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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Definition of the hessian as a bilinear functional on the tangent space

In Milnor's Morse Theory, the Hessian of a smooth function $f : M \to \mathbb R$ defined on a manifold $M$ at a critical point $p$ is the bilinear functional on $T_p M$ defined as follows: $$f_{**}(v,...
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Surgery or Partial Whitney Trick in McDuff and Salamon proof of Gromov squeezing

In the proof of Gromov squeezing in McDuff and Salamon's epic J-Holomorphic Curves and Symplectic Topology, the authors use a surgery argument without any explicit construction. In the following, $\...
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Degree of Map between Pseudomanifold

There are two different ways to define a degree of map. Let $M$, $N$ be smooth, and $M$ is compact, $N$ is connected. If $f\in C(M,N)$, we define $\deg f$ by smooth approximation. [Milnor/Topology ...
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Orientability of integrable plane fields

Let $\xi \subset \text{T}M$ be a integrable plane field on a smooth 3-manifold (i.e. the tangent field of a foliation). Is it true that $\mathcal{F}$ is coorientable?
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Realizing a Contact Structure on S^1 x S^2 via an Open Book Decomposition

I am trying to learn about Contact Geometry and Open Book Decompositions. I went through the example of the Hopf Fibration for $S^3$ and how you can see a contact structure. I am now trying to do ...
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Prove that $f$ map a neighborhood of $Z$ diffeomorphically on to a neighborhood of $f(Z)$( more detail)

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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59 views

Something about Degree of Map

I have been told that degree of map can be defined by the combinatorial method instead of the differential one. Assume $M$, $N$ is orientiable pseduomanifold, and $M$ is compact, $N$ is connected. Let ...
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Is a diffeomorphism's image automatically open?

Sorry if this question is trivial, I am new to smooth manifold theory. Let $\varphi : I \times \mathcal S^{n-1} \to X$ be a diffeomorphism. $I=(0,1)$, $\mathcal S^{n-1}$ is the unit sphere in $\...
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113 views

Tubular neighborhood of $X^k$ compact submanifold with normal bundle $\perp X$ trivial

For $X^k\subset M^n$ compact submanifold with $\perp X$ trivial and set $S^k$ the $k$-sphere. Then there is a function $f:M^n\rightarrow S^k$ such that $X$ is the preimage for a regular value. My ...
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52 views

A function from a smooth manifold with boundary to $[0,\infty)$

Suppose $M$ is a smooth manifold with boundary, show that there exists a smooth function $f: M \rightarrow [0, \infty)$ such that $\partial M = f^{-1}(0)$. My attempt is that given a chart $(U_\alpha,...
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218 views

Connected sum of orientable manifolds

I was reading through Lee's Smooth Manifolds on the part regarding orientations and I was wondering if the connected sum preserves the orientability of manifolds. Intuitively it seems to be true, but ...
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41 views

Arbitrary Smooth structure

Is it possible to give a smooth structure to any objects? Say two lines intersecting at a point. It seems there is a smooth structure though at the intersecting point it is not locally euclidean if ...
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1answer
38 views

holomorphic function and simple zeros

How can I prove this? If $f$ is a holomorphic function in a domain $U$ and $f'(z)\neq0$ for all $z\in U$ then every zero of $f$ are simple and positive. Definition: $q\in U$ is a simple ...
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1answer
103 views

Does the compact manifold $f=0$ resist small perturbations?

Suppose we have a compact manifold of the form $\left\{f=0\right\}$ where $f:\mathbb R^n\to\mathbb R$ is a smooth Morse function. I am interested in showing that the manifold is topologically ...
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The index of a smooth vector field is well-defined

How can I prove that the index of a smooth vector field is well-defined? All I know that it is locally constant. Definition. Given an open set $U\subset\Bbb{R^m}$, and a smooth vector field $v:U\to\...
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1answer
265 views

Closed 1-forms implies exact 1-forms

I have two problems, the first one I think I've proved, but I have problems on the second one. Let $\omega$ a closed $1$-form defined on a open $U\subset \mathbb{R^2}$ and let $\gamma:[0,1]\rightarrow ...
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180 views

Prove that any map $f\colon S^1 \to S^1$ mapping antipodal point to antipodal point has $\deg_2(f)=1$ by a direct computation.

