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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A question related to the topology of the level sets of a particular type of smooth functions $f:\mathbb{R}^2\to \mathbb{R}$.

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a smooth function without critical points; i.e. such that $\nabla f(x)\neq (0,0)$, for all $x\in\mathbb{R}^2$. Is it true or false that all the level curves of $f$ ...
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40 views

Problem solving strategies in differential topology

I was wondering if there is a bag of tricks somewhere for differential topology and smooth manifold problems just like there is for analysis by prof. Tao ...
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1answer
88 views

Prove that $\deg_2 (f) \equiv q \mod 2$

Let $f:S^1→S^1$ be any smooth map. There exists a smooth map $g:\mathbb R \to \mathbb R$ such that $f(\cos(t),\sin(t) )=(\cos(g(t)),\sin(g(t) )$ and satisfying $g(2π)=g(0)+2πq$ for some integers $q$. ...
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2answers
100 views

Prove that there exists a smooth map $g:R→R$ such that $f(cos(t),sin(t) )=(cos(g(t)),sin(g(t) )$ and satisfying $g(2π)=g(0)+2πq$ .

Let $f:S^1→S^1$ be any smooth map. Prove that there exists a smooth map $g:R→R$ such that $f(\cos(t),\sin(t) )=(\cos(g(t)),\sin(g(t) )$ and satisfying $g(2π)=g(0)+2πq$ . The book told me to show that ...
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0answers
79 views

Prove that $N(Z;Y)$ is trivial if and only if there exists a set of $k$ independent functions $g_1,…,g_k$ for $Z$ on some set $U$ in $Y$.

Prove that $N(Z;Y)$ is trivial if and only if there exists a set of $k$ independent global defining functions $g_1,…,g_k$ for $Z$ on some set $U$ in $Y$. That is $Z=\{y∈U:g_1 (y)=0,…,g_k (y)=0\}$ ...
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Prove that there exists a diffeomorphism from an open neighborhood of $Z$ in $N(Z;Y)$ onto an open neighborhood of $Z$ in $Y$.

Prove that there exists a diffeomorphism from an open neighborhood of $Z$ in $N(Z;Y)$ (normal bundle of $Z$ in $Y$) onto an open neighborhood of $Z$ in $Y$. $\epsilon$ neighborhood theorem: For a ...
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1answer
108 views

Smoothing Lemma

Given a $C^0$ function $g:[a,b]\to \mathbb{R}$ that is smooth everywhere except at $c$ (where $a<c<b$), and has positive derivative everywhere except at $c$, the claim is that there exists a ...
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2answers
86 views

Pathwise, simple connectedness of real Grassmannian $G(2, 4)$

Let $G(2, 4)$ denote the space of two dimensional planes in $\mathbf R^4$. I have found that the integral homology is the following: $H_0 = \mathbf Z, H_1 = \mathbf Z / 2 \mathbf Z, H_2 = 0, H_3 = ...
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1answer
71 views

Submersion preserves openness

Can you help me with this, but please don't post solutions, just give hints :) $M, N$ are manifolds, $f : M → N$ is a submersion, and $U \subset M$ is open, then $f(U)$ is open in $N$.
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133 views

Tangent bundles of smooth manifolds

Using the identity $T(M \times N) = T(M) \times T(N)$, it is easy to construct the tangent bundles for various smooth manifolds such as the n-dimensional sphere $S^{n}$. However, I could not figure ...
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59 views

Operations on a smooth vector bundle

On a smooth vector bundle, one often defines addition and scalar multiplication to form a vector space. However, doesn't one need to show that these operations are smooth? Is this trivial or is there ...
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17 views

Topology of the intersection of toric arrangement

Hope someone will help me in the solution of the following question. I'm working on some topological problem involving the topology of the intersection of some characters of the torus. I want ot find ...
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72 views

Existence of transversal intersection for $M$ submanifold and of some hyperplane.

