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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isotopy of homeomorphisms of a torus

Let's consider some homeomorphism of a torus which is isotopic to identity. Is it possible to construct an explicit isotopy?
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Embedding vs embedding as a closed subset

A version of Whitney's Theorem state that any $n-$dimensional manifold can be embedded in $\mathbb{R}^{2n+1}$ as a closed subset. Is there an example of $n-$manifold which can be embedded in ...
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Edited: Proper nonsingular smooth map between connected manifolds is a covering map

Can you help me with this problem? Thanks Let $f:M->N$ be a proper nonsingular smooth map between connected manifolds. Dim(M) = dim(N). Show f is a covering map. Edit: So here is what I have so ...
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Prove existence of trajectory on $\mathbb{R}^2$

This question is asked on my differential topology mock mid-term, but I can't figure out what to do: Consider smooth curves $\gamma_i: \mathbb{R} \to \mathbb{R}^2, i = 1, . . . , n$ which ...
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About the term $-\nabla_{[u,v]}w$ in the definition of Riemann curvature tensor

As we know, in the definition of Riemann curvature tensor, we require $$ R(u,v)w=\nabla_u\nabla_v w-\nabla_v\nabla_u w-\nabla_{[u,v]}w $$ Could somebody tell me why we need $-\nabla_{[u,v]}w$ ...
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Cutting a torus enough times disconnects it

I am interested in showing that if you cut a torus too many times it becomes disconnected. Let $\mathbb T^n$ be the standard $n$-dimensional flat torus. Let $M_1, \ldots, M_k$ be $k$ disjoint smooth ...
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Show that S is homeomorphic to a Klein Bottle

I've been struggling quite a bit with this question. Any hints/help would be greatly appreciated! Consider the quotient S = R^2/G where G = Z^2 acts by (n, m) • (x, y) = ((−1)mx + n, y + m) on R^2 , ...
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Tangent Bundles to manifolds

I am having trouble trying to visualize exactly what a tangent bundle to the klein bottle is spuposed to look like. Is it possible for one to decompose it as a direct sum of simpler bundles?
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The tangent space of the boundary of a manifold with boundary is a subspace of the tangent space

I was trying to understand the following sentence in some notes I am reading: Let $X$ be a manifold with boundary. At any point $p \in {\partial}X$ there is a canonical subspace $T_{p}({\partial}X) ...
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Show that the central circle $X$ in the open Mobius band has mod 2 intersection number $I_2(X,X)=1$

Show that the central circle $X$ in the open Mobius band has mod 2 intersection number $I_2(X,X)=1$ Like in this picture http://i58.tinypic.com/2dkjwug.png Boundary Theorem: suppose that $X$ is ...
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Show that if $X$ may be deformed into $Z$ then $X$ and $Z$ are cobordant.

Show that if $X$ may be deformed into $Z$ then $X$ and $Z$ are cobordant. Deformation definition: deformation of a submanifold $Z$ in $Y$ is a smooth homotopy $i_t:Z\to Y$ where $i_o$ is the ...
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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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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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42 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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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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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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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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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
27 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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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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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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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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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
29 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
36 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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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, ...
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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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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 ...
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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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Why Shape space is manifold?

In Shape analysis, often shape is considered as continuous parametrized closed curve and the shape space as Hilbert Riemannian manifold. Can any one help me to understand, why the shape space is ...
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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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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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27 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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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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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
53 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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47 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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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'\\ ...
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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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26 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
47 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
20 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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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, $m\geq r,n\geq r$, $F$ ...
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1answer
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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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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 ...
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Retract of a free $\Omega(\mathbb{R})$-module

Can an open subset X of $\Omega(\mathbb{R}^2)$ be an $\Omega(\mathbb{R})$-module retract of some free $\Omega(\mathbb{R})$-module? Here $\Omega(\mathbb{R}^n)$ denotes the usual topology of ...
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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), ...
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91 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 ...
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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 ...