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7 votes
1 answer
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Fibers of generic smooth maps between manifolds of equal dimension

I have heard that the following is a "well-known" Claim. Let $M$ and $N$ be smooth manifolds with equal dimensions and $M$ compact. Then a generic smooth map $f\colon M\to N$ has finite ...
Matthew Kvalheim's user avatar
8 votes
0 answers
229 views
+300

Maps with small fibers between manifolds of equal dimension

The following question is an attempt to revise this one into what I intended. Important revisions are shown in bold. Are there any known examples of a compact Riemannian manifold $M$ with (possibly ...
Matthew Kvalheim's user avatar
11 votes
2 answers
303 views
+500

Cohomology of foliations and closed forms along the leaves

Let $M$ be a manifold equipped with a codimension one, transversely orientable, regular foliation $F \subset M$. Let $\alpha\in \Omega^k(M)$ be a differential form on $M$ that is not closed on $M$ ...
Bilateral's user avatar
  • 2,816
4 votes
0 answers
177 views

Basis of topology on space of properly embedded smooth manifolds

In A Short Exposition of the Madsen-Weiss Theorem, Hatcher discusses (starting at p.6) a basis for a topology on the space $\mathcal{C}^n$, the space of all smooth oriented $d$-dimensional ...
jasnee's user avatar
  • 141
5 votes
0 answers
122 views

Explicit parallelization of an exotic sphere

I asked this question on MathStackExchange a week ago (see here), but, despite a few upvotes, I received no comments or answers. Ideally, I would love a detailed answer, but a yes/no would do the job! ...
User175a23's user avatar
1 vote
1 answer
94 views

Linear combinations of smooth sections

I am dealing with smoothness issues for which I do not even know a successful approach, so any help or reference would be welcome. Let $M$ be a manifold, $E\to M$ a smooth vector bundle, and let $\...
CuriousUser's user avatar
  • 1,452
7 votes
1 answer
201 views

Lipschitz bounds and homotopy groups of diffeomorphism groups

Let $M$ denote a closed Riemannian manifold. Let $\mathrm{Diff}_0^L(M)$ denote the supspace of the identity component of the diffeomorphism group $\mathrm{Diff}_0(M)$ of diffeomorphisms with Lipschitz ...
ThorbenK's user avatar
  • 1,174
0 votes
0 answers
42 views

Fiber-wise mappings composed with projection map $\pi$

Let $M^2=(0,1)^2$. Recall that a chart is a diffeomorphism $\varphi:M^2 \to M^2$. Given a chart $\varphi:(M^2,g_0)\to (M^2,g_0)$ for $g_0$ the Euclidean metric, consider the curves $\varphi^{-1}(u,t)=\...
John McManus's user avatar
2 votes
0 answers
82 views

Is isoperimetric hypersurface unique up to homeomorphism?

Is there a Riemannian structure on $\mathbb{R}^n $with two non homeomorphic compact hypersurfaces $M,N$ such that both satisfy the isoperimetric inequality. I precisely meanthe following: $$\...
Ali Taghavi's user avatar
3 votes
1 answer
203 views

Cohomology of the complex of differential forms with Schwartz coefficients

Let $U$ be an open manifold (say an open subset of $\mathbb{R}^n$ for simplicity). Denote by $\mathscr{S}(U)$ the space of Schwartz functions on $U$. Schwartz functions are defined as usual to be ...
Grisha Taroyan's user avatar
5 votes
1 answer
393 views

No analytic surjection $f:M \to N$ when $\dim(M) >\dim(N)$

Inspired by comment discussions in this MO post smooth version of splitting principle we ask: Are there two compact real analytic manifolds $M,N$ of dimension $m>n$ such that there is not any ...
Ali Taghavi's user avatar
1 vote
0 answers
87 views

A counterexample to the extendibility property for submanifolds

Let $(M,g)$ be a Riemannian manifold. We say that a bounded embedded submanifold $S\subset M$ of class $C^k$ has the extendibility property if there exists a larger embedded submanifold $\tilde{S}\...
trenghia's user avatar
1 vote
0 answers
240 views

Smooth version of the splitting principle

Inspried by this MO question A manifold whose tangent space is a sum of line bundles and higher rank vector bundles we pose the following question as a possible smooth version of the splitting ...
Ali Taghavi's user avatar
4 votes
0 answers
116 views

Do any Legendrian knots in standard contact 3-space have big tubular neighborhoods?

