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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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Weak Whitney topology

I'm trying to figure Whitnney topology out. For example I need to prove that $C^0(\mathbb{R},\mathbb{R})$ is path connected in Weak Whitney topology. I think that path from $f$ to $g$ is $l(t)=gt+f(1-t)=t(g-f)+f$, $t∈[0,1] $. But how to prove that…
Kartan12
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If $f : M \to \mathbb{R}^N$ is an immersion then $\exists \delta > 0$ such that $f$ is injective on all balls of radius $\delta$?

Theorem: Let $M$ is a smooth compact manifold with complete metric $\rho$. Let $f : M \to \mathbb{R}^N$ be an immersion. Then there exists a $\delta > 0$ such that for any $m_1,m_2 \in M$ that satisfy $\rho(m_1,m_2) < \delta$, it follows $f(m_1) =…
Allen Hart
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Global Frobenius Theorem- Warner's Proof(Theorem 1.64, Page 48-49)

I am reading Frank Warner's "Foundations of Differentiable Manifolds and Lie Groups" I am trying to prove the existence of unique maximal connected integral manifold of a $k$-distribution $\mathfrak{V}$ passing through a point $p \in M$($Theorem…
permutation_matrix
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diffeomorphic closed manifolds are cobordant

Two closed oriented $n-1$ dimenisonal manifolds $A,B$ are cobordant if there is a compact $n$ dimensional manifold $M$ such that $A \sqcup -B$ diffeomorphic(orientation-preserving) to boundry of $M$. If there is a diffeomorphism between $A$ and $B$…
iefjkfdhfure
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Is there a proof that the degree of the map $f: S^1 \to S^1$ given by $f(z) = z^n$ is $n$ which doesn’t use homology?

This was a proposed exercise in some lecture notes on differential topology I’m reading. Some proofs for this result are given in the answers to this question. However, they assume previous knowledge of algebraic topology, which the lecture notes…
dahemar
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$S^2$ Exact form in Geometry, Topology and Physics Example 6.7

I'm currently reading Nakahara's Geometry, Topology and Physic and I didn't understand the exemple 6.7 about the $S^2$ sphere. Exemle 6.7 Geometry, Topology and Physics I didn't understand the last sentence, why $\Omega$ isn't an exact form sinc…
2PiOmega
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Getting an intution behind smooth manifold theory.

So I recently started studying smooth manifold theory. I understood the definition of a manifold. It is basically union of sets homeomorphic to open sets in $\mathbb{R}^n$. Now I learnt about cotangent spaces, tangent spaces, induced map between…
zero2infinity
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A way to show that the Brouwer degree is constant on connected components

Let $M, N$ be two oriented manifolds and let $A \subset M$ be an open subset s.t. $\overline{A}$ is compact. Let $\phi: \overline{A} \to N$ be continuous and smooth on $A$. For $y \in N \backslash \phi(\partial A)$, we have defined the degree of…
gogoog
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Prove closed of dimension one of $X\times I$.

Suppose $F(x,t): X\times I \rightarrow R$ is a homotopy of Morse functions. That is, $f_t: X \rightarrow R$ is Morse for every $t$. Show that the set $C = \{(x,t)\in X\times I : d(f_t)_x = 0\}$ forms a closed, smooth submanifold of dimension one of…
1LiterTears
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Families of Morse functions

I don't have a clue with this problem. Thank you very much for your help & guidance. (a) Suppose $F(x,t): X\times I \rightarrow R$ is a homotopy of Morse functions. That is, $f_t: X \rightarrow R$ is Morse for every $t$. Show that the set $C =…
1LiterTears
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Is there a branched cover of $T^n \to S^n$?

Taking $T^2 = \mathbb{R}^2/\mathbb{Z}^2$, we see there is an involution $(x,y) \mapsto (-x,-y)$ which has 4 fixed points; $(0,0), (1/2,0), (0,1/2), (1/2,1/2)$. Indeed, on $T^n$, the map $v \mapsto -v$ has $2^n$ fixed points. Now, if I'm not…
inkievoyd
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Almost every vector space $V$ of any fixed dimension $l$ in $\mathbb R^N$ intersects $X$ transversally.

It is a problem from Guillemin and Pollack. Suppose that $X$ is a submanifold of $\mathbb R^N$. Show that almost every vector space $V$ of any fixed dimension $l$ in $\mathbb R^N$ intersects $X$ transversally. [HINT: The set $S\subset (\mathbb…
Jale'de jaled
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On the density of vector fields with only nondegenerate zeroes

Suppose we have a manifold $X$ embedded in $\mathbb{R}^n$. Define the vector field $v_u(p) : X \rightarrow TX$ by taking the point $u \in \mathbb{R}^n$ to its natural (orthogonal) projection onto $TX_p$. How can we show that the set of $u \in…
Vladimir Chernousov
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Proof of the Hopfs Lemma

In his book Milnor proof the following Lemma: If $v:X\to \Bbb{R}^m$ is a smooth vector field with isolated zeroes, and if $v$ points out of $X$ along the boundary, then the index sum $\sum{\iota}$ is equal to the degree of the Gauss mapping from…
Frederick Manfred
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Solution verification in a Differential topology exercise

I did the following differential topology exercise and I wanted to make sure everything was alright , the exercise is from Hirsch The limit set $L(f)$ of $f:M\rightarrow N$ is the set of $y\in N$ such that $y=\lim_{n\rightarrow \infty}f(x_n)$ where…
Someone
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