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13
votes
2answers
454 views

Are there results from gauge theory known or conjectured to distinguish smooth from PL manifolds?

My question begins with a caveat: I sometimes spend time with topologists, but do not consider myself to be one. In particular, my apologies for any errors in what I say below — corrections are ...
10
votes
3answers
548 views

Characterizing Hessians among symmetric bilinear tensors

I apologize in advance if this is somewhat elementary, but: Let $(M,g)$ be a compact Riemannian manifold. Is there a "characterization" of which symmetric bilinear tensors $B\in Sym^2(M)$ are ...
0
votes
1answer
215 views

special Lagrangian n-Torus has Tubular neighbourhood?

Let $\imath :T^{n}\rightarrow X$ is a special Lagrangian n-Torus so that $\imath(T^{n})=L$ and all small special Lagrangian deformations of $L$ are flat then why $L$ has Tubular neighbourhood which ...
2
votes
2answers
397 views

When a Riemannian manifold is of Hessian Typ

When a Riemannian manifold is of Hessian Type (i.e., a Riemannian manifold which its metric is Hessian)
2
votes
0answers
63 views

Are there a group of mappings from (n-1)-dim space to an (n-1)-sphere guaranteeing the orthogonality of images?

Hello, everyone. As we know that in an $n$-dimensional Euclidean space $\mathbb{R}^n$, there exists a continuous bijective mapping from a subset $V^{n-1}\subseteq\mathbb{R}^{n-1}$ to a unit ...
30
votes
3answers
2k views

Connected sum of topological manifolds

A definition of the connected sum of two $n$-manifolds $M$ and $M'$ begins by considering two $n$-balls $B$ in $M$, $B'$ in $M'$, and glueing the varieties $M\setminus \mathring B$ and $M'\setminus ...
11
votes
2answers
931 views

homotopy type of embeddings versus diffeomorphisms

Previously, I asked a question on mathoverflow comparing smooth embeddings and diffeomorphisms, which received a very interesting and somewhat unexpected answer by Agol. I now ask a further question ...
12
votes
1answer
458 views

Essential uniqueness of the real-analytic structure on $\mathbb R$

It is well-known that any $C^k$-smooth $1$-manifold homeomorphic to $\mathbb R$ is $C^k$-diffeomorphic to $\mathbb R$. The cases of $k\in{\mathbb N}\cup$ {$\infty$} may all be handled similarly by ...
4
votes
1answer
622 views

an extended question of Gromov: Every **generalized open almost complex manifold** admits a **generalized symplectic structure**? [closed]

Definition (Open Manifolds):An open manifold is a manifold without boundary with no compact component. For a connected manifold, "open" is equivalent to "without boundary and non-compact. we know that ...
10
votes
1answer
215 views

orbit space of a topological manifold

Given a compact Lie group G acting freely on a topological manifold M, is it true that the orbit space M/G is also a topological manifold? If so, why?
15
votes
2answers
471 views

If $X$ is a simplicial complex, is their a characterization of the links of the vertices of $X$ that is equivalent to the statement "$|X|$ is a manifold

We have a characterization when we want $|X|$ to be a PL-manifold, in particular that the links of all the vertices are themselves (PL) spheres. If we are in the category of PL- spaces then this is a ...
8
votes
1answer
407 views

rational homotopy of a manifold

Given a finite dim rational homotopy type satisfying Poincaré duality, what is the best reference to when it is the rational homotopy type of a fin dim manifold?
15
votes
2answers
727 views

Are homeomorphic open subsets of $\mathbb{R}^n$ also diffeomorphic?

Let $U_1, U_2$ be open subsets of $\mathbb{R}^n$. Both are naturally differentiable submanifold, getting the differentiable structure from $\mathbb{R}^n$. Further, both are natural topological ...
0
votes
1answer
175 views

Length of intersection of intervals

Can anyone prove this statement? It seems true, but I'm finding it tricky to give a concise proof. Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. Define $B(c,r)\equiv[c-r,c+r]$, where $[\cdot, ...
4
votes
1answer
353 views

What manifolds can have a (non-piecewise) linear structure?

By the definition I'm using, all manifolds are Hausdorff and second countable. For all non-negative integers $n$, I define $B_n$ to be $\bigl\{ \mathbf{v} \in \mathbf{R}^n : \lVert\mathbf{v}\rVert ...
6
votes
1answer
512 views

ANOTHER Exterior differential system on $SO(3;\mathbb R) \times \mathbb R$

I have another exterior differential system for one forms $U^i$, where the $\theta^i$ are a cotangent basis on $SO(3)$, i.e. they satisfy $d \theta^i = \epsilon_{ijk} \theta^j \wedge \theta^k$ for the ...
12
votes
1answer
564 views

The geometry of crinkled aluminum foil

I wonder if the geometry of crinkled aluminum foil has been studied?            The above is a photo of foil I flattened to reuse. It might be ...
1
vote
1answer
177 views

Exterior differential system on $SO(3;\mathbb R) \times \mathbb R$

I have the following exterior differential system for one forms $\alpha, \beta, \gamma$, where the $\theta^i$ are a cotangent basis on $SO(3)$, i.e. they satisfy $d \theta^i = \epsilon_{ijk} \theta^j ...
7
votes
0answers
137 views

PL surface projections - is there a theory of folds and cusps?

