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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Lemma 6.3 of Milnor's Lectures on the h-cobordism theorem

Milnor's statement is: "Let $M^r$ and $N^s$ be sub-manifolds of $V^{r+s}$ which are all smooth, compact, oriented and without boundary. If $p$ is a point of $M^r$ contained in an $r$-cell $U$, ...
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smooth function $\mu:\mathbb{R}\rightarrow\!\mathbb{R}$ with $\mu(0)>\varepsilon$, $\:\mu_{[2\varepsilon,\infty)}=0$, $\:-1<\mu'\leq 0$

How can I prove (preferrably without the use of any heavy theorems) the existence of a smooth function $\mu\!:\mathbb{R}\rightarrow\!\mathbb{R}$ with properties $\mu(0)\!>\!\varepsilon$, ...
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340 views

Explain the Stokes -theorem from differential from into Integral form

I want to understand the Stokes -theorems deeper. I am trying to understand the operation from $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$$ to ...
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2answers
70 views

Homology of submanifold

Let $M$ be a manifold space, Let $H^1(M,\mathbb R)=\{0\}$, then is it possible to get a submanifold $S$ of $M$. such that $H^1(S, \mathbb R)\neq \{0\}$. If $M$ is simply connected then we can ...
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70 views

domain of surface of revolution

Let $0<b<a,(u,v) \in \mathbb{R} \times \mathbb{R}$. Then the map $g(u,v):=((a+b\cos u)\cos v,(a+b\cos u)\sin v,b\sin u)$ defines a torus. I wonder for $g$ to be a surface does it really need ...
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96 views

map that is the time 1 of a flow

I have a very general question. If I have a smooth map $\phi:X\to X$ with $X$ compact, what kind of strategics should I try to prove that $\phi$ is the time 1 of a flow in $X$? Any information or ...
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292 views

Morse function with indices of only $0$ and $n$

Q1: If a Morse function on a smooth closed $n$-manifold $X$ has critical points of only index $0$ and $n$, does it follow that $X\approx \mathbb{S}^n\coprod\ldots\coprod\mathbb{S}^n$? I think the ...
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87 views

Grassmanians $Gr_k(\mathbb R^n) \cong Gr_{n-k}(\mathbb R^n)$

I am trying to prove that the Grassmanians $Gr_{n-k}(\mathbb R^n)$ and $Gr_{k}(\mathbb R^n)$ are homeomorphic. Intuitively, this makes sense; specifying a $k$-dimensional subspace is equivalent to ...
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107 views

$P^1$ not a regular level surface of a $C^1$ function on $P^2$

I'm working through the first chapter of Morris Hirsch's "Differential Topology". On Chapter 1, section 3 exercise 11, I encountered the following question. "Regarding $S^1$ as the equator of $S^2$, ...
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$\{(x,y)\!\in\!\mathbb{B}^n; -\varepsilon\leq-\|x\|^2\!+\!\|y\|^2\leq\varepsilon\}\approx\mathbb{B}^k\!\times\!\mathbb{B}^{n-k}$

The question is motivated by the notion of handle attachment, Morse theory, critical points of index $k$, Morse lemma, sublevel sets, etc. For $0\!\leq\!k\!\leq\!n$ and ...
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Variants of isotopy extensions

I am interested in slight variations of the usual isotopy extension theorems. In short, my question is the following : Can one extend isotopies of $C \subseteq M$, where $C$ is compact and $M$ is a ...
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3k views

Computing the homology and cohomology of connected sum

Suppose $M$ and $N$ are two connected oriented smooth manifolds of dimension $n$. Conventionally, people use $M\#N$ to denote the connecte sum of the two. (The connected sum is constructed from ...
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217 views

Morse theory and homology of an algebraic surface (example)

Let $T_n$ denote the $n$-th Chebyshev polynomial and define $f_n(x,y,z)\!:=\!T_n(x)\!+\!T_n(y)\!+\!T_n(z)$ and $$Z_n:=\mathcal{Z}(f_n) \subseteq \mathbb{R}^3,$$ the Bachoff-Chmutov surface, where in ...
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295 views

Morse functions are dense in $\mathcal{C}^\infty(X,\mathbb{R})$.