Without using the Borsuk-Ulam theorem. Prove that any map $f\colon S^1 \to S^1$ mapping antipodal point to antipodal point has $\deg_2(f)=1$ by a direct computation. I know that $f$ map anitpodal ...
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1answer
104 views

Question on the proof of the Poincare-Hopf theorem

OP is reading Milnor's celebrated Topology from the Differentiable Viewpoint's 6th chapter, where he deals with indices of vector fields and in particular the Poincare-Hopf theorem. A lemma he used is ...
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1answer
196 views

Poincare Hopf Theorem

I'm trying to apply the Poincare-Hopf theorem for a vector field over a closed disk. The vector fields sometimes have zeros on the boundary (if number of zeros is infinite, then it's zero over the ...
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1answer
89 views

Piecewise differentiable homotopy

Let $\gamma_0$ and $\gamma_1$ be piecewise differentiable closed curves in an open set $U$ in the plane $\mathbb{C}$. (Of course, you may consider a more general setting if relevant.) Suppose that ...
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Prove that the Borsuk-Ulam theorem and following ttheorem are equivalent

Prove that the Borsuk-Ulam theorem and following ttheorem are equivalent Borsuk-Ulam theorem: Let $f:S^k \to R^{n+1}$ be a smooth map whos image does not contain the origin, and supposed that $f$ ...
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Is $M$ is a compact manifold?

Let $M$ be a manifold of $m$--dimensional, and $M\subset \mathbb{R}^k$. Assume $m>n$. If every smooth function $f:M\longrightarrow \mathbb{R}^n$ has regular values that form an open subset of $\...
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64 views

show that $R^n -X$ has at most 2 connected components.

show that $R^n -X$ has at most 2 connected components. Theprem: Suppose that $X$ is the boundary of $D$, a compact manifold with boundary and let $F:D \to R^n$ be a smooth map extending $f$; ...
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every $f \in F^k_p$ has a Taylor expansion

$F_p$ is the set of germs of functions on a manifold M which vanish at $p \in M$. Let $F^k_p$ be the ideal of $C^\infty(p)$ generated by $f_1,... \,f_k$, where $f_i \in F_p$. (i.e $F^k_p$ is $\sum ...
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contour which can be homeomorphic?

If I have a function $\phi:\mathbb{R^{2}}\rightarrow\mathbb{R}$ which is $C^{\infty}$ without critical points, can I assure that all the contour are homeomorphic?
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Wedge product = set intersection?

In a research article [1] I found the following formulation: The wedge product may be considered as set intersection. For example, surfaces of constant $f(x,y,z)$ and surface of constant $g(x,y,...
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Diffeomorphism from level set onto $S^2$

I'm given a map $\phi : R^4 \rightarrow R^2$ defined by $\phi(x,y,s,t) = (x^2 + y, x^2+y^2+s^2+t^2+y)$. It's easy to show that the level set $C = \phi^{-1}(0,1)$ is a smooth submanifold of $R^4$ with ...
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Are $C^\infty$ exotic spheres $C^k$ exotic?

The only theory of exotic spheres that I know is of $C^\infty$ structures on them; that is, that there are plenty of spheres (in dimensions $n \geq 7$ that are homeomorphic but not diffeomorphic. To ...
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Show that there exist a compact manifold with boundary $W$ in $Y \times I$ such that $\partial W= X\times \{0\} \cup Z \times \{1\}$

I found this question had been posted by someone, but got no answer, so I hope I will have better luck. Show that if $X$ may be deformed into $Z$ then $X$ and $Z$ are cobordant. Deformation ...
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$\mathbb{C}P^1$ diffeomorphic to $S^2$

I am trying to show that the complex projective line is diffeomorphic to the 2-sphere. I'm using the $C^{\infty}$ structure on $\mathbb{C}P^1$ given by $U_1 = \{ [z_1 : 1], z_1 \in \mathbb{C} \}$, $\...
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Lie algebras of vector fields over $\mathbb{R}$ and over $S^1$

An exercise in the book "Moonshine beyond the Monster" has me stumped. It asks whether the real Lie algebras of smooth vector fields over the reals $V(\mathbb{R})$ and over the circle $V(S^1)$ are ...
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Using the results of the local immersion/submersion theorems on manifolds

When $X,Y$ are $k$- and $l$-manifolds, we can have a function $f:X\rightarrow Y, x\in X$ such that $f$ is an immersion resp. submersion at $x$. The local immersion/submersion theorem now says: There ...
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Intuition about Morse functions

We have defined: a Morse function on $X$ is a smooth function $f:X\rightarrow\mathbb R$ with only non-degenerate critical values. I tried to get some intuition about this, and found the section Basic ...
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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? Edit: It's well-known statement that a homoemorphism of a torus is ...
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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. Another version states that that any $n-$dimensional manifold can be ...
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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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1answer
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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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55 views

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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1answer
69 views

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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42 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 http://terrytao.wordpress.com/2010/10/21/245a-...
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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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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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1answer
112 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 ...