Questions tagged [differential-topology]
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”.
1,741
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An example to show that when $P$ is a complex polarization the subbundle $P+ \bar P$ is not necessarly involutive
I start my question with some motivation. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian
P involutive
dim$P\cap\bar P ...
4
votes
1
answer
448
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Codimension zero embeddings and diffeomorphism groups
Let $V$ be a smooth manifold obtained by attaching the ``open collar'' $[0,1)\times \partial N$ to a compact smooth manifold $N$ along the boundary. Let $\mathrm{Emb}(N, V)$ be the space of embeddings ...
-1
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1
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469
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Differentiable maps between topological spaces [closed]
Is it possible to define differentiable maps between topological spaces without using the idea of manifolds? I mean with using just the topological structure (open sets or neighborhoods).
9
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1
answer
545
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Pontryagin numbers on manifolds with an $S^1$-action
Let $M$ is a smooth compact manifold with an $S^1$-action with isolated fixed points. Suppose the representation of $S^1$ at tangent spaces at all fixed points is known. Can one then find all ...
1
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0
answers
411
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How to show the Euler Characteristic is equal to self-intersection number of zero-section [duplicate]
myThe definition of the Euler characteristic (given in Guillemin and Pollack's "Differential Topology") of a compact oriented manifold $X$ is the self-intersection number of the diagonal $\Delta$ in $...
10
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4
answers
1k
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Cohomology classes represented by submanifolds
Let $Y\subset X$ be a codimension $k$ proper inclusion of submanifolds. If we choose a coorientation of $Y$ inside of $X$ (that is, an orientation of the normal bundle), then we get a class $[Y]\in H^...
2
votes
2
answers
303
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evaluation map $ev_t$ on loop space
Considering parameter of $S^1$ as $t$, we define.
$$ev_t: C^\infty(S^1, \mathbb R^n)\to \mathbb R^n$$
$$ev_t(\gamma):=\gamma(t)$$
I am looking for a possible topology on $C^\infty(S^1,\mathbb R^n)$ ...
1
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0
answers
308
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Smooth structures on quotient space
Suppose $G$ be a discreat group acting on $\mathbb R^n$ freely via two different actions $\rho_1$ and $\rho_2$. Suppose that $\mathbb R^n/\rho_1$ is homeomorphic to $\mathbb R^n/\rho_2$. However the ...
0
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4
answers
451
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An isomorphism on space of smooth sections
Let $M$ be a smooth complex manifold and $L$ be a complex line bundle over $M$. Let $\Gamma(M,L)$ be the space of smooth sections. Why $\Gamma(M,L)$ is it isomorphic to
$$A=\{f:L^{\times}\to \mathbb{...
39
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10
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Are there some other notions of "curvature" which measure how space curves?
I am learning differential geometry and have a few questions on curvature. -- Background:
Gauss invented "Gauss curvature" to measure how surface curves.
Riemann gives an ingenious generalization of ...
3
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0
answers
301
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Diffeomorphism between open annuli preserving common symmetries
Suppose $A$ and $B$ are subsets of $\mathbb{R}^2$ homeomorphic (and thus $C^\infty$ diffeomorphic) to the open annulus (punctured $\mathbb{B}^2$) and let $G$ be the group of isometries of ${\mathbb R}^...
4
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6
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4k
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Good books on Geometric Theory of Dynamical Systems
I am looking for a good book on Geometric Theory of Dynamical Systems . I found Geometric Theory of Dynamical Systems by Jr. Palis myself,but it's very old, anyway i would like to find a pure ...
4
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0
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407
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The $\Omega$-Stability Theorem
I'm currently studying the $\Omega$-Stability Theorem:
Theorem: If $\mathcal{R}(f)$ has a hyperbolic structure then $f$ is $\mathcal{R}$-stable.
Some explanations about the statement: $f$ is a $C^1$ ...
6
votes
1
answer
494
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consequence of Novikov conjecture
Novikov conjeture is a famous open problem in Geometric topology.It predicts that higher signature is oriented-homotopy invariant.
http://en.wikipedia.org/wiki/Novikov_conjecture
I am a student ...
2
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1
answer
341
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(n-1)-dimensional normal currents and Smirnov's paper
I don't know much about currents, but I saw a paper of Smirnov which seems relevant to a problem I am working on. In the very last paragraph of page 848 of the following paper
http://www.unige.ch/~...
1
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0
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151
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Remark in paper by Lima about $\epsilon$-neighborhood type function on a closed manifold
This is a closing remark in a paper of Elon Lima, Separation Theorem for Smooth Hypersurfaces.
He proves a well known lemma: Suppose $M\subset\mathbb{R}^m$ is a compact, orientable, smooth ...
