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 knot which intersects $S^2$ transversely once in 3-connected manifold

I am reading the paper,ON ATTACHING 3-HANDLES TO A 1-CONNECTED 4-MANIFOLD by BRUCE TRACE here.He says in this paper that we need only construct a knot $K\subset \partial W^4$ which meets $Σ ^2$ ...
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142 views

If $f:M\rightarrow N$ is $C^{\infty}$, bijective, and everywhere non-singular, then $f$ is a diffeomorphism

I am not able to solve this problem: Prove that if $f:M\rightarrow N$ is $C^{\infty}$, one-to-one, onto, and everywhere non-singular, then $f$ is a diffeomorphism. This $f$ is a diffeomorphism ...
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224 views

Differentiable structure on the real line

The usual differentiable structure on real line was obtained by taking ${F}$ to be the maximal collection containing the identity map, Let ${F_1}$ be the maximal collection containing $t\mapsto t^3$. ...
5
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219 views

Any manifold admits a morse function with one minimum and one maximum

I have heard the claim: "Any closed manifold admits a Morse function which has one local minimum and one local maximum" often used in talks without a reference. This does not seem to be very easy to ...
5
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190 views

Why does the Gauss-Bonnet theorem apply only to even number of dimensons?

One can use the Gauss-Bonnet theorem in 2D or 4D to deduce topological characteristics of a manifold by doing integrals over the curvature at each point. First, why isn't there an equivalent theorem ...
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103 views

associated disk bundle of a sphere bundle over $S^4$

We are given an $R^4$ bundle $\xi$ over $S^4$, whose total space is E, and we know that the associated sphere bundle is the Hopf fibration $S^3 \rightarrow S^7 \rightarrow S^4$. How can we show that ...
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89 views

About manifolds after attaching handles.

I'm reading R.E. Gompf and A.I. Stipsicz, 4-Manifolds and Kirby Calculus. I don't understand Remarks 4.4.1 on page 116-117 Google books here. At first I can't understand why we take immersed disk $D ...
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755 views

Question about Milnor's talk at the Abel Prize

I don't quite follow the rough outline Milnor gives of the fact that the 7-sphere has different differentiable structures. The video is available here, and the slides he used can be found here. ...
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137 views

About Kirby Diagrams

I'm reading R.E. Gompf and A.I. Stipsicz, 4-Manifolds and Kirby Calculus. There is something I don't understand on page 116 (Google Books link to page 116; alternatively, here are images of page 115 ...
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1answer
107 views

total spaces of S3 bundles over S4 which are homotopic to S7

Milnor showed that if the Euler class of an $S^3$ bundle over $S^4$ is $\pm 1$, then the total space is a homotopy sphere. How many $S^3$ bundles over $S^4$ do we have with the total space is ...
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180 views

Gaussian curvature in $S^3$

I'm trying to read a survey paper on the Willmore conjecture and I'm missing a lot of basic knowledge. In particular, let $u: \mathcal{M} \rightarrow S^3 \rightarrow \mathbb{R}^4$ be a smooth ...
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377 views

Product of spheres embeds in Euclidean space of 1 dimension higher

This problem was given to me by a friend: Prove that $\Pi_{i=1}^m \mathbb{S}^{n_i}$ can be smoothly embedded in a Euclidean space of dimension $1+\sum_{i=1}^m n_i$. The solution is apparently ...
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2answers
343 views

Regarding Legendre transform from tangent bundle to cotangent bundle

(I'm a complete beginner at differential geometry) I'm studying about constrained systems, in which we "map a Lagrangian system from a tangent to a cotangent bundle. Hamiltonian dynamics then appears ...
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116 views

Smooth deformation retracts

Under what circumstances can it be concluded that if two items from the smooth category are related by a topological relationship, then they are also smoothly related in the corresponding way? For ...
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311 views

A proof of the Morse Lemma

On page 7 of Milnor's Morse Theory is part of a proof of the Morse Lemma: Suppose by induction that there exist coordinates $u_1, \ldots, u_n$ in a neighbourhood $U_1$ of $0$ so that $$f = ...
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2answers
278 views

Signature of a manifold as an invariant

Could you help me to see why signature is a HOMOTOPY invariant? Definition is below (from Stasheff) The \emph{signature (index)} $\sigma(M)$ of a compact and oriented $n$ manifold $M$ is defined as ...
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2answers
148 views