For smooth surfaces, the generic singularities of a map of one surface to another are folds and cusps (Whitney). It is a standard result in singularity theory that the generic isotopy of such a map is ...
0
votes
2answers
385 views

Interpolating a “manifold” between two points

Edit: I have reworded the question. This may be a basic question but I am having trouble figuring out the correct answer. I want to find a local coordinate chart that fits a d-dimensional ...
11
votes
1answer
586 views

Manifolds with prescribed fundamental group and finitely many trivial homotopy groups

Fix $G$, a finitely generated presented group. It is known that for every $k > 3$ there is a closed $k$-manifold whose fundamental group is $G$. Similarly, there is a topological space with ...
3
votes
1answer
260 views

Smooth functions tangent to the leaves of a foliation

Given two smooth manifolds $M$ and $N$, it is known that if $M$ is compact, then $C^\infty(M,N)$ is a Fréchet manifold whose tangent space at $f \in C^\infty(M,N)$ is the space $$T_f C^\infty(M,N) = ...
1
vote
1answer
300 views

When are $k$-sectors of a Lie groupoid a manifold?

Let ${\mathcal{G} = \lbrace s,t:G_1 \to G_0 \rbrace}$ be a Lie groupoid. Define $$(\mathcal{G}^k)_0:=\lbrace (a_1,\dots,a_k) \in G_1^k\mid s(a_1)=t(a_1)=\dots=s(a_k)=t(a_k) \rbrace$$ (This is the ...
5
votes
2answers
328 views

Vector space structure on velocity space of manifold

Let $M$ be $C^{\infty}$-manifold and $x\in M$. We define $(k,r)$-velocity space at x as $(T_k^rM)_x:=J_0^{r}(\mathbb{R}^k,M)_x$.Can we define vector space structure on $(T_k^rM)_x$?
10
votes
2answers
538 views

When do blowups ''commute''?

Let $M$ be a manifold (variety, scheme, your favorite object) and let $N_1,N_2$ be two submanifolds (subvarieties, closed subschemes, ideal sheafes, etc.) such that $N_1 \cap N_2 \neq \emptyset$. ...
21
votes
2answers
1k views

Non-regular Connected Hausdorff Banach Manifold

After reading this MO post, I am wondering: Is every (connected) Hausdorff Banach manifold a regular space? Though unjustified, page 53 of this paper nonchalantly states: "Note that a Hausdorff ...
4
votes
0answers
179 views

Mean number of $n$-simplices per $(n-2)$-simplex in a triangulated $n$-manifold

Work by Tamura (extending results by Luo and Stong) shows the following. Theorem: For any closed 3-manifold $M$ and any rational number $4.5 < r < 6$ there is a triangulation $T$ of $M$ for ...
6
votes
1answer
260 views

What is known about the distribution of average edge-degrees for 3-manifold triangulations (with the number of 3-simplices less than a fixed constant)?

This is my first question on mathoverflow! It relates to a project I'm undertaking with a student. Work by Tamura (extending results by Luo and Stong) shows that for any closed 3-manifold $M$ and any ...
8
votes
3answers
721 views

Proving the existence of good covers

Usually one proves the existence of good covers in compact manifolds by Riemannian methods: we pick an arbitrary Riemannian metric, prove that geodesically convex neighborhoods exist, that they are ...
0
votes
1answer
453 views

sign of the First chern class fundamental group of Kahler Manifolds

We know by some facts from Kobayashi, if the Kahler manifold $M$ has positive first Chern class, i.e., $c_1 (M)>0$ then $M$ is simply connected. So if $c_1 (M)<0$ under which assumption on $M$ ...
17
votes
1answer
751 views

isotopy inverse embeddings vs. diffeomorphisms

I would like to find an example, if one exists, of manifolds $M$ and $N$ with embeddings $f:M\to N$ and $g:N\to M$ such that $f\circ g$ and $g\circ f$ are both isotopic (i.e. homotopic through ...
0
votes
1answer
268 views

Is it possible to classify the boundaries of a manifold?