4
votes
0
answers
283
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Looking for a necessary and sufficient condition for the polarization $\mathbb{P}$ being positive
My question is about positivity of polarization in Geometric quantization theory. Let $\mathbb{P}$, be a complex polarization on symplectic manifold $(M,\Omega)$. For every $m\in M$, we can define a ...
7
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1
answer
557
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how to obtain a generalized Morse function out of a fiber bundle?
I already asked this question in MSE but did not get any answer/comment yet.
Let $M\to E\to B$ be a smooth fiber bundle. In "Parametrized Morse Theory and Its Applications,(Proceedings of the ICM, ...
32
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2
answers
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Converse to Stokes' Theorem
Does satisfying Stokes' Theorem imply that a form is linear?
Let $M$ be an $n$-manifold. A differential $k$-form $\omega \in \Omega^k M$ assigns to each point $x \in M$ a function $\omega_x : \Lambda^...
22
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1
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The cone on a manifold
I believe that I have run across the statement that if $X$ is a compact smooth manifold and $CX$ is the cone on $X$, i.e. $[0,1] \times X$ modulo $(0,x)\sim(0,y)$ for all $x,y \in X$, then $CX$ admits ...
4
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2
answers
700
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on Brieskorn Manifolds
Brieskorn showed that $X_{k}=\{(z_0, \dots, z_k)\in
\mathbb{C}^{k+1}| -z_{0}^{3}+\sum_{i=1}^{k} z_{i}^{2}=0\}$(k odd, k>2) is a
topological manifold. Is it a smooth manifold?
In general, let $a_1, \...
3
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0
answers
105
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Analytic stuctures on $\mathbb R^n$ and the nilpotent ideal of supermanifolds
I have two questions which are somewhat related:
(a) It is a well known result (of Freedman?) in differential topology that $\mathbb R^4$ has exotic smooth structures. Apparently, it is known that ...
0
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1
answer
94
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Connecting two hypersurfaces in R^{n+1} by embedded curves
Let $M^n$ be a smooth closed embedded hupersurface in $\mathbb R^{n+1}$.
Denote by $D$ the bounded connected component of $\mathbb R^{n+1}\backslash M$.
We assume that $\mathbb R^{n+1}\backslash D$ is ...
1
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0
answers
208
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Fractional degree of a map?
Is there some natural notion of a fractional degree of a map?
The degree of a map is a generalization of the winding number,
and fractional winding numbers appear in the (mathematical physics)
...
4
votes
2
answers
339
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good reference on brieskorn manifold
I am trying to learn something on the Brieskorn manifold (interested in the topological property)
Can the Mathoverflow Experts give me some good refencece (in English)?
By the way,is there an ...
4
votes
0
answers
146
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Are Sasakian metrics (associated to bumpy metrics) bumpy?
Some background :
(1) Let $(M,g)$ be a smooth Riemannian manifold. Let $LM$ be the free loop space, the space of loops in $M$ of Sobolev class $W^{1,2}$. There is the energy functional $E:LM\to \...
2
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1
answer
342
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Is it possible to approximate an area-preserving diffeomorphism by a sequence of conjugates of periodic rotations?
Is it possible to approximate an area-preserving diffeomorphism $T$ of the disk $\mathbb{D}^2$ by a sequence of conjugates of periodic rotations $B_n^{-1} S_{\frac{p_n}{q_n}} B_n$, where $ S_{\frac{...
1
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1
answer
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Is a convex function continuous and almost everywhere differentiable? [closed]
is this statement true ?
assume $f:D\rightarrow \mathbf{R} $ is a convex function where $D\subset \mathbf{R}^n$ is a convex set. $f$ is continuous and almost everywhere differentiable and in class $C^...
1
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0
answers
92
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Pseudo-Euclidean orbifolds
Are there any papers (reviews) devoted mainly to pseudo-Euclidean orbifolds in mathematics and physics (e.g. string theory)? A more specific question is related to orbifolds of type $\mathbb R^{1,4m-3}...
3
votes
1
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454
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Spin structures and divisibility of cohomology classes
Let me begin with some motivation. In calculating the Chern-Simons invariant of a $U(1)$ connection $A$ on a 3-manifold $M$, we can proceed by picking a bounding 4-manifold $X$ with $\partial X = M$ ...
1
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2
answers
398
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even dimensional manifold homotopic to a symplectic or complex manifold
I have the following question: Let $M$ be an even dimensional Riemannian manifold. Under which conditions does there exists a homotopy to some symplectic manifold? is there any chance that such a ...
12
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1
answer
840
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Embedding of products of projective spaces into a projective space
Does anybody know estimates on the minimal dimension $k$ for which the product $P^n \times P^m$ can be smoothly embedded into $P^k$? I am interested in projective spaces over $\mathbb RR$ and $\mathbb ...
1
vote
0
answers
176
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Proving that two given functionally structured spaces are isomorphic
The relevant definitions are listed below. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups; and Section 2, Chapter II of Bredon's Topology and ...