Details on the Hopf Foliation

I am trying to understand the Hopf foliation better....that is, the foliation of the 3-sphere induced from the Hopf fibration. Start with the 3-sphere $\mathbb{S}^3=\{(z_1,z_2)\in ...
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176 views

Hairy ball theorem: references to applications

I'm looking for references to applications of the Hairy ball theorem. I already visited wikipedia and cited references, but I need a little more explanation in both meteorology and applications in ...
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211 views

Question on “up to isotopy” when attaching two spaces

Let $M$, $N$, $A$, $B$ be topological spaces (or manifold) such that $A$ and $B$ are subspaces in $M$, $N$ respectively. Let $f: A \to B$ and $g:A \to B$ be homeomorphism and assume that $f$ and $g$ ...
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98 views

How to show that a topological submanifold is a retract of an open set?

Suppose $M \rightarrow N$ is a continuous embedding of a topological (not necessarily smooth) manifold $M$ as a closed subset of a smooth manifold $N$. Do you know a nice way to see that $M$ is a ...
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604 views

Homology and cohomology: why does Poincaré duality fail for domains with boundary?

Poincaré duality says that for a compact, orientable manifold without boundary the $k$th and $(n-k)$th homology groups are isomorphic. For domains with boundary, it's easy to construct examples where ...
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330 views

Submanifold of $\mathbb R^n$ : projections onto tangent spaces

Let $M$ a submanifold of $\mathbb R^n$, for all $x$ in $M$, let $\pi_x:\mathbb R^n\rightarrow T_xM$ the orthogonal projection onto the tangent space $T_xM$ of $M$ at $x$. How could you show that for ...
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116 views

Holomorphic Poincaré conjecture

Is there a notion of n-sphere for complex manifolds so that one could formulate a Holomorphic Poincaré conjecture ?
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404 views

Mapping Degree of a Smooth Map from a Compact Manifold without Boundary

There is a comment in Milnor's "Topology from a Differentiable Viewpoint," that I don't quite understand: Let $f$ be a smooth map from $M$ to $N$, where $M$ is compact without boundary, and $N$ is ...
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Are these equivalent characterizations of closed manifolds?

Let $M$ be a connected smooth manifold without boundary. Are the following equivalent? $M$ is compact $M$ cannot be realized as a proper open subset $M\subset N$ of another connected manifold $N$. ...
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105 views

Proving that a particular submanifold of the cotangent space is Lagrangian

I have the following problem in my differential topology class: Let $M$ be an $n$-dimensional manifold and let $\omega$ denote the standard symplectic form on the cotangent space $T^*M$. Let $f \in ...
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1answer
69 views

Variations in a Riemannian Manifold

Let be $M$ a Riemannian manifold and $X,Y$ vector fields over $M.$ Now take $p\in M$ arbitrarily, my question is, how construc a variation $f:U\to M,$ $$U\subset ...
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126 views

Compute $\chi(\mathbb{C}\mathrm{P}^2)$.

I need to compute $\chi(\mathbb{C}\mathrm{P}^2)$ using techniques from differential topology. I cannot think of any theorems that are particularly useful for this computation, so I think that I will ...
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301 views

Characterization of gradient vector fields

Let $V$ be a vector field on a smooth manifold $M$. Are there nice conditions under which there exists a (Riemannian) metric on $M$ such that $V$ is the gradient of some smooth function on $M$? One ...
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279 views

Definition of Reshetikhin-Turaev TQFT

I am studying Reshetikhin-Turaev TQFT. In their paper or in the book " Quantum invariants of knots and 3-manifolds", they first define an invariant $\tau(M)$ for a closed orientable 3-manifold $M$ and ...
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246 views

Surgery link for lens spaces

Let $p$ and $ q$ be a relatively prime integers. I want to know how to prove that a Hopf link with framing $-p$ and $-q$ is a surgery link for a lens space $L(p,q)$. The lens space is first a result ...
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281 views

The cone is not immersed in $\mathbb{R}^3$

The problem is to show that the cone $ z^2= x^2+y^2$ is not an immersed smooth manifold in $\mathbb{R}^3$.
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1answer
77 views

Smooth functions on $\mathbb{R}^k$ as a subset of $\mathbb{R}^l$ are the same as usual.

I am reading a book on differential topology and the first question in it has me confused. If $k < l$ we can consider $\mathbb{R}^k$ to be the subset $\{(a_1, \cdots,a_k, 0, \cdots, 0)\}$ in ...
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130 views

Energy displacement of a cylinder is at most $\pi r^2$.

