $-1.$ I believe ...
2
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2answers
32 views

Homeomorphism in the definition of a manifold

Many texts will define a manifold as "a second-countable Hausdorff space that is locally homeomorphic to Euclidean space". By definition of homeomorphism, shouldn't this really and officially read as ...
4
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1answer
51 views

Redundance of the Smoothness of the Inversion Map in the Definiton of a Lie Group.

$\DeclareMathOperator{\inv}{inv}$ I am trying to understand the proof of the following from this document: Let $M$ be a smooth manifold which admits a group structure such that the multiplication ...
2
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41 views

There is no smooth submersion from $S^2$ to $S^1$.

Show that there is no smooth submersion from $S^2$ to $S^1$. I know of one algebraic topology proof which I think is not the shortest one. That submersion is an open map should be a useful fact in ...
1
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1answer
15 views

Problem about perfectly convex set and convex set.

Let $W$ be a subspace of a Banach space $X$. Which of the following are true. a. W is closed then it is perfectly convex b. W is perfectly convex then it is closed. Definition of perfectly convex ...
1
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0answers
38 views

A problem possibly using the technique which has been used to prove the Whitney Embedding Theorem.

Problem. (1) Suppose $F:S^1 \to \mathbb{R}^3$ is a smooth immersion (i.e. $dF$ is nowhere zero). Prove that there is a unit vector $\mathbf v$ in $\mathbb{R}^3$ such that $\pi_{\mathbf v} \circ F : ...
8
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1answer
40 views

Is it possible to have a sphere $S^m$ equidistant to sphere $S^n$ in $R^k$?

Is it possible to place a sphere $S^m$ and another sphere $S^n$ in Euclidean $k$-dimensional space $R^k$ in such a way that the distance from any point of the first sphere to any point of the second ...
4
votes
2answers
103 views

Classify sphere bundles over a sphere

Problem (1) Classify all $S^1$ bundles over the base manifold $S^2$. (2) Do the same question for $S^2$ bundles. Moreover, does there exist a universal method to solve this kind of ...
0
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1answer
28 views

Confused with The Transversality Theorem when all manifolds are boundaryless

In Guillemin-Pollack's book Differential Topology, the Transversality theorem states that The transversility Theorem. Suppose that $F:X \times S \to Y$ is a smooth map of manifolds, where only $X$ ...
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0answers
22 views

Isometry of covering space [closed]

Let $M$ be a compact Riemannian manifold. Consider a covering space $N$ of $M$, with the pull-back metric from $\pi : N \to M$. Given a point $x \in M$, and a couple of points $y, z \in \pi^{-1}(x) ...
1
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0answers
14 views

What is an “essential 2-sphere” in a 4-manifold?

I am aware what an essential 2-sphere in a 3-manifold is. But in several Articles by Fintushel and Stern essential 2-spheres in 4-manifolds occur. The articles are: "Immersed Spheres in 4-manifolds ...
15
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0answers
197 views

When is there a submersion from a sphere into a sphere?

(Edit: Now posted to MO.) That is: For which positive integers $n, k \ge 1$ does there exist a submersion $S^{n+k} \to S^k$? The discussion at this math.SE question has narrowed it down to the ...
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2answers
71 views

Why is the Hopf link the only link with knot group $\mathbb{Z} \oplus \mathbb{Z}$?

We can use the Loop Theorem to show that if $\Sigma$ is a minimal-genus Seifert surface for a link $L$, then $\pi_1(\Sigma)$ injects into the knot group $\pi_1(S^3 \setminus L)$. An orientable ...
2
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0answers
63 views

Lie groups and Lie algebras for matrices

Recently, I stumbled over a few things in very basic Lie group / Lie algebra theory concerning matrix groups. Basically, my question is: Is there a way to canonically understand all the Lie groups ...
-1
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0answers
25 views

how to embed a square into $R^2$?