Let $M^n\subset\mathbb{R}^P$ submanifold, show that there exist a hyperplane $H^{p-1}$ in $\mathbb{R}^P$ sucht that $H^{p-1}$ intersect $M^n$ tranversally. This problem is I prove using the next: ...
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1answer
157 views

Whitney sum of smooth vector bundles

I was reading through Lee's smooth manifolds book, in his chapter on vector bundles. Upon reading about smooth vector bundles and its definition, I was wondering if the whitney sum of two smooth ...
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1answer
49 views

Let $\pi: E \to M$ a vector bundle. Is $E$ a direct summnad of $M\times\mathbb{R}^{d}$, for some $d$?

Let $\pi: E \to M$ a vector bundle over a smooth manifold $M$. $E$ is direct summand of $M\times\mathbb{R}^{d}$, if there exist a vector bundle morphisms $f:E\to M\times\mathbb{R}^{d}$ and ...
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1answer
47 views

cup product of stiefel-whitney class

Let $\xi$ be a vector bundle. Let $w(\xi)$ be the total Stiefel-whitney class. Let $\bar w$ be the dual Stiefel-whitney class. In John Milnor's Characteristic class book, page 40-41 Chap.4, ...
4
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2answers
52 views

Definition of topological manifold

This might be a stupid question, but I was wondering why we define the topological manifold to be Hausdorff and Second countable? Thanks :)
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1answer
180 views

Partial derivatives on Manifolds

Let $F : A \times B \to C$ be a map of smooth manifolds. Define the following maps ("partial derivatives"): $E_1 F: TA \times B \to TC$ $E_1 F(a,v,b) = D_a F(-,b) v $ where $v \in T_a A$ $E_2 F: A ...
9
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1answer
168 views

Show every $f_t$ is Morse for $t$ is sufficiently small

Let $f$ be a Morse function on the compact manifold $X$. Let $f_t$ is a homotopic family function with $f_0=f$. Show every $f_t$ is Morse for $t$ is sufficiently small Here is my argument, but my ...
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2answers
155 views

Find a map of the solid torus into itself having no fixed point. Where does the proof of the Brouwer theorem fail.

Find a map of the solid torus into itself having no fixed point. Where does the proof of the Brouwer theorem fail. I know that the proof is fail because the torus has a hole, so we can't construct ...
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1answer
111 views

Prove that the Brouwer theorem is false for the open ball $|x|^2 <a$

Prove that the Brouwer theorem is false for the open ball $|x|^2 <a$ Brouwer Theorem: Any smooth map $f$ of the close unit ball $B^n \subset R^n$ tin to it self must have a fixed point. I need to ...
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1answer
39 views

Prove isomorphism of fundamental groups

Hei, guys! I'm having some problem solving the next exercise: Let $f: M -> N$ be a homeomorphism. Define a map $f*:π_1 (M, x_0) → π_1 (N, f(x_0 ))$ such that $f*([\gamma])=[f∘\gamma]$. Show that ...
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1answer
87 views

Prove that$ H_x (X)$ does not depend on the choice of local parametrization.

Suppose that $X$ is a manifold with boundary and $x∈∂X$. Let $ϕ:U→X$ be a local parametrization with $ϕ (0)=x$ where $U$ is an open subset of $H^k$. Then $dϕ_0:R^k→T_x (X)$ is an isomorphism. Define ...
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1answer
66 views

Hausdorff property of $\mathbb{RP}^n$ from unusual definition

Rather than defining the topology on $\mathbb{RP}^n$ as the quotient $(\mathbb{R}^{n+1}\backslash\{0\})/$~ or $S^n/$~ in the usual way, suppose you use these equivalence relations simply to define a ...
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1answer
74 views

Canonical projection on submanifold $M^k $ over a hyperplane $H^{n}$ is immersion

Let $M^k \subset \mathbb{R}^{n+1}$, $M$ compact and $2k\leq n$. Show tha exist a $n$-hyperplane $H^n\subset \mathbb{R}^{n+1}$ such that if $\pi:H^{n}\oplus (H^n)^{\perp}\rightarrow H^{n}$ is the ...
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1answer
106 views

Apply the theorem of tubular neighborhood

$M$ is a connected manifold, $N\subset M$ is a connected submanifold with nontrivial normal bundle, and dimM-dimN=1. How to prove $M-N$ is connected? There is a hint to use the tubular neighborhood ...
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3answers
114 views

does a commutative diagram implies pull-back?

