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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Counterexpample for solution for D.E. of second order

Set $M$ a $n$-manifold with $1\leq n$. Show that not every curve in $M$ is the solution for a differential equation of second order. A curve on $M$ is a differentiable fuction ...
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Visualizing sums of bundles

So I was wondering about what the whitney sums of various line bundles would look like in general, since it is possible to visualize such sums. I know that the sum of two mobius bundles is just the ...
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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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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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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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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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Prove that the index of a 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.
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Closed 1-forms implies exact 1-forms

I have two problems, the first one I think I´ve prove it, 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 ...
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stable splittings of projective space

On Hatcher's book Algebraic Topology, page 468 Prop. 4I.3, For prime number $p$, can we decompose $\mathbb{C}P^\infty$ in a similar way?
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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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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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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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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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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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32 views

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 ...
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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. 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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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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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: ...