1
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0
answers
194
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Extension of diffeomorphisms preserving bilateral bounds of the derivatives
Suppose $f$ is a $C^k (1\leq k\leq\infty)$ function from the unit ball $\mathcal{B}$ in $\mathbb{R}^n$ to itself, which is a diffeomorphism from the domain to its image, with the upper and lower ...
8
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0
answers
213
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A classification of smooth $S^1$-actions on $\mathbb CP^3$?
Question 1. Is there a classification of smooth $S^1$-actions with isolated fixed points on the standard $\mathbb CP^3$?
Question 2. What if one additionally imposes the condition that the action ...
2
votes
1
answer
201
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Understanding maps from M to R^n, for n>dim M
I am interested in "approximating" smooth maps from a compact smooth manifold $M$ of dim $m$ into $\mathbb R^n,$ for $n>m,$ by "nice" maps, with properties similar to those of Morse functions. Of ...
6
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3
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539
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Cobordism and finite sheeted covers of manifolds
Let $M$ be an oriented manifold, not necessarily compact. Let $M'$ be a (finite) $k$-sheeted cover and let $\pi:M'\longrightarrow M$ be the covering map.
Question 1 : Is it true that $M'$ is (...
9
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1
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Do partitions of unity exist if we impose additional conditions on the derivatives?
Let $ ~~\cup_{k=-1}^{\infty} U_k = \mathbb{R} $ be an open covering of
$\mathbb{R}$. It is a well known fact that partitions of unity subbordinate to
the cover exists, i.e. there exists smooth
...
5
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0
answers
333
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Stratification of a smooth map
So, this is an exercise. But from math.stackexchange I have been suggested to post this question here.
To find the Thom-Boardman stratification of the smooth map
$f(x,y,a,b,c,d)=x^2y+y^3+a(x^2+y^2)+...
7
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3
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smooth manifolds as real algebraic set (continued)
There are several ways of producing manifolds,say:
1.orbits space of group action
2.connected sum of manifolds
3.underlying topological space of nonsingular algebraic set
....
here,i am ...
5
votes
2
answers
520
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Physical meaning of the integral cohomology condition in Souriau-Kostant pre-quantization?
The question is in the title. The form of the condition looks like the Bohr-Sommerfeld quantization formula of angular momentum, is there a link between the two formulas?
0
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1
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464
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Does Frobenius theorem apply to vector-valued function?
We know Frobenius theorem handle pde systems like
$\{Xf=0, Yf=0\}$
requiring Lie bracket $[X,Y]\equiv 0 \mod X, Y$ for completely integrability of the system. However, how to handle systems like $\{Xf+...
2
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3
answers
429
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Fatou sets and topological entropy
Let us consider a diffeomorphism of a compact real manifold (complex manifold defined over the reals), and let us say that the diffeomorphism is birational. Hence, it extends to a birational map from ...
7
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1
answer
559
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Iterated Milnor fibrations and Thom's a_f condition
Ok so there's a lot of litterature about nearby cycles functor since it was introduced by Grothendieck and Deligne but I couldn't find any clear answer to the following natural question:
Problem: Let ...
8
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2
answers
557
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Exotic 4-manifolds with even positive partial betti number
It seems that usually smooth structures on compact 4-manifolds are distinguished by Seiberg-Witten/Donaldson invariants. And I don't know another direct way to do that. But in the case when $b_2^+$ is ...
4
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1
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239
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dimension of a union of grassmannians
I'm working on a dynamical systems problem and have arrived at a naive question in differential topology? geometric measure theory?
I have a smooth path $\gamma\colon \mathbb R\to\mathcal G_2(\...
10
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1
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478
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Is there an "exponential law" for differentiable maps between smooth manifolds?
Although it seems like a textbook question, I was not able to find a textbook or even a research article answering the following question:
Let $M$, $N$ and $P$ be finite-dimensional smooth manifolds ...
3
votes
1
answer
231
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Lipschitz Approximation to a PW Smooth Map
Suppose I have a triangulated smooth manifold, $\tau : |K| \rightarrow M$ (so that $\tau | _{\sigma}$ is smooth for each $\sigma \in K$), and a piecewise smooth map, $f: M \rightarrow \mathbb{R}^n$. ...
0
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0
answers
147
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Extending a 2-frame field - manifolds with boundary
If $M$ is an orientable compact 3-manifold with boundary such that there is defined $s\in\Gamma(\partial M,V_2(TM))$ a section of the 2-frame bundle of $TM$, then $s$ extends to $\tilde s\in\Gamma(M,...
3
votes
2
answers
352
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intersection of Whitney stratifications
Let $X$ be an oriented smooth manifold with dimension $n$. If $U$ and $V$ are two oriented closed submanifolds of $X$ and $U$ is transverse to $V$ in $X$. Then $U\cap V$ (suppose the intersection is ...