I want to show that the energy displacement of $Z^{2n}(r)$ is at most $\pi r^2$. In the textbook of Mcduff and Salamon they write that I should identify the two dimensional ball (with radius $r$) ...
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279 views

$4$-form $ \omega \wedge \omega$ vanishes on $S^4$

If $\omega$ is a closed $2$-form on $S^4$, how can I show the $4$-form $ \omega \wedge \omega$ vanishes somewhere on $S^4$? I am guessing that the fact we're talking about the $2$-form being ...
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61 views

How to find the boundary of a $\mathcal{C}^1$ manifold?

Let $U$ be a bounded open convex set of $\mathbb{R}^d$, and $\Phi$ a differentiable map from $U\times \mathbb{R}$ to $\mathbb{R}^d$. The object of interest for me is $R=\Phi(U\times \mathbb{R})$. The ...
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1answer
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How to detect a twist or framing in a 3-manifold.

This question is somewhat a continuation of the question Gluing a solid torus to a solid torus with annulus inside. If we consider a genus one handlebody $U$ with an (nontwisted)annulus inside the ...
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How do different definitions of “degree” coincide?

I've recently read about a number of different notions of "degree." Reading over Javier Álvarez' excellent answer for the thousandth time finally prompted me to ask this question: How exactly do ...
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Embedding of $T^{2}$ in $S^{1}\times S^{2}$.

Let $i:T^{2}\rightarrow S^{1}\times S^{2}$ be an embedding map and $i_{*}:\pi(T^{2})\rightarrow\pi(S^{1}\times S^{2})$ be the homomorphism induced by $i$. If $i(T^{2})$ is separable in $S^{1}\times ...
3
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1answer
697 views

orientation preserving map

Let $f:X\rightarrow Y$ be a diffeomorphism between connected oriented manifolds. $f$ is orientation-preserving at $p\in X$ if the induced map $df_{p}:T_{p}X\rightarrow T_{f(p)}Y$ is ...
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Visualizing identity $m\le3n-6$ for simple connected finite planar graphs

How can I visualize the identity $m\leq3n-6$ (where $m$ is the number of edges, $n$ the number of vertices) for simple connected finite planar graphs?
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1answer
308 views

diffeomorphism of derivative map at tangent space level

$f: X\rightarrow Y$ is a diffeomorphism, then at each $x$ its derivative $df_x$ is an isomorphism of tangent spaces.could you please give me proof and insight of this result?
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617 views

Pulling back vector fields

I want to find conditions under which one can pull-back vector fields (if it is at all possible). Let $F:M \to N$ be a smooth surjective map between two $C^{\infty}$ manifolds of the same dimension. ...
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Showing that some symplectomorphism isn't Hamiltonian

I have the next symplectomorphism $(x,\xi)\mapsto (x,\xi+1)$ of $T^* S^1$, and I am asked if it's Hamiltonian symplectomorphism, i believe that it's not, though I am not sure how to show it. I know ...
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Derivative/Chain Rule (for MANLYfolds) Computation

Embarrasingly, I can't compute the following derivative. $dh(X)=\left.\frac{d}{dt}h(e^{XT}]\right|_{t=0}$, where $X$ resides in the lie algebra of $\rm SL(3,\Bbb C)$ [ie $\mathfrak{sl}(3,\Bbb C)$] ...
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1answer
192 views

Smooth structures on compact manifolds

I am currently reading some notes where an operator, originally defined on functions over Euclidean Space is now transferred to the setting of a compact, smooth Riemannian manifold. There is a ...
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150 views

Poincaré-Hopf theorem using Stokes

The wiki entry on the Poincaré-Hopf theorem claims that it "relies heavily on integral, and, in particular, Stokes' theorem". However, in the sketch of proof given there which is more or less the one ...
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118 views

Giving tangent space a vector space structure

As I understand it, the tangent space $T_{p}(M)$ to a manifold is given a vector space structure by taking a chart $\varphi:U\rightarrow V\subset\mathbb{R}^{n}$ and making the identification via the ...
2
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1answer
95 views

normal bundle of a boundary

let $X$ and $Y$ be compact, oriented manifolds and assume that $\partial X=Y$. Is it true that the normal bundle of $Y$ in $X$ is trivial? if it is the case, is there a simply explaination? Thanks
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Two Grassman manifold problems from Milnor and Stasheff's book

I'm stuck on the following two problems in Milnor and Stasheff's book Characteristic Classes. Really can't get my head around this material and I'm hoping that more worked examples would help. Even ...