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”.

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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 ...
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4 votes
1 answer
448 views

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 ...
Igor Belegradek's user avatar
-1 votes
1 answer
469 views

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).
Abdullah Almariah's user avatar
9 votes
1 answer
545 views

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 ...
aglearner's user avatar
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1 vote
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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 $...
Spanky's user avatar
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10 votes
4 answers
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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^...
Nicholas Proudfoot's user avatar
2 votes
2 answers
303 views

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 $C^\infty(S^1,\mathbb R^n)$ ...
Jonujohn's user avatar
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1 vote
0 answers
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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 ...
J. GE's user avatar
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0 votes
4 answers
451 views

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{...
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39 votes
10 answers
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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 ...
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0 answers
301 views

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}^...
Pedro Teixeira's user avatar
4 votes
6 answers
4k views

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 votes
0 answers
407 views

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 $C^1$ ...
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6 votes
1 answer
494 views

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 ...
student's user avatar
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1 answer
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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/~...
Matchmaticians's user avatar
1 vote
0 answers
151 views

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 ...
Yong Pan's user avatar
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0 answers
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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 ...
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7 votes
1 answer
557 views

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, ...
iamamtb's user avatar
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32 votes
2 answers
2k views

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^...
Tim Campion's user avatar
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22 votes
1 answer
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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 ...
Ben McKay's user avatar
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4 votes
2 answers
700 views

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, \...
sife's user avatar
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3 votes
0 answers
105 views

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 ...
Valerie's user avatar
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1 answer
94 views

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 ...
Entaou's user avatar
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1 vote
0 answers
208 views

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) ...
Joseph O'Rourke's user avatar
4 votes
2 answers
339 views

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 ...
student's user avatar
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4 votes
0 answers
146 views

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 \...
Somnath Basu's user avatar
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2 votes
1 answer
342 views

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{...
Mostafa's user avatar
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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^...
behrad mahboobi's user avatar
1 vote
0 answers
92 views

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}...
Vladimir's user avatar
  • 359
3 votes
1 answer
454 views

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 $\partial X = M$ ...
Ryan Thorngren's user avatar
1 vote
2 answers
398 views

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 ...
mirta's user avatar
  • 29
12 votes
1 answer
840 views

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 ...
Matthias Kreck's user avatar
1 vote
0 answers
176 views

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 ...
John's user avatar
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1 vote
0 answers
194 views

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 ...
Horizonto's user avatar
8 votes
0 answers
213 views

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 ...
aglearner's user avatar
  • 14k
2 votes
1 answer
201 views

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 ...
Adam's user avatar
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6 votes
3 answers
539 views

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 (...
Somnath Basu's user avatar
  • 3,403
9 votes
1 answer
905 views

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 ...
Ritwik's user avatar
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5 votes
0 answers
333 views

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)+...
PepeToro's user avatar
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7 votes
3 answers
1k views

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 ...
sara's user avatar
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5 votes
2 answers
520 views

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?
Issam Ibnouhsein's user avatar
0 votes
1 answer
464 views

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+...
Shuchang's user avatar
  • 270
2 votes
3 answers
429 views

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 ...
Jérémy Blanc's user avatar
7 votes
1 answer
559 views

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 ...
AFK's user avatar
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8 votes
2 answers
557 views

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 ...
Misha's user avatar
  • 143
4 votes
1 answer
239 views

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(\...
Anthony Quas's user avatar
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10 votes
1 answer
478 views

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 ...
Stephan Mescher's user avatar
3 votes
1 answer
231 views

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, $f: M \rightarrow \mathbb{R}^n$. ...
Danny Brown's user avatar
0 votes
0 answers
147 views

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,...
Karthik C's user avatar
  • 261
3 votes
2 answers
352 views

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 ...
yangyang's user avatar
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