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Results tagged with differential-topology
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user 99042
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
5
votes
1
answer
390
views
Stable normal bundle and immersions
Corollary 9 in these notes by Ralph Cohen has grabbed my attention.
I do not undestand how to show that if we have a rank $k$ bundle which is stably isomorphic to the stable normal bundle then there …
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5
votes
1
answer
314
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Is identification of double points of an immersion smooth?
Let $f:M^m\to N^n$ be a generic map between smooth manifolds $n>m$. Depending on the pair $(m,n)$ generic maps will have a singular set of double points $\Sigma_2\subset M$.
Let $\phi:\Sigma_2\to \Sig …
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7
votes
0
answers
1k
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Quotient by a non-free action of a Lie group and manifolds with corners
The quotient manifold theorem says that
If $G$ is a Lie group acting freely and properly on a smooth manifold $M$ then $M/G$ has a (unique) smooth structure such that the projection $\pi:M\to M/ …
- 2,533
4
votes
0
answers
192
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Constructing new complex manifolds out of old
It is not difficult to build new manifolds out of old in the smooth category, for example
taking the direct product or constructing a fiber bundle,
taking the level set of a regular value of a smoot …
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6
votes
0
answers
166
views
Elliptic operators with with same index but non homotopic symbols
Let $\mathcal{D}:\Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $k$.
Where $E,F$ are $\mathbb{C}$-vector bundles over $X$, a compact smooth manifold.
In Atiyah-Singer "the index of ellipti …
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4
votes
1
answer
291
views
On the proof of the surgery step in Wall's book
This question regards a part of the proof of the so called surgery step, in Wall's book "surgery on compact manifolds", Theorem 1.1.
Setting
$M^m$ smooth manifold, $X$ CW complex, $\phi :M\to X$ cont …
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0
votes
0
answers
112
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May this slice disk for the unknot be pushed into the boundary?
Write the 4-ball as $\mathbb{D}^4=\mathbb{D}^2\times \mathbb{D}^2$.
Then its boundary $\mathbb{S}^3\simeq \mathbb{S}^1\times \mathbb{D}^2\cup \mathbb{D}^2\times \mathbb{S}^1$. We will use implicitely …
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8
votes
2
answers
585
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Which curves are boundary of pseudoholomorphic curves?
I have posted it on Mathstackexchange but nobody replied.
Consider a loop $\gamma:\mathbb{S}^1\to M^{2n}$ in a symplectic manifold $(M^{2n},\omega)$. Let $J$ be an $\omega$-compatible almost complex …
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3
votes
1
answer
115
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Index formula for elliptic operators acting on Sobolev sections vanishing on the boundary (s...
Given a first order elliptic operator $D:\Gamma(X; E)\to \Gamma(X; F)$ where $X$ is a closed manifold, and $E\to X, F\to X$ vector bundles, we know that $D$ induces a Fredholm operator
between the sp …
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9
votes
2
answers
538
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Rational slice knot that is not slice
Does there exists a knot $K\subset \mathbb{S}^3$ such that
$K$ is not slice
$\exists W^4$, $\partial W = \mathbb{S}^3$ rational homology ball
$\exists $ properly embedded smooth disk $(D,\partial D)\ …
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15
votes
3
answers
2k
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Examples of odd-dimensional manifolds that do not admit contact structure
I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure.
Can someone provide me with some examples?
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7
votes
1
answer
397
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Characteristic classes of the bundle of trace free, skew adjoint endomorphisms
In "Floer Homology groups in Yang-Mills theory", Donaldson says that if we take an $U(2)$-vector bundle $E$ and we construct the bundle $\mathfrak{g}_E$ of trace-free, skew adjoint automorphisms of $ …
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6
votes
1
answer
545
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Akbulut's cork involution
Akbulut's cork is the Mazur manifold $W$ shown in the picture below,
This manifold carries an involution of it's boundary $f:\partial W\to \partial W$ that exchanges a meridian of the 0-framed curve …
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5
votes
1
answer
757
views
Kirby diagrams: sliding 1-handles over 1-handles and ribbon disks
Consider the Kirby diagram $ D$ given by a 2-component unlink, both dotted circles.
In general, when performing a 1-handle slide over another 1-handle, the band chosen must not link any dotted circle …
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3
votes
1
answer
357
views
Homotopy class of maps into Stiefel manifolds
Motivation
Hopf theorem, asserts that $C^0$-maps $f:M^n\to \mathbb{S}^n$ from an orientable, closed n-manifold into an n-sphere are classified up to homotopy by their degree $deg(f)$.
The theorem no …
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