Let $\xi=(E,p,B),\xi'=(E',p',B')$ be fibre bundles. Let $f: B\to B'$, $\bar f: E\to E'$ be maps such that the diagram commutes $\require{AMScd}$ \begin{CD} E @>\displaystyle\bar f>> E'\\ ...
2
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1answer
57 views

Can a smooth function $f\colon\partial D^n\to\partial D^n$ be extended to a smooth function $\hat{f}\colon D^n\to D^n$?

Suppose for $n\geq 1$, you have a smooth map $f\colon S^{n-1}\to S^{n-1}$. Viewing $S^{n-1}=\partial D^n$, is it possible to extend $f$ to a smooth map $\hat{f}\colon D^n\to D^n$, $D^n$ being the ...
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1answer
69 views

Signature of a finite covering space

Suppose $\tilde{M}\rightarrow M$ is a finite (k-fold) covering of the smooth, oriented, compact 4-manifold $M$. Is there a relation between the signatures ...
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1answer
66 views

Is cone not a topological manifold?

Is the cone = X a Hausdorff, second-countable topological space that is not a topological manifold? Since the open subsets $U_{\alpha}$ do not cover the vertex of the cone, so $U_{\alpha}$ is not a ...
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1answer
76 views

Submersion and immersion

I googled wiki about submersion and immersion. Wiki states that submersion is dual to immersion. I wonder where this duality relationship comes from.
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276 views

The intuition of the rank theorem

In Rudin's Principle of Math Analysis, I know the proof of the rank theorem. However, I fail to understand its content. 9.32 Theorem: Suppose $m,n,$ are nonnegative integers, $m\geq r,n\geq r$, $F$ ...
3
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1answer
197 views

Finding bump function on a smooth manifold using partitions of unity.

Let $M$ be a smooth manifold. Let $A$ and $B$ be disjoint closed sets of $M$. Show there exists a smooth function $f$ such that $f^{-1}(0)=A$ and $f^{-1}(1)=B$. This is my idea so far, Since $A$ ...
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0answers
108 views

Exponential map on Diffeomorphism group of $S^1$

I am reading Segal book on Loop groups, and he mentions the following theorem: $$ \exp: Vect(S ^1) \rightarrow Diff(S ^1) $$ the map taking a vector field to the diffeomorphism obtained by flowing ...
3
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1answer
159 views

A question about contractibility of a Lie group

In Lawson's book Spin Geometry, chapter II, before Proposition 1.11, it mentions a fact that if the 1st, 2nd and 3rd homotopy groups of a Lie group $G$ vanish (although $\pi_2(G)=0$ automatically), ...
2
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1answer
122 views

Prove that $f$ map a neighborhood of $Z$ diffeomorphically on to a neighborhood of $f(Z)$

Supposed that the derivative of $f:X\to Y$ is an isomorphism whenever $x$ lies in the sub-manifold $Z \subset X$, and assume that $f$ maps $Z$ diffeomorphically onto a $f(Z)$ . Prove that $f$ map a ...
4
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0answers
146 views

De Rham Cohomology is Sheaf Cohomology

I'm trying to prove that de Rham cohomology computes the cohomology of the constant sheaf $\mathbb{R}$ of real valued smooth functions on a manifold. I would like to do this without using Čech ...
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2answers
110 views

Prove that if $S^k$ has a non vanishing vector field, then its antipodal map is homotopy to the identity if $k$ is even.

