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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.

7287 questions
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Tubular neighbourhood theorem

in your opinion is it possible to get the existence of a tubular neighborhood for a manifold M even if it not embeds smoothly (but only topologically) in some R^N? Thank you!
Flux
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Give an example of two 3-manifolds with different second de rham groups

I am asked to construct two 3-manifolds $M_1$,$M_2$ both covered by two open sets $U$,$V$ (different for each manifold) s.t. the intersection is diffeomorphic to $\mathbb S^1 \times\mathbb R^2$ but the second de Rham group of $M_1$ and $M_2$ is…
gene wolfe3
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Can a non-compact contractible bounded submanifold of the Euclidean space be extended to a compact submanifold?

Let $M$ be a non-compact contractible bounded submanifold of the Euclidean space. Is it possible to find a compact submanifold $S$ of the Euclidean space such that $M$ is a submanifold of $S$? As a positive example, let $M = \{(x,y) \in…
DavideL
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Is $S^1/ \sim $ homeomorphic to $S^1$?

I was reading in a lecture notes the proof which says that the real projective space $\mathbb{R}P^n$ is homeomorphic to $S^n/ \sim$, where $p \sim q$ iff $p = \pm q$. Just after this proof, there is an exercise which ask to prove $\mathbb{R}P^1$ is…
Asma
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Show that the Euler characteristic of $O[3]$ is zero.

Show that the Euler characteristic of $O[3]$ is zero. Consider a non zero vector $v$ at the tangent space of identity matrix. Denote the corresponding matrix multiplication by $\phi_A$. Define the vector field $F$ by $F(A)=(\phi_A)_*(v)$. Where…
1LiterTears
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The global Lefschetz number of $f$ vanishes. - Is this conterexample work?

I am hoping someone will be willing to help me take a look at if this conterexample works? Let $X$ be an oriented compact manifold and $f : X \to X$ a map. Suppose $W$ is a compact oriented manifold with boundary $\partial W = X$ and $F : W \to…
1LiterTears
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For which values of $n$ does there exist a compact, oriented 3-manifold $X$.

So I could barely understand the problem statement ("oriented boundary given by a surface $F$ having the map $f$"), nor how to proceed. Can I get some hints? Thank you. Consider the smooth map $f: F \to S^2$ of degree $n$. For which values of $n$…
1LiterTears
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Assume that $f: X \rightarrow Y$ is a smooth map between two smooth manifolds.

Assume that $f: X \rightarrow Y$ is a smooth map between two smooth manifolds. Must there exist a smooth manifold $Z$, a submersion $g:X \rightarrow Z$, and an immersion $h:Z \rightarrow Y$ such that $f = h \circ g?$ Why or why not?
okipik
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Show that $f$ extends to a smooth map.

Identify $\mathbb{R}^2$, with coordinates $x, y$, with $\mathbb{C}$, with coordinate $z = x + iy$. Likewise, identify a copy of $\mathbb{R}^2$ with coordinates $u, v$ with $\mathbb{C}$ with coordinate $w = u+iv$. Let $f : \{\mathbb{R}^2 - (1, 0) -…
WishingFish
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form exact $\Leftrightarrow$ pull-back exact

Is the form exact $\Leftrightarrow$ pull-back exact? Since $$f^*\omega = \omega \circ df,$$ which seems irrelavant. Because the composition with $df$ does not change $\omega$ is exact or not. The definition I was trying to use: Suppose $A: V \to W$…
WishingFish
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$S^3$ and $T^3$ are not diffeomorphic.

Let $f : T^2 \to S^3$ be the smooth map of a 2-torus into $S^3$, therefore $$\int_{T^2}f^*\omega = 0.$$ There is a closed $2$-form $\beta$ on $T^3 = S^1 \times S^1 \times S^1$ and a map $g : T^2 \to T^3$ such that $$\int_{T^2}g^*\beta \neq…
WishingFish
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Is volume form equal to the top-dimensional form with coefficient $1$?

Wikipedia says a volume form on a differentiable manifold is a nowhere-vanishing top-dimensionial form. And Guillemin and Pollack says $f: V \to U$ is a diffeomorphim of two open sets in $\mathbb{R}^k$ and $\omega = dx_1 \wedge \cdots \wedge dx_k$…
WishingFish
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Notation on the tangent space.

Consider $Y$ an element of the $n$-dimensional tangent space $T_yY$. The the canonical basis is $(\frac{\partial}{\partial y^1}, \cdots, \frac{\partial}{\partial y^n}).$ Then should I write $$Y = (Y^1\frac{\partial}{\partial y^1}, \cdots,…
WishingFish
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Show that $deg(f)=I(graph(f), M \times \{y\})$

Let $f: M \rightarrow N $ be a smooth map with $M,N$ smooth compact oriented manifolds, without boundary and with the same dimension $n$ with $N$ connected show that $$deg(f)=I(graph(f), M \times \{y\})$$ for any $y \in N$ First i know that…
Nick_W
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Winding number of $f$ is equal to $\frac{1}{2\pi}\int_{S^1}f^*(d\theta).$

Let $f: S^1 \to \mathbb{R}^2 - \{0\}$ be a smooth map. Define the winding number of $f$ about 0 and prove that it equals $\frac{1}{2\pi}\int_{S^1}f^*(d\theta).$ I have no clue, except for the winding number of $f$ around 0 is just the degree of…
Tumbleweed
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