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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80 views

Need help understanding proof about critical values of the determinant map.

In , problem 5 the author shows that the differential of the determinant map $d(\det)_A$ for an invertible matrix $A$ is nonsingular by only showing that $d(\det)_A(A) \ne 0$. I don't really see why ...
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55 views

Uniform Poincaré-Wirtinger constant for diffeomorphic domains?

Consider a bounded and connected open set (with regular boundary) $\Omega\subset\mathbb{R}^d$ and a family of $\mathscr{C}^1$-diffeomorphisms $(\varphi_t)_{t\in[0,\alpha]}$ all in ...
3
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2answers
148 views

Question about definition of a smooth manifolds (do all transition maps have to be smooth)?

I need to clear up my confusion on the definition of a smooth manifold. So we say that $M$ is a smooth manifold (of dimension $n$), if $M$ is Hausdorff and if every $x \in M$ is contained in a ...
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4answers
206 views

How do you imagine the shape of a manifold $S^2 \times S^1$?

In 3-dimensional manifold theory, I have encountered the manifold $S^2 \times S^1$ many times. (The following story can be applied not only this manifold but also for any 3-dimensional manifold.) But ...
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0answers
53 views

How to prove $[\partial M, N]=[\partial N, M]$?

Consider $M,N$ submanifolds of some manifold $X$ such that $\dim M+\dim N=\dim X$. For $x\in M\cap N$, let $\langle M_{x},N_{x}\rangle$ denote the index by matching local orientation of ...
2
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1answer
77 views

How to show $\mathbb{R}^{n}+\mathbb{S}^{n}=\mathbb{R}^{n}$?

I want to ask how to show $\mathbb{R}^{n}+\mathbb{S}^{n}\cong \mathbb{R}^{n}$ as connected sums where the isomorphism is a differeomorphism between $\mathbb{R}^{n}$ and $\mathbb{R}^{n}$. The proof in ...
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37 views

Existence of a smooth function with a given kernel

Let $K\subset \mathbb{R}^n$ be a closed set, then is there existing a smooth function $f\in C^{\infty}(\mathbb{R}^n,\mathbb{R})$, such that $$ (1)\quad f\ge 0, $$ $$ (2) \quad f^{-1}(0)=K. $$
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137 views

Prime decomposition of 3-manifolds

Let $H_g$ be a three dimensional handlebody bounded by a genus $g$ surface. Let $M_g$ be a manifold obtained by gluing two copies of $H_g$ via an orientation reversing homeomorphism of the surface of ...
4
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1answer
78 views

Gluing axiom of a TQFT

In the book, Lectures on tensor categories and modular functors by Bakalov and Kirillov they construct a TQFT. When they come to prove the gluing axiom, they just mention that "...This statement is ...
5
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1answer
162 views

On the converse of Sard's theorem

Let $f: M \rightarrow N$ be a smooth map between two submanifolds of $\mathbb{R}^{m}$, $\mathbb{R}^{n}$ respectively. Sard's famous theorem asserts that the set of critical values $C$ of $f$ has ...
3
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1answer
47 views

Possible lengths of geodecics

Let $M$ be a compact manifold (w/o boundary). Suppose that there is no closed geodesics on $M$ of length precisely $C$. I am trying to prove that there is an open cover $\{U_j\}$ of $M$ and $\epsilon ...
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1answer
494 views

Prerequisite for Differential Topology and/or Geometric Topology

What are the prerequisites to learning both or one of the items? Consider that one will have done some of the "core" classes like Differential Geometry, Real Analysis, Abstract Algebra and POint-Set ...
2
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115 views

Image of smooth manifold is a submanifold

It's know that if $M$ is a compact, smooth manifold of dimension $n$ and the map $f: M \to \mathbb{R^m}$ is injective, smooth, $n \le m$ and $Jf(a)$, the Jacobian, has rank $n$ for every $a \in M$, ...
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62 views

Differential of a Sobolev map between manifolds

Let $\Sigma, M$ be compact Riemannian manifolds. By embedding $M$ isometrically into $\mathbb{R}^N$, one can define the Sobolev spaces $W^{k,p}(\Sigma, M)$ by $$W^{k,p}(\Sigma,M) = \{ u \in ...
2
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2answers
73 views

Is there any notion for a certain type of embedding of a smooth curve in a 2-d euclidean space?

There is a smooth 1-manifold (a smooth curve of infinite arc length) embedded in a 2 dimensional euclidean space. This curve (of infinite arc length) is such that, there is one and only one point $P$ ...
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3answers
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Good textbook or lecture notes on Seiberg-Witten theory.

I am looking for a good introductory book for Seiberg-Witten theory. The only textbook I have now is Morgan's "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds". ...
0
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1answer
124 views

$f$ a differentiable map between manifolds of same dimension; $df(p)$ is nonsingular - show $f$ is an open map

Let $f: X \to Y$ be a differentiable map of manifolds where $dim \;X = dim\;Y = n$. If $df(p)$ is nonsingular for all $p \in X$, show $f$ is an open map. So here is what I was thinking: As $df(p)$ ...
2
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1answer
74 views

The decomposition of open set

If $U$ is an open set in $\mathbb R^n$, then there exists a sequence of open sets $\{U_i\}$, such that a.$U_i\subset \subset U_{i+1}$ (that is, ${\overline U _i}$ is compact and ${\overline U _i} ...
2
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1answer
191 views

Why don't we have many differential topologist

I am interested in learning differential topology as Milnor, Guillemin, Pollack, Hirsch, Kosinski, etc.. did it. However, I am in University of Toronto as an undergraduate and none of colleges (or ...
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1answer
71 views

A question on the kernel of the differential of two transverse maps

Let $P,Q,M$ be smooth manifolds, $f:P \rightarrow M$, $g:Q \rightarrow M$ be two smooth maps, which are transverse to each other. Denote by $Z$ the fiber product of $f$ and $g$, $Z=\{(p,q) \in P ...
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1answer
227 views

Method for showing a quotient map is locally homeomorphic

I'm new to Topology and need a little help in working out a general intuition of how to build proofs. I have a question that is asking me to show that a quotient map (from a topology onto it's ...
5
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219 views

Self Intersection and Euler characteristic

Reading the "Differential Topology" of V.Guillemin and A.Pollack, i found a definition of the Euler Characteristic different from the other one using the simplicial complex and betti number (ex. for ...
4
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1answer
189 views

The first fundamental form can NOT be identity

Let $f:V \to S^{2} \subset \mathbb{R}^{3}$ be a surface whose image is an open subset of the usual unit sphere $S^2$. Prove that there is NOT a change of variables $\phi: U \to V$ such that in the new ...
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1answer
220 views

Transversality condition

From my textbook: "If we examine linearizations of $f_\mu (x)$, (where $\dot{x}=f_\mu(x) \ x\in \mathbb{R}^n, \mu\in \mathbb{R}^2$) at the equilibria $f_\mu(x)=0$, then we can formulate a ...
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2answers
655 views

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 $f(x)=0$ ...
4
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1answer
129 views

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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1answer
92 views

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 ...
4
votes
1answer
316 views

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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2answers
338 views

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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1answer
180 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 ...
3
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1answer
248 views

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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144 views

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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1answer
112 views

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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2answers
294 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
61 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 ...
2
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1answer
66 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 ...
4
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1answer
79 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 ...
4
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2answers
265 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 ...
4
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1answer
82 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 ...
4
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2answers
93 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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1answer
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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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1answer
197 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 ...
5
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1answer
214 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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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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85 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 ...
9
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1answer
122 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 ...
10
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1answer
494 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 ...