The open ball is a manifold, and the closed ball is a compact manifold-with-boundary which extends the open ball, in which the open ball is dense, in which all the new points are boundary points. Is ...
5
votes
2answers
571 views

Pursuit-Evasion on a Manifold

I know pursuit-evasion has been studied in many contexts, including on a manifold (e.g., Melikyan, "Geometry of Pursuit-Evasion Games on Two-Dimensional Manifolds"), but I have not seen this version: ...
11
votes
0answers
480 views

Codimension Two Embeddings in Goodwillie-Weiss Manifold Calculus, and the Difficulty of Fundamental Groups

In manifold calculus, there are various analyticity estimates which run into trouble for codimension two embeddings. For instance, the functor $\operatorname{Emb}(M,N)$ is analytic in $M$ if $\dim M ...
31
votes
6answers
3k views

Status of PL topology

I posted this question on math stackexchange but received no answers. Since I know there are more people knowledgeable in geometric and piecewise-linear (PL) topology here, I'm reposting the question. ...
14
votes
3answers
3k views

What is the difference between holonomy and monodromy?

And what is the simplest example in which one is trivial and the other is not?
12
votes
6answers
878 views

Does every vector bundle allow a finite trivialization cover?

Suppose there is a vector bundle (smooth, with constant rank finite-dimensional fibres) over a (smooth, second-countable, Hausdorff, not necessarily connected) manifold $B$ of dimension $n$. (a) Is ...
8
votes
2answers
734 views

When is the connected sum of manifolds orientation-independent?

Given $M$ and $N$, two connected orientable manifolds of the same dimension, when is $M$ # $N$ diffeomorphic to $M$ # $\overline{N}$, where $\overline{N}$ is $N$ with the orientation reversed? If $N$ ...
4
votes
5answers
725 views

Good overview of singularity theory

Can anyone recommend a good overview of singularity theory? In particular, quotient singularities... Thanks!
5
votes
1answer
360 views

Triangulation of Surfaces without Jordan-Schoenflies

Does anyone know of a proof of the fact that any 2-manifold can be triangulated that does not use the Jordan-Curve Theorem or the Jordan-Schoenflies Theorem? Thanks for your help
6
votes
3answers
940 views

When is a finite cw-complex a compact topological manifold?

I think the statement of the question is pretty straightforward. Given a finite $n$-dimensional CW complex, are there necessary and sufficient conditions for determining that it is also a compact ...
2
votes
1answer
125 views

Decomposition of straight line between points on a manifold

In an article by Lubich, I came across a decomposition for points on the straight line between two points lying in an embedded submanifold $M$ of $R^{n}$. To be precise, it is proposed that for $X$, ...
13
votes
2answers
638 views

Uniqueness of compactification of an end of a manifold

Let $M$ be an $n$-dimensional manifold (smooth or topological). I call $\bar{M}$ a compactification of $M$ if it is an $n$-dimensional compact manifold with boundary $\partial \bar{M}$, an ...
1
vote
2answers
323 views

epsilon-Manifold with curvature at one point

I remember briefly hearing about this notion (stated in the title), of a manifold where there is a nonzero curvature at precisely one point (a delta-function distribution), and such that there is a ...
0
votes
0answers
335 views

Locally flat submanifold

Recently I found the next definition: Let $M^n$ be an $n-$dimensional topological manifold. Then $N^k\subseteq M^n$ is a locally flat submanifold if for every $x\in N$ there exists an open set $U$ in ...
2
votes
1answer
334 views

Symbol map in Getzler calculus

I hope someone can help me, although this question is rather specific. I am reading John Roe's chapter on Getzler symbols in "Elliptic operators, topology and asymptotic methods" to understand the ...
11
votes
1answer
541 views

Stable normal bundle of a manifold

Hi, in bordism-theory and many bordering areas one has the following construction: Given a manifold M (say closed for the purposes of this discussion and k-dimensional), we embed it into some ...
3
votes
1answer
407 views

Cartan-Weil model for Equivariant Cohomology

Let $\mathfrak{g}$ be the Lie algebra of a Lie group $G$ which acts on a manifold $M$. It is quite standard that the basic forms in $\Omega^*(M) \otimes W(\mathfrak{g}^*)$ form a model for the ...
1
vote
2answers
235 views

Elements of finite order in mapping class groups of high dimensional manifolds

Let $M$ be a manifold with boundary. Consider the following groups: (1) $\pi_0(\operatorname{Diff}(M,\partial M))$. (2) $\pi_0(\operatorname{Homeo}(M,\partial M))$. (3) ...
5
votes
4answers
2k views

Examples of non-simply connected manifolds with trivial H^1

It is known that, if a topological space is simply connected,its first homology group vanishes. The converse is not true, since for every presentation of a (say, finite) perfect group G we can ...