Consider $\mathbb{R}^3$ with the standard contact structure $\ker(dz-y\,dx)$. According to the contact version of Weinstein's theorem, any Legendrian knot $L\subset \mathbb{R}^3$ has a tubular ...
Matthew Kvalheim's user avatar
0 votes
0 answers
133 views

What do associated fibre bundles have in common?

Two fibre bundles are said associated if they have isomorphic associated principal bundles. I understand that this means they are defined by the same transition functions, but still is there some more ...
Lefevres's user avatar
0 votes
0 answers
232 views

What is the adjoint bundle of groups $P\times_{G}G$?

It is said that G acts on itself by conjugation. I am familiar with another type of adjoint bundle in which a representation of G on a vector space is given. Can someone explain the differences and ...
Lefevres's user avatar
5 votes
1 answer
298 views

Schoenflies problem in PL setting

What is the status of the Schoenflies problem in the PL category? In other words, given an injective PL map $f:S^{n-1} \hookrightarrow S^n$, is it always PL equivalent to the equatorial inclusion? (I ...
Victor's user avatar
  • 1,875
3 votes
0 answers
87 views

Are these contact structures on the open solid torus diffeomorphic?

Let $M=S^1\times \mathbb{R}^2$ and $\alpha_1, \alpha_2$ be a pair of contact one-forms on $M$ such that the restrictions $\alpha_1|_{S^1\times \{0\}}$, $\alpha_2|_{S^1\times \{0\}}$ coincide and ...
Matthew Kvalheim's user avatar
6 votes
0 answers
129 views

Are there isospectrally equivalent exotic spheres?

Let $X$ and $Y$ be two different exotic spheres. Are there metrics $g$ and $h$ on $X$ and $Y$, respectively, such that the laplacians of $(X,g)$ and $(Y,h)$ have the same spectrum? I would be happy ...
discretephenom's user avatar
0 votes
0 answers
27 views

Heuristics for constrained maximal volumes in hypercubes as $n \to \infty$

It can be shown that there is a unique maximal surface of revolution with constant positive Gaussian curvature embedded in $[0,1]^3$ with a pair of antipodal points as cone points which attain the ...
John McManus's user avatar
0 votes
1 answer
91 views

On nontrapping manifolds

Suppose that $(M,g)$ is a compact connected smooth Riemannian manifold without boundary. Let $U \subset M$ be a smooth submanifold of codimension zero with smooth boundary and assume that $U$ is ...
Ali's user avatar
  • 4,135
3 votes
1 answer
367 views

Topology and local isometry, spinning cosmic string

Suppose one is given the spacetime $(M,g)$ where $M$ is a fixed differentiable manifold and $g$ is a Lorentzian metric whose local expression is: $$g= -(dt + a \, d \phi)^2 + d\rho^2 + \kappa^2 \rho^2 ...
Bastam Tajik's user avatar
7 votes
2 answers
614 views

Locally conformally flat

Is there any example of a locally conformally flat manifold that is neither a space form nor a product of space forms?
Sayoojya's user avatar
2 votes
1 answer
108 views

Finite group extensions of lattices

I'm currently reading the proof of Geroch's conjecture in Lawson-Michelsohn's Spin Geometry book and in the proof of Proposition IV.5.8 that every Ricci-flat enlargeable manifold is flat the following ...
pizzalberto's user avatar
2 votes
1 answer
122 views

Existence of a spin map from a standard sphere to any closed Riemaninan manifold with nonnegative curvature operator

Let $S^m$ be a standard sphere of dimension $m=n+4k$, and let $M$ be any closed Riemaninan manifold of dimension $n$ with nonnegative curvature operator. My question: Is there always a smooth spin map ...
Radeha Longa's user avatar
2 votes
0 answers
96 views

Differential operators and iterations of tangent bundle

Is there a relationship between higher order differential operators and higher tangent bundle viewed as bundle on the base manifold?
Lefevres's user avatar
0 votes
1 answer
155 views

Vector bundles over a homotopy-equivalent fibration

I think this question is related to what is known as "obstruction theory", but I'm not very familiar with this field of mathematics, so I am asking here. Let $\pi:N\rightarrow M$ be a smooth ...
Bence Racskó's user avatar
2 votes
0 answers
46 views

Under what conditions principal directions define an integrable distribution?

