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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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Partition of Unity involved in proving that $C^{\infty}(U,V)$ is dense in $C_S^r(U,V)$, differential topology

I have been reading Hirsch's Differential Topology , and here is giving a proof that if $U\subset \mathbb{R}^m$ and $V\subset\mathbb{R}^n$ are open sets then $C^{\infty}(U,V)$ is dense in $C^r_S(U,V)$. Now for this in the proof he claims that if we…
Someone
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Extending curves

I have the following situation, $N$ is a $k$-manifold, $X$ a compact $(k+1)$-manifold and $F:X \to N$ a smooth map. Let $y$ be a regular value of both $F$ and $F|_{\partial X}$, hence $F^{-1}(y)$ is a compact $1$-manifold. Let $J$ be a connected…
Vinicius M.
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About the Brouwer degree of a mapping between subsets of the real line.

I am looking for examples of the the Brouwer degree of a smooth map $f:M \rightarrow N$, where $M$ and $N$ are subsets of the real line $\mathbb{R}$. In particular, there is a seemingly nice example on page 244 of this pdf:…
Mark
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What's meant by the number of "distinct $C^k$ differential structures" other than the amount of distinct maximal atlases?

When reading the Wiki page on differential structures, I'm struck by the exceptional case of $R = 4$. However, the definition of differential structure leaves me nonplussed, as it seems to just be another name for "maximal atlas" in the context of…
Patrick James
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Tangent vector definition

I'm reading the book "Semi-Riemannian Geometry with Applications to Relativity", which defines a tangent vector at a point $p$ in a manifold $M$ to be a function $v : \mathcal{F}(M) \to \mathbb{R}$ (where $\mathcal{F}(M)$ is all smooth functions $M…
David Dima
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Degree (oriented) of $\bar{z}^m$

Let $f: S^1 \to S^1$ defined by $f(z) = \bar{z}^m$. I need to calculate the degree ofthis map, where the degree is defined by $I(f, \{\omega\})$ - the oriented intersection number-, where $\omega$ is any element of $S^1$. I was thinking of…
Nuntractatuses Amável
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How to prove that the map is open?

I am trying to prove that $\phi$ is homeomorphism,where $U=\{[1,u,v]|u,v\in \mathbb{R}\}\subset{\mathbb{R}P^{2}}$,and $\phi$:$U\rightarrow\mathbb{R}^{2}$ given by $\phi([1,u,v])=(u,v)$,$\mathbb{R}P^{2}$ is the usual quotient topological space made…
C Weid
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Can we say that the moduli of differentiable structures of a topological manifold is discrete?

Differentiabl structure is defined by tangent bundle and any bundle determines a homotopy class of a map from the manifold to the Grassmanian manifold of the planes whose dimension is the rank of the tangent bundle. Because, I am not sure but..., I…
Camford Oxbridge
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Submersion and local section.

A surjective submersion (surmersion) between smooth manifolds $ \pi : M \rightarrow B $ is a fiber bundle if it is proper (Ehresmann theorem). In particular, there always exists local sections of $\pi$. One can try to glue this local section in a…
Quentin
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Suspension of $\mathbb{R}P^2$ Contractible?

Is the suspension of $\mathbb{R}P^2$ Contractible? And if it is, How would you prove it. Thank you!
Susan
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Are level surfaces of differentiable functions differentiable?

Suppose I have a differentiable function $f: {\rm I\!R}^m \rightarrow {\rm I\!R}^n$. I obtain a level surface of $f$ by taking the set of points where $f$ is some constant vector: $S = \{\mathbf{x} | f(\mathbf{x}) = \mathbf{c}\}$. Since $f$ is…
Him
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Vector Fields Question 4

I am struggling with the following question: Prove that any left invariant vector field on a Lie group is complete. Any help would be great!
James
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Degree-1 map from connected sum $M\#T^n$ to torus $T^n$

Let $M^n$ be a closed oriented smooth $n$-manifold, denote by $M\#T^n$ its connected sum with the $n$-torus. (How) can I get a smooth degree-1 map $f: M\#T^n \rightarrow T^n$? Are any additional assumptions on $M$ needed in order for $f$ to exist?
user729518
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Every hypersurface is orientable?

I think that every hypersurface is orientable. Suppose I have a hypersurface $X$ that sits inside the ambient Euclidean space $\mathbb{R}^n.$ Then for every $x \in X, T_x(X)$ is an $n - 1$ dimensional vector space. Hence, we can find a normal…
green frog
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proof of the existence of the Tubular neighborhood

In Lee's book, the following reasoning is used to prove the existence of a Tubular neighborhood: Suppose that $M$ is an embedded submanifold of $\mathbb R^n$. Then we can define $NM : = \{(x,v)|v \bot T_xM\}$ where orthogonality is defined using…
Keith
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