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Questions tagged [differential-topology]

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.

Differential topology is the field dealing with differentiable functions on differentiable manifolds. Somewhat simplified view of this field is that it describes the setting to which the notion of differentiable function can be generalized from the more familiar case of functions $\mathbb R^n\to\mathbb R^k$. Differential topology is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

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How to prove that the complex in Morse homology is isomorphic to the one in cellular homology

Since every stable submanifold with orientation in Morse homology is actually a cell in cellular homology, it suffices to prove the two boundary map coincide. Intuitively one may accept it is true by noticing the degree of a map between two cpt…
Honglu
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Diffeomorphisms are either both orientation-preserving or both orientation reversing.

Let $F, G : M \to N$ be diffeomorphisms of compact, connected, oriented, $n$-manifolds. If $F$ and $G$ are smoothly homotopic, prove that $F$ and $G$ are either both orientation-preserving or both orientation reversing. Degree is homotopic…
1LiterTears
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Sphere turned inside out

How can I turn a sphere inside out? I saw this video on YouTube and I didn't understood how can i turn a sphere inside out. any help will be appreciated. http://www.youtube.com/watch?v=R_w4HYXuo9M
Gil
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Define pull-back on a manifold by pull-back on the linear space.

It appears to me that pull-back on a manifold If $f: X \to Y$ is a smooth map and $\omega$ is a $p$-form on $Y$, define a $p$-form $f^*\omega$ on $X$ as follows: $$f^*\omega(x) = (df_x)^*\omega[f(x)].$$ is defined by pull-back on the linear…
WishingFish
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Proof the degree of a reflection through a hyperplane is −1.

This seems intuitively true, but it seems impossible to work it out according to the definition on Guillemin and Pollack's Differential Topology Page 108, tracing back t0 107 and 100. Page 108: We define the degree of an arbitrary smooth map $f: X…
WishingFish
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If $f\in C(\mathbb R^n,\mathbb R^n)$ satisfies $\lim_{\|x\|\to+\infty}\frac{(f(x),x)}{\|x\|}=+\infty$, then $f(\mathbb R^n)=\mathbb R^n$.

Let $(\cdot,\cdot)$ be the standard inner product on $\mathbb R^n$. Suppose that $f\in C(\mathbb R^n,\mathbb R^n)$ satisfies $$\lim_{\|x\|\to+\infty}\frac{(f(x),x)}{\|x\|}=+\infty.\tag{$*$}$$ Prove: $f(\mathbb R^n)=\mathbb R^n$. This is an…
Feng
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The proof in textbook on The Transversality Theorem

I was especially thrown out by the proof on The Transversality Theorem on the point "we want to exhibit a vector $v \in T_x(X)$ such that $df_s(v) - a \in T_z(Z).$" I understand so far that in order to show that $f_s \pitchfork Z$ we need to show…
1LiterTears
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If $f(x)$ is an extreme value, why $f$ cannot be a coordinate function?

In the text, it says: Consider smooth function on a manifold $f: X \to \mathbb{R}$. If $f(x)$ is an extreme value, then $f$ cannot be a coordinate function near $x$, so $df_x$ must be zero. I don't understand here - if $f(x)$ is an extreme value,…
1LiterTears
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GP 1.3.9(b) Every manifold is locally expressible as a graph.

This is exercise 1.3.9(b) on Guillemin and Pollack's Differential Topology I believe I am pretty much done with this problem, but I still do not understand why the last step shows the existence, and what is $g$? Is it $\varphi \circ…
WishingFish
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Finitely many Lefschetz fixed points

The questions is Show that if $X$ is compact and all fixed points of $X$ are Lefschetz, then $f$ has only finitely many fixed points. n.b. Let $f: X \rightarrow X$. We say $x$ is a fixed point of $f$ if $f(x) = x$. If $1$ is not an eigenvalue of…
1LiterTears
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Morse Function Definition: Does it implies Morse function is $C^2$?

In my understanding, Morse function just means the determinant of Hessian matrix is nonsingular at critical points. So my claims are: the function itself should be continuous the reference to Hessian matrix in the definition implies Morse functions…
1LiterTears
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Convergence of sequences in the strong Whitney topology

Let $\{f_n\}_{n \in \mathbb{N}}$ a sequence of $C^r(M, N)$ functions of paracompact manifolds converging to $g$ in the strong Whitney topology. Prove that there exist a compact $K$ of $M$ such that $f_n$ and $g$ agree every where except on $K$,…
Alexandre Sallinen
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does the linear isomorphism between the tangent space at a point and $ \mathbb{R}^n $ depend on charts?

I am taking a class in analysis and differential topology. We were learning about the tangent bundle $$TM := \sqcup_{p \in M}T_pM $$ of a $C^1$ manifold M. In class our definition of the tangent space $T_pM$ at a point $p \in M$ was the definition…
Josh Messing
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Does a diffeomorphism between the interiors of two manifolds extend to the manifolds?

Let $M,N$ be manifolds with boundary. Let $f:\mathring{M}\to \mathring{N}$ be a diffeomorphism of their interiors. Does it extend to a diffeomorphism $M\to N$? I suppose we can look at the problem locally, and then the problem is: if $U,V\subset…
Bruno Stonek
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Understanding the proof of the theorem that image of an embedding is a submanifold

I am trying to understand the proof of the following theorem in Differential Topology by Guillemin and Pollack (page 17). Theorem. An embedding $f:X\to Y$ maps $X$ diffeomorphically onto a submanifold of $Y$. The "difficult" part of the proof shows…
user648097
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