Consider a hypersurface $M^n \subset \mathbb{R}^{n+1}$ which is compact without boundary. Assume that its second fundamental form $A$ has distinct eigenvalues $\lambda_1<\ldots<\lambda_k$ (with $...
Dorian's user avatar
  • 363
0 votes
0 answers
77 views

Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations

Given a smooth nested set of "partial" foliations $\mathcal F_{\alpha}=\big\lbrace e^{\frac{\alpha}{\log x}}: \alpha \in (1/k,k), x\in(0,1),k\in [1,\infty) \big\rbrace$ of $X^2=(0,1)^2$ with ...
John McManus's user avatar
-4 votes
1 answer
328 views

Does a coarser topology lead to a non-Hausdorff topological manifold? [closed]

Take a topological manifold $M$. Suppose one considers a strictly coarser topology than the manifold topology. Can such topology result in a non-Hausdorff topological manifold? NOTE: PLEASE avoid the ...
Bastam Tajik's user avatar
2 votes
1 answer
320 views

How to chart tubes around manifolds with boundary/corners?

Let $M \subset \mathbb{R}^d$ be a manifold with boundary/corners. For example, a piece of curve with endpoints or a $2d$ unit square in $\{ z = 0 \}$. I am interested in introducing local coordinates ...
tsnao's user avatar
  • 620
3 votes
1 answer
200 views

Does the group of compactly supported diffeomorphisms have the homotopy type of a CW complex?

It is known that the group of diffeomorphisms of a compact manifold with the natural $C^{\infty}$ topology has the homotopy type of a countable CW complex. See for instance this thread: Is the space ...
Yasha's user avatar
  • 491
2 votes
0 answers
208 views

Classification of bundles with fixed total space

I am aware of classification theorems for principal bundles, vector bundles, and covering spaces $\pi:E\to B$ over a fixed base space $B$. Principal and vector bundles over $B$ are classified by ...
Matthew Kvalheim's user avatar
1 vote
1 answer
85 views

When is the real Abel-Jacobi-Albanese map injective?

$ \def\tMA{\tilde{M}_a} \def\tomega{\tilde{\omega}} \def\tx{{\tilde{x}}} \def\tzeta{\tilde{\zeta}} \def\T{{\mathbb T}} \def\R{{\mathbb R}} \def\Z{{\mathbb Z}} \def\raw{\rightarrow} $ I want to work ...
Philip Boyland's user avatar
0 votes
0 answers
85 views

Existence of covering space with trivial pullback map on $H^1$

I have seen somewhere the following claim (but can't remember where): let $M$ be a connected orientable closed smooth manifold with $b_1(M)=1$, then there exists a connected covering space $p:\tilde{M}...
F. Müller's user avatar
1 vote
0 answers
153 views

Poincaré-Hopf Theorem for domains with a point of vanishing curvature

Consider $\Omega \subset \mathbb{R}^2$ a convex planar domain having positive curvature on the boundary except for a point $p \in \partial \Omega$ where the curvature vanishes. I would like to know ...
DrHAL's user avatar
  • 111
5 votes
1 answer
415 views

A question about the existence of spin maps

Let $M, N$ be two smooth manifolds, not necessarily spin. My question is the following: How can we construct a non-constant spin map $f:M\to N$ of degree zero? Here spin map means that $f$ preserves ...
Radeha Longa's user avatar
0 votes
1 answer
181 views

Formulating a 3D "twist" transformation for a unit circle into a lemniscate while preserving arc length

I'm exploring the transformation of a 2D unit circle into a lemniscate (infinity symbol) by fixing two antipodal points and "twisting" the circle (in the 3rd dimension) such that the ...
swami's user avatar
  • 375