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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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Why the matrix of $dG_0$ is $I_l$.

I am reading the proof of Local Immersion Theorem in Guillemin & Pallock's Differential Topology on Page 15. But I got lost at the following statement: Define a map $G: U \times \mathbb{R}^{l-k} \rightarrow \mathbb{R}^{l}$ by $$G(x,z) = g(x) +…
WishingFish
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The mod 2 degree of a function when the image space N has a boundary

I was flipping through Milnor's "Topology from the Differentiable Viewpoint," and I came upon a sentence concerning the mod 2 degree of a function from M to N. It essentially says: "We may as well assume also that N is compact without boundary, for…
user8946
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intersection number of a zero dimensional manifold

I am working on the following problem in p.83 of Guillemin-Pollack: Prove that intersection theory is vacuous in contractible manifolds: if $Y$ is contractible and $\dim Y > 0$, then $I_2 (f, Z) = 0$ for every $f: X \to Y$, $X$ compact and $Z$…
bvbrst
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Problem 5 of Milnor's "Topology From The Differentiable Viewpoint"

I'am trying to come up with a solution to the referred problem which, by the way, states the following: If $m
user255306
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Topology on function space between smooth manifolds

I'm having trouble reconciling my intuitive notion of "functions close to $f$" with the seemingly very technical definition of the weak and strong topologies on $C^r(M,N)$ in Hirsch's Differential Topology. Hirsch's definition of the weak…
whippedcream
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Exercise $2.1.1$ - Differential Topology by Guillemin and Pollack

If $U \subset \mathbb{R}^k$ and $V \subset H^k$ are neighborhoods of $0$, prove that there exists no diffeomorphism of V with U. Here, $H^k$ is simply the upper half-space. I tried to solve this problem with the continuity, but it is a dead-end. I…
user230283
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Is a cube a smooth manifold?

Is the unit square $\partial I^2$ (i.e. the square with vertices $(0,0), (0,1), (1,0), (1,1) \in \mathbb R^2$) a smooth manifold? I guess it shouldn't be smooth because it has "corners", but i have trouble actually finding an explicit atlas which…
Benno
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Show that if $X$ may be deformed into $Z$ then $X$ and $Z$ are cobordant.

Show that if $X$ may be deformed into $Z$ then $X$ and $Z$ are cobordant. Deformation definition: deformation of a submanifold $Z$ in $Y$ is a smooth homotopy $i_t:Z\to Y$ where $i_o$ is the inclusion map $Z\to Y$ and each $i_t$ is an…
XiaoXiao Zhen
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Prove that $\deg_2 (f) \equiv q \mod 2$

Let $f:S^1→S^1$ be any smooth map. There exists a smooth map $g:\mathbb R \to \mathbb R$ such that $f(\cos(t),\sin(t) )=(\cos(g(t)),\sin(g(t) )$ and satisfying $g(2π)=g(0)+2πq$ for some integers $q$. Prove that $\deg_2 (f) \equiv q \mod 2$ Mod 2…
XiaoXiao Zhen
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Prove that there exists a diffeomorphism from an open neighborhood of $Z$ in $N(Z;Y)$ onto an open neighborhood of $Z$ in $Y$.

Prove that there exists a diffeomorphism from an open neighborhood of $Z$ in $N(Z;Y)$ (normal bundle of $Z$ in $Y$) onto an open neighborhood of $Z$ in $Y$. $\epsilon$ neighborhood theorem: For a compact boundaryless manifold $Y$ in $R^M$ and…
Diane Vanderwaif
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Definition of topological manifold

This might be a stupid question, but I was wondering why we define the topological manifold to be Hausdorff and Second countable? Thanks :)
user181662
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Prove that$ H_x (X)$ does not depend on the choice of local parametrization.

Suppose that $X$ is a manifold with boundary and $x∈∂X$. Let $ϕ:U→X$ be a local parametrization with $ϕ (0)=x$ where $U$ is an open subset of $H^k$. Then $dϕ_0:R^k→T_x (X)$ is an isomorphism. Define the upper half space $H_x (X)$ in $T_x (X)$ to be…
Diane Vanderwaif
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why is there no non-degenerate 2-forms on 4-sphere?

The question is in the title. I have been told that there are actually no non-degenerate 2-forms on $S^{2n}$ for $n \neq 1,3$. I have found the following question: No symplectic structure on $S^{2n},\ n>1$ but it only eliminates the possibility of…
Dima Sustretov
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What is a coordinate function $x^i$ of a manifold, given a chart $(U,x)$?

I am trying to understand the notes here: http://unapologetic.wordpress.com/2011/04/13/cotangent-vectors-differentials-and-the-cotangent-bundle/. Specifically, this sentence: If we have local coordinates $(U,x)$ at $p$, then each coordinate function…
Jean Valjean
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Question about a lemma from Milnor's Topology from the Differentiable Viewpoint

I've been reading John W. Milnor's Topology from the Differentiable Viewpoint for some time and currently I'm stuck at a little lemma. I would appreciate if someone can clarify it to me. The details are as follows: Define $H^m = \{(x_1,\ldots, x_m)…
Sayantan
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