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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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Trivial tangent bundle of sphere with handles

I am wondering if there is a simple proof of this statement: A sphere with $g$ handles has trivial tangent bundle iff $g=1$ I know that it is a corollary of Poincaré-Hopf theorem, but it seems to be too hard for this problem.
evgeny
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Compensation of the anticommutativity of wedge product.

In Guillemin and Pollack, Differential Topology Page 166, The automatic appearance of the compensating factor $\det (df)$ is a mechanical consequence of the anticommuntative behavior of $1$-forms: $$dx_i \wedge dx_j = - dx_j \wedge…
1LiterTears
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The tangent space to the preimage of $Z$ is the preimage of the tangent space of $Z$.

More generally, let $f: X \to Y$ be a map transversal to a submanifold $Z$ in $Y$. Then $W = f^{-1}(Z)$ is a submanifold of $X$. Prove that $T_x(W)$ is the preimage of $T_{f(x)}(Z)$ under the linear map $df_x:T_x(X) \to T_{f(x)}(Y)$. ("The tangent…
1LiterTears
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Immersive Injections whose images are Embedded Submanifolds

Let $M,N$ be smooth manifolds where the dimension of $M$ is less than or equal to the dimension of $N$. Suppose that $F: M \rightarrow N$ is an injective immersion and $F(M)$ is an embedded submanifold, is it the case that $F$ is a smooth…
JSchlather
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Question about the Structure group of a circle bundle over a Riemann surface

The last two pages of Appendix C in Milnor's Characteristic Classes gives an example of a flat bundle with nonzero Euler class. I have a question about the structure group of this bundle. The example starts with a Riemann surface of genus larger…
marc gordon
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Extension of a smooth function on a set of a manifold to an open nbd of the set.

Exercise 1.74 of the book Manifolds and differential geometry of Jeffrey M. Lee says: Show that if a function is smooth on an arbitrary set $S\subseteq M$ as defined earlier, then it has a smooth extension to an open set that contains…
Zero
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Integral form of Euler characteristic

There is a known formula for Euler characteristic in terms of Ricci scalar: \begin{equation} \chi(M)=\frac{1}{4\pi} \int_M \sqrt{g} \,R\,d^2x\,. \end{equation} I am sure that this formula holds for two dimensional manifolds, but what about higher…
MEDVIS
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Prove that the diagonal $∆$ in $S^2×S^2$ is not globally definable by $2$ independent function.

Prove that the diagonal $∆$ in $S^2×S^2$ is not globally definable by $2$ independent function. In contrast, show that the other standard copies of $S^2$ in $S^2×S^2$ – ie, $S^2×\{a\}$ for $a∈S^2$ are definable. Definition: We say $Z$ is globally…
Diane Vanderwaif
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Prove that $N(Z;Y)$ is trivial if and only if there exists a set of $k$ independent functions $g_1,…,g_k$ for $Z$ on some set $U$ in $Y$.

Prove that $N(Z;Y)$ is trivial if and only if there exists a set of $k$ independent global defining functions $g_1,…,g_k$ for $Z$ on some set $U$ in $Y$. That is $Z=\{y∈U:g_1 (y)=0,…,g_k (y)=0\}$ Local submersion theorem: Suppose that $f:X\to Y$ is…
Diane Vanderwaif
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Existence of transversal intersection for $M$ submanifold and of some hyperplane.

Let $M^n\subset\mathbb{R}^P$ submanifold, show that there exist a hyperplane $H^{p-1}$ in $\mathbb{R}^P$ such that $H^{p-1}$ intersect $M^n$ tranversally. In this problem I prove using this: Almost every vector space $V$ with dim$V=l
Donyarley
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Find a map of the solid torus into itself having no fixed point. Where does the proof of the Brouwer theorem fail.

Find a map of the solid torus into itself having no fixed point. Where does the proof of the Brouwer theorem fail. I know that the proof is fail because the torus has a hole, so we can't construct the retraction to its boundary. I think the only…
Diane Vanderwaif
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A doubt from Milnor's "Topology from a Differentiable Viewpoint".

This is a doubt from Milnor's "Topology from a Differentiable Viewpoint". For a smooth $f:M\to N$, with $M$ compact, and a regular value $y\in N$, we define $n(f^{-1}(y))$ to be the number of points in $f^{-1}(y)$. It is easy to see that…
user67803
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$S^2 \times S^2$ is diffeomophic to $G_2(\mathbb{R}^4)$

$G_2(\mathbb{R}^4)$ is the Grassmannian manifold of two-dimensional subspaces of $\mathbb{R}^4$. I would like a detailed proof. Can it be done explicitly? I mean, showing the map and checking its differentiability using charts?
Ugo Iaba
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Why do we integrate 1-forms?

So integration of a 1-form $\omega$ over a path $\gamma$ is defined to be the integral of the pullback of $\omega$. Why does this make sense? Why don't we integrate over a vector field instead, like in vector calculus, and define integration of a…
user031849012903123
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