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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Prerequisites for studying smooth manifold theory?

I am attending first year graduate school in about three weeks and one of the courses I am taking is an introduction to smooth manifolds. Unfortunately, my topology knowledge is minimal, limited to ...
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Averaging differential forms

Let $M$ be a manifold with a circle action, i.e. with a 1-parameter group of diffeomorphisms $\phi_t:M\to M$ of period 1. I think of the average of a differential form $\omega \in \Omega^n(M)$ with ...
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235 views

Retracts are Submanifolds

Looking over some old qualifying exams, we found this: Let $A\subseteq M$ be a connected subset of a manifold $M$. If there exists a smooth retraction $r:M\longrightarrow A$, then $A$ is a ...
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508 views

Continuous maps between compact manifolds are homotopic to smooth ones

If $M_1$ and $M_2$ are compact connected manifolds of dimension $n$, and $f$ is a continuous map from $M_1$ to $M_2$, f is homotopic to a smooth map from $M_1$ to $M_2$. Seems to be fairly basic, ...
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120 views

A Couple of Normal Bundle Questions

We are working through old qualifying exams to study. There were two questions concerning normal bundles that have stumped us: $1$. Let $f:\mathbb{R}^{n+1}\longrightarrow \mathbb{R}$ be smooth and ...
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96 views

Algebraic topology involved in the 1/4-pinched sphere theorem?

Can anyone familiar with this theorem and its proof let me know how much algebraic topology is involved, and where specifically? I am familiar with a lot of differential geometry, but not many of the ...
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216 views

Orthogonal complement of a vector bundle

Let $E \rightarrow X$ be a vector bundle with an inner product. If $F$ is a sub-bundle, we can define an orthogonal complement bundle $F^\perp$ (see http://www.math.cornell.edu/~hatcher/VBKT/VB.pdf ...
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60 views

Unit partition to produce smooth function from continuous ones

Given a positive continuous function (except on closed set, where is zero ) on a smooth manifold how to find a smooth function under the same conditions being less (or equal) than this one ...
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Uncountable disjoint union of $\mathbb{R}$

I'm doing 1.2 in Lee's Introduction to smooth manifolds: Prove that the disjoint union of uncountably many copies of $\mathbb{R}$ is not second countable. So first, let $I$ be the set over which we ...
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202 views

A curve in a submanifold with a tangent vector not necessarily in the submanifold's tangent space

I came across this exercise in Warner's Foundations of Differential Manifolds and Lie Groups, Let $N \in M$ be a submanifold. Let $\gamma \colon (a,b) \to M$ be a $C^{\infty}$ curve such that ...
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1answer
318 views

Vector Field of Torus

Explicitly construct a differentiable vector field $W$ in the torus. Meridians of $T^2$ parameterized by arc length, for all $p \in T^2$, define $W (p)$ as the velocity vector of the meridian ...
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83 views

Thom space 2 definitions

For a vector bundle thom space $T$ is defined as $T=E/A$, where $E$ is the total space and $A$ is the set of vectors in $E$ of length $\geq 1$. Alternatively, $T$ is the mapping cone of the associated ...
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Poincaré Lemma Contractible Hypothesis

Poincaré's Lemma is often stated as saying that a closed differential form on a star-shaped domain is exact. More generally, it is true that a closed differential form on a contractible domain is ...
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2answers
106 views

non-equivalent bundles

Is it possible to find a specific example of two fiber bundles with the same base, group, fiber and homeomorphic total spaces but these bundles are not equivalent/isomorphic, if so should I find a ...
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1answer
60 views

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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0answers
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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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2answers
245 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 $t\mapsto t^3$. ...
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1answer
272 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 ...
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243 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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111 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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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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2answers
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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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1answer
150 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
108 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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197 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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466 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
409 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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122 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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392 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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313 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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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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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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1answer
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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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0answers
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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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665 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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1answer
372 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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1answer
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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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455 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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1answer
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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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2answers
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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
73 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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1answer
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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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345 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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0answers
311 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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258 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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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
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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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131 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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1answer
289 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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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 ...