By Whitney embedding theorem you can embed a smooth 1-manifold in $\mathbb{R}^2$. Now if you give the unit square a smooth structure(for example by inducing the unit circle's smooth structure on it), ...
18
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2answers
260 views

Existence of submersions from spheres into spheres

I would like to know if there exist submersions $f\colon \mathbb{S}^4\to \mathbb{S}^2$ and $g\colon\mathbb{S}^6\to \mathbb{S}^2.$ It is simply a question of curiosity. Any suggestion or ideas are ...
2
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1answer
39 views

Restriction of the projection from compact manifold onto hyperplane is a smooth embedding

Problem: Let $M \subset \mathbb{R}^{n+1}$ be a compact submanifold ($\dim M=k$) and $n \geq 2k + 1$. Show that, for the projection $\pi : \mathbb{R}^{n+1} \longrightarrow H^n$ onto a suitable ...
0
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0answers
30 views

Transversality of graphs of functions

Consider the $C^1$ function $f: [0,1] \to \mathbb{R}$. I understand that a curve in the plane that intersects the graph of $f$ non-transversally would be tangent to it at a point of intersection. I ...
0
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0answers
30 views

What is an affline manifold? How do check if a space (?) is affline manifold. Is it related to vector space?

I have no idea about differential geometry or topology because I am engineer. I need the conditons to check the applicability of a theorem to particular case.From my search I could only find a wiki ...
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0answers
38 views

Unknown proof Lie Algebra

I have a calculation where I do not know what it actually shows. I think it tells me that for right invariant vector fields, the commutator is again right invariant. Maybe somebody here could help me ...
6
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1answer
61 views

When does covering preserve rational cohomology?

Let $X$ and $Y$ be compact manifolds. $p:X \rightarrow Y$ is a covering. Generally it is not true that $$ H^* (X , \mathbb{Q} ) = H^* (Y , \mathbb{Q} )$$ For instance, if $Y$ is a sphere with $y$ ...
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20 views

Degenerate subspace

A null vector is a nonzero vector that is orthogonal to itself. If W is a subspace of V,let $W^{\perp}$ = [$v{\in}$ W : $v{\perp}$W]. $W^{\perp}$ is a subspace of V called W perp. A subspace W of ...
4
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4answers
74 views

Can someone illustrate the definition of manifold with a simple example?

In my text the definition of a differential manifold is given as follows: A subset $M$ of $\mathbb{R}^n$ is a $k$ dimensional manifold if $\forall x \in M$ there are open subsets $U$ and $V$ of ...
0
votes
1answer
45 views

Volume element and orientabality

A volume element on an $n$-dimensional semi-Riemannian manifold $M$ is a smooth $n$-form $w$ such that $w(e_1,\cdots, e_n) = \pm1$ for every frame on $M$. How do I prove A semi-Riemannian ...
1
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0answers
32 views

Topological properties of regular and critical points and values

Let $f\colon M\rightarrow N$ be a smooth map between smooth manifolds. Consider the following two statements, the second one under the assumption The set of regular points of $f$ are open in $M$, ...
4
votes
1answer
61 views

Poincaré Lemma, differential forms and I do have troubles

I think I need some hints about a proof I am currently reading in order to understand it. This question is similar to the construction used in Lemma 17.9 in the book "Introduction to smooth manifolds" ...
0
votes
1answer
26 views

What is the meaning of $\Omega^o_n$?

First, write $M^n \sim M^n$ cobordant, if $M^n$ # $M^n = \partial W^{n+1}$. Where # represent connected sum. Then define $ \Omega^o_n = \{ \textrm{closed manifolds} \} / \sim$ From $M^n$ # $S^n$ ...
2
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0answers
25 views

Constructing commutative diagram up to homotopy from a smooth map.

This is from the proof of Theorem 20.7 in "Characteristic classes" by Milnor and Stasheff. Let $f : M^n \rightarrow S^r$ be a smooth map where $M$ is smooth $n$-dimensional manifold and $S^r$ is ...
5
votes
2answers
50 views

Even number of points in the preimage of a regular value of a map $f:M^n \to \mathbb{R}P^n$

Let $M^n$ be a closed manifold (I think we can also assume it is connected, though it isn't explicitly stated). Let $f:M^n \to \mathbb{R}P^n$ be a smooth map, and let be $a \in \mathbb{R}P^n$ be a ...
1
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0answers
38 views

Existence of Immersion of a manifold in Euclidean space?

