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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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Counter-example to G&P exercise

Local diffeomorphism is diffeomorphism provided one-to-one. Isn't this false, though? For instance, consider the map of $(0,1)\to S$ where $S\subset\mathbb{R}^2$ is the set indicated by the black line in the picture Follow up: Is this not a counter…
J. Moeller
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Is the Transversality Homotopy Theorem wrong as it is stated in Lee‘s Introduction to smooth manifolds?

In Lee‘s Introduction to smooth manifolds he states the Transversality Homotopy Theorem as follows: Suppose $M,N$ are smooth manifolds and $X \subset M$ is an embedded submanifold. Every smooth map $f:N\to M$ is homotopic to a smooth map $g: N \to…
Frieder Jäckel
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Show that the map $p$ is a submersion except at finitely many points.

Show that the map $p:\mathbb{C}\rightarrow \mathbb{C}$ where $p(z)=z^m+a_1z^{m-1}+...+a_{m-1}z+a_m$ is a submersion except at finitely many points. My Attempt: I calculated $df_{z_0}(w)$ and finally got that $\displaystyle…
Extremal
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Understanding Stokes's Theorem in terms of cohomology

Let $M$ be a compact orientable smooth manifold with boundary. Stokes's Theorem states $\int_{\partial M} \omega = \int_M d\omega$, where $\omega$ is an $n-1$ form on $\partial M$. Via the difference map in the long exact sequence for de Rham…
maxematician
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Framed cobordism

I have an exercise as follows: "Let $M\subset \mathbb{R}^k$ be a smooth, connected, oriented, compact manifold without boundary of dimension $p$. Let $\Omega^{Fr}_n(M)$ be the set of all equivalence classes of framed cobordism of submanifolds of $M$…
mapping
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Is h-cobordism theorem true on smooth category?

On h-coobrdism page of Wikipedia, it says that h-cobordism theory is true for smooth category. As far as I know, it would imply that smooth Poincare conjecture is true for dimension greater than 6, which is not true by milnor. Is the Wikipedia page…
topologisttt
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Smooth fibre bundles as manifolds

After reading some introductory texts i've got a question on smooth fibre bundles. I understand that we require both the base space and the typical fibre to be smooth manifolds. But then i am not completely sure why the total space becomes a smooth…
NDewolf
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Equivalence of different definitions of the linking number

I would like to show that 2 definitions of the linking number are equivalent. Let $J$ and $K$ be compact, oriented differential manifolds of dimension 1 embedded in $\mathbb{R}^{3}$. Suppose $J \cap K =\emptyset$. Consider $\lambda:J \times K…
PP123
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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 Kosinski's book is not readable. Sorry if this…
Bombyx mori
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Germs of a $C^{\infty}$ function at $p$, and the derivation at $p$.

Loring Tu defines $C^{\infty}_p(M)$ to be the germs of a $C^{\infty}$ function at $p$, which is the equivalence class of functions defined in a neighborhood of $p$ and agree on some possibly smaller neighborhood of $p$. Then he defines a derivation…
Keith
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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 $g$, $Z=\{(p,q) \in P \times Q \; | \; f(p)=g(q)\}$. Since $f$ and $g$ are…
Nils Matthes
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Help with Milnor's proof that inverse of regular value is submanifold

My question is the same as that asked in this post, namely, about the statement Note that $f^{-1}(y)$ corresponds, under $F$, to the hyperplane $y\times \mathbb{R}^{m-n}$. The answer given in the linked thread seems inadequate. To prove that…
J. Dong
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Basis for $T_x \mathbb{R}$

Consider the manifold $\mathbb{R}$ with coordinate chart $(\mathbb{R}, \psi)$ where $\psi(x)= x^3$. I am looking at the tangent space of $\mathbb{R}$ with respect to the chart $\psi$. Lee states that for any $x \in \mathbb{R}$,…
user7090
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Orientation of a manifold and conformal structure

If $M$ is a n-manifold, I've the definition that: $M$ is orientable if $\forall i, j$ the composition of the charts $\phi_{i}^{-1}\circ \phi_{j}$ has a positive Jacobian determinant (here $\phi_{\alpha}:U_{\alpha} \subset \mathbb{R}^{n} \rightarrow…
Reinaldo R.
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Diffeomorphism between rotational hyperboloid and $S^1 \times \mathbb{R}$

Premise: I know this post exixts Manifold diffeomorphic to $\mathbb{S}^1\times\mathbb{R}$., but I'm asking for something more. I need to prove that $A=\{(x,y,z) \in \mathbb{R}^3 : x^2+y^2-z^2=1 \}$ is diffeomorphic to $S^1 \times \mathbb{R}$. $A$…
DavideL
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