In Shastri's Elements of Differential Topology, p.210-211, there is written: Why do we get a Morse function $f_u$ on $X$? We know that for any $f\!\in\!\mathcal{C}^\infty(X,\mathbb{R})$, there is ...
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241 views

Pullback of differential form on the double covering

On a double covering there is a differential form $\omega$ arises by the pullback of a differential form under the projection iff it is the pullback of $\omega$ under the map $i$, where $i$ is the map ...
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663 views

A fiber bundle over Euclidean space is trivial.

What's the easiest way to see this? The only thing I could think to do was try to patch together trivializations. I couldn't find a way to make that work. Thank you! edit: For the record, here's why ...
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111 views

Neighborhood Retraction of Boundary

Here is the problem: If $M$ is a manifold with boundary, then find a retraction $r:U \rightarrow \partial M$ where $U$ is a neighborhood of $\partial M$. I realize that the collar neighborhood ...
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157 views

Are close maps homotopic?

Consider a smooth manifold $M = M^m$ and a smooth submanifold $N = N^n \subset M$. Suppose that two maps $f, g: M \to N$ are close to each other, in the sense that there exists $\epsilon > 0$ such ...
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573 views

If $f\!: X\simeq Y$, then $X\!\cup_\varphi\!\mathbb{B}^k \simeq Y\!\cup_{f\circ\varphi}\!\mathbb{B}^k$.

How can I prove that if two spaces $X$ and $Y$ are homotopy equivalent, then the corresponding spaces obtained by gluing a $k$-cell are also equivalent? In detail, if ...
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459 views

A star shaped open set in $\mathbb{R}^n$ is diffeomorphic to $\mathbb{R}^n$

After googling this fact (which is used in Bott and Tu to show the existence of good covers of manifolds) I've gotten the impression this is somewhat difficult to prove. But I also came across this ...
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206 views

Manifold/Topology Notation

I have a basic notation related doubt as follows: Let $M\subset \mathbb{R}^N$ be a manifold. What does $C^\infty(M)$ denote in $f \in C^\infty(M)$?
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104 views

Showing a certain form is exact

I'm trying to solve the following: Let $f: S^{2n - 1} \rightarrow S^n$ be a smooth map, and let $\omega$ be an n-form on $S^n$ such that $\int_{S^n} \omega = 1$. Show that $f^*\omega$ is exact, and ...
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163 views

Strong deformation retraction $\mathbb{I}\!\times\!\mathbb{B}^l \longrightarrow \{1\}\!\times\!\mathbb{B}^l\cup \mathbb{I}\!\times\!\{0\}$?

In Shastri's Elements of Differential Topology, p. 225, there is written: I don't understand this map $R$. Why is $\theta\!\in\!\mathbb{S}^{l-1}$, shouldn't we have $\theta\!\in\!\mathbb{B}^l$? ...
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1answer
133 views

Every $1$-manifold is orientable

How to prove that every $1$-manifold is orientable? Can I use Zorn's Lemma and produce a maximal orientable manifold that will have to be all M?
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891 views

Geometric meaning of a nondegenerate critical point

Let $f\!:M\!\rightarrow\!\mathbb{R}$ be a smooth function on a manifold and $p\!\in\!M$. Is there any way to geometrically/visually characterize the conditions $p$ is a critical point (i.e. ...
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203 views

Transform flat surface into paraboloid

Is it possible to transform a flat surface into a paraboloid $$z=x^2+y^2$$ such that there is no strain in the circular in the circular cross section (direction vector A)? If the answer is yes, is ...
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281 views

Why is there no foliations of the 2-sphere, or a genus two surface?

I'm trying to see why there is no (one-dimensional) foliation of $S^2$ or an orientable surface of genus two. Originally I was thinking that such a foliation could give me a non-vanishing vector ...
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193 views

How to calculate the degree of this Gauss map?

In reviewing the familiar Poincare-Hopf theorem I come across the following question: Suppose $x$ an isolated 0 of $V$. Pick up a disk around $x$ in its neighborhood. Calculate the degree of the map ...
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Confusion on Cech cohomology

From Harvard math qualification exam, 1990. Let $X$ be a smooth manifold with an open cover $N<\infty$ sets $\{B_{n}\}^{N}_{1}$ which are contractible. Assume that $$\pi_{0}(B_{n}\cap B_{m})\le ...
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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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1answer
162 views

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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265 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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636 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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132 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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100 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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260 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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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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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
383 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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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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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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68 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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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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291 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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329 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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290 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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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 ...