I am trying to prove the following claim which I saw in some paper: Let $M$ be an $n$-dimensional smooth, oriented, simply connected manifold, which is homeomorphic to a bounded subset of ...
3
votes
1answer
64 views

Is there an infinite dimensional Lie group associated to the Lie algebra of all vector fields on a manifold?

Since the space $\Gamma(TM)$ of all vector fields on a smooth manifold $M$ is a real Lie algebra with respect to the usual commutator bracket, I was curious if in fact it is the Lie algebra of some ...
5
votes
1answer
56 views

Does every even-dimensional sphere admit an almost complex structure?

We know that there is an almost complex structure on $S^6$ which is not integrable. Is it always possible to find almost complex structures on $S^{2n}$? In particular does $S^4$ admit one?
0
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0answers
11 views

If $U$ is an open subset of the manifold $X$, check that $T_x(U)=T_x(X)$ for $x \in U$

Know that I'm already aware that there is a similar question on the forum. However, comments do not allow to clarify my ideas. Let $ϕ:W∈R^k→X$ be a parametrization of $X$ around $x$ so that $ϕ(W)∈X$ ...
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1answer
31 views

Poincaré invariant corresponds to area?

In this paper(equation 34) click me it says that if we have a symplectic egg, then the area of a $(x_i,p_i)$ clone intersected with the egg is given by the poincaré invariant $\int p dx = \int_0^{2 ...
2
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0answers
33 views

Proving Sard's theorem

Theorem: (Sard) Let $f:U\to \mathbb R^p$ be a smooth map with $U$ open in $\mathbb R^n$, and let $C$ be the set of all critical points of $f$. Then $m([f(C)]=0$ where $m$ is the Lebesgue measure of ...
3
votes
1answer
103 views

Perturb a given smooth function to a Morse function relative to fixed level sets, which are already fine.

Let $M$ be a manifold (not necessarily compact) , for the sake of clearness embedded in $\mathbb{R^n}$ and $f\colon M\rightarrow \mathbb{R}$ a smooth function. The theorem of Sard gives us that ...
2
votes
1answer
37 views

Is level set near a maximum value a circle?

Let $f$ be a $C^2$ function defined on $[0,1] \times [0,1]$. Let $0 \leq f(x) < 1$ on $[0,1] \times [0,1] \setminus \left(\frac{1}{2},\frac{1}{2}\right)$ and $f(\frac{1}{2},\frac{1}{2})=1$. It has ...
0
votes
2answers
60 views

What is the inverse limit?

In his book of Differential Topology, Hirsch starts a little detour in his theory in order to present a way to see things in a general perspective. The precise fragment I'm referring to is the ...
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1answer
36 views

About the Chern class of the determinant line bundle

Is it true that the first Chern class of a rank $k$ complex line bundle is equal to the first Chern class of its determinant line bundle?
3
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0answers
33 views

Error in Hirsch?

Consider the following lemma: In the fragment: Write $\displaystyle X= \bigcup_{1}^{\infty} X_j$ where each $X_j$ is a compact subset of a ball $B$ as above. why is he allowed to do that? If ...
1
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1answer
26 views

Obstruction to the existence of constant-rank sections of $T^*M\odot T^*M$

If $\alpha$ is a section of $T^*M\odot T^*M$, where $M$ is a smooth manifold, the rank of $\alpha$ at $m\in M$ is the codimension of the kernel of $\alpha_m$, i.e. the subspace of vectors $v_m\in ...
0
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0answers
23 views

The pullback bundle

Consider a smooth fibre bundle $P:E\to B$ over a base $B$ with fiber $F$, and a trivializing family $\{(\phi_a:P^{-1}(U_a)\tilde =U_a\times F\}_{a\in A}$. Let the clutch functions of this family be ...