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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Transversal intersection.

In my textbook, it says: "Consider two curves in the plane, one of which is the x-axis, the other being the graph of a function $f(x)$. The two curves intersect transversally at a point x if $f(x)=0$ ...
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Highest DeRahm Cohomology

Let $X$ be a $C^\infty$ manifold, compact oriented and connected of dimension $n$. How do you prove that the integration map $$\int_X: \omega \mapsto \int_X \omega $$ from $H^n_{DR}(X)$ to ...
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Local compactness of $C^k(M,N)$ strong space.

Let $M$ and $N$ be two $C^k$ manifolds with $k\geq 1$, with $M$ non compact. I know that $C^k(M,N)$ with its strong (Whitney) topology isn't metrizable and that it's a Baire space. Can I prove that ...
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Intuition about pullbacks in differential geometry

I am struggling to understand the role of pullbacks after noticing that they are used when defining an integral of $k$-forms on a manifold. Let $F:M \to N$ be a map between differentiable manifolds. ...
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Smooth closed real plane curve intersecting itself at infinitely many points

Can a smooth closed real plane curve intersect itself at infinitely many points? It seems intuitively obvious that the answer should be no, yet I have no idea how to prove this or construct a ...
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181 views

Taking Differential Topology concurrently with Analysis

So I'm trying to finalize my schedule for this semester. I can't decide whether I should enroll in a grad level Differential Topology (Milnor) class or just the undergrad general topology one. The ...
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How to show the height function of a torus has 4 critical points

In $\mathbb{R}^3$, let $h$ be the height function of a torus standing vertically on the top of the table. A critical point of a function is those point where the differential of the function is a zero ...
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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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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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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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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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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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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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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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$P^1$ 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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...