Prove that if $S^k$ has a non vanishing vector field, then its antipodal map is homotopy to the identity if $k$ is even. Here is what I got so far. Suppose we have an antipodal map $x \to -x$ of ...
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2answers
94 views

Prove this plane algeraic curve is not a differentiable manifold

Prove the algebraic curve $\{(x,y)~|~x^2(x+1)-y^2=0\}$ in $\mathbb{R}^2$ is not a differentiable manifold. Remark: It is evident that the given cubic curve has a singularity at $(0,0)$ which disable ...
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1answer
97 views

Show that if $f_0$ is Morse in some neighborhood of a compact set $K$, then so is every $f_t$ for $t$ is sufficiently small.

Suppose that $f_t$ is a homotopic family of function on $R^k$. Show that if $f_0$ is Morse in some neighborhood of a compact set $K$, then so is every $f_t$ for $t$ is sufficiently small. I know that ...
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1answer
82 views

Prove that $f$ is Morse function if an only if $det(H)^2 + \sum_{i=1}^k (\frac {\partial f}{\partial x_i})^2>0$

Let $f$ be a smooth function on an open set $U\subset R^k$. For each $x \in U$ let $H(x)$ be the Hessian Matrix of $f$, whether $x$ is critical point or not. Prove that $f$ is Morse function if an ...
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1answer
88 views

Existence of smooth diffeomorphism $f$ of open ball onto itself with $f(0) = p$.

I am trying to show that for every point $p$ of the open $n$-disk $B^n$, there exists a smooth diffeomorphism $B^n \to B^n$ sending $0$ to $p$. Certainly, it seems intuitively obvious for points $p$ ...
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1answer
207 views

Naturality of the pullback connection

I'm completely stuck proving the naturality of the pullback connection. The strategy suggested is a follows: We let $\phi: (M,g) \to (\tilde{M}, \tilde{g})$ be an isometry, with connections ...
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1answer
64 views

Noncomplete riemannian manifolds

In Lee's Riemannian manifolds text, he claims that "on $\mathbb{R}^n$ with metric $(\sigma^{-1})^* g$ obtained from the sphere from stereographic projection, there are geodesics that escape to ...
2
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1answer
186 views

Divergence in terms of Levi-Civita connection

The divergence of a vector field $X$ on a manifold $M$ is defined usually as the function $\text{Div}(.)$ such that $(\text{Div} X) \;\mu =L_X \mu$ for $\mu$ a volume form. I know that there is also ...
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1answer
54 views

The derivative as a linear function

In Milnor's Topology From the Differentiable Viewpoint, the derivative of a smooth map $f: U\to V$ is defined as $$ \mathrm{d}f_x: \mathbb{R}^k \to \mathbb{R}^l $$ $$ h\mapsto \lim_{t\to0} ...
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0answers
81 views

Diffeomorphism between $\mathbb{R}^{2}/\sim$ (Torus ) and $\mathbb{S}_{1}\times \mathbb{S}_{1}$

This is homework so no answers please. Consider the quotient map $\pi:\mathbb{R}^{2}\to \mathbb{R}^{2}/\sim$ ,where $(x_{1}y_{1})\sim (x_{2},y_{2})$ iff $x_{1}-x_{2},y_{1}-y_{2}\in \mathbb{Z}$. I ...
0
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1answer
457 views

Tangent bundle to 2-sphere isnt trivial as a vector bundle?

I read many quiestion about $TS^{2}\ncong S^{2}\times\mathbb{R}^{2}$ where the hint is use Hairy ball theorem and directly is done. My question is: how do I proof that $TS^{2}\ncong ...
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1answer
59 views

Calculate Brieskorn Manifold? [duplicate]

I need show that Brieskorn Manifold is submanifold with dimension $2n-1$ and calculate specifically for $d=2$ and $n=1$ $W(d)=\lbrace (z_{0},z_{1},...,z_{n})\in \mathbb{C}^{n+1}\vert$ $ ...
3
votes
1answer
179 views

A question regarding the proof of Hopf's theorem.

This is a question regarding the proof of the Hopf theorem given in "Topology from a Differential Viewpoint" by Milnor: If $v:X\to \Bbb{R}^m$ is a smooth vector field with isolated zeroes, and if ...