**0**

votes

**1**answer

188 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, ...

**7**

votes

**1**answer

437 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

**1**answer

540 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

**1**answer

606 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

**1**answer

199 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

**0**answers

157 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

**2**answers

512 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 ...

**13**

votes

**1**answer

742 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

**1**answer

303 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

**1**answer

329 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

**2**answers

367 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 $(T_k^rM)_x$?

**12**

votes

**2**answers

637 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 $N_1 \cap N_2 \neq \emptyset$. ...

**22**

votes

**1**answer

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

**0**answers

190 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

**1**answer

280 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

**3**answers

1k 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

**1**answer

482 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 $M$ ...

**18**

votes

**1**answer

816 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 ...

**1**

vote

**1**answer

281 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

**2**answers

592 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:
...

**13**

votes

**0**answers

513 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 ...

**35**

votes

**6**answers

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. ...

**30**

votes

**3**answers

5k views

### What is the difference between holonomy and monodromy?

What is the difference between holonomy and monodromy?
And what is the simplest example in which one is trivial and the other is not?

**14**

votes

**6**answers

1k 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

**2**answers

915 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

**5**answers

756 views

### Good overview of singularity theory

Can anyone recommend a good overview of singularity theory? In particular, quotient singularities...
Thanks!

**5**

votes

**1**answer

411 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

**10**

votes

**4**answers

2k 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

**1**answer

136 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

**2**answers

735 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 ...

**2**

votes

**2**answers

335 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 ...

**4**

votes

**0**answers

189 views

### Manifolds with a lower degree of regularity

I've been reading a paper about regularity theory for a P.D.E in a non-smooth domain(see the reference below).
There, the authors consider domains of $R^n$ with regularity of class $W^2 ...

**0**

votes

**0**answers

442 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

**1**answer

398 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

**1**answer

640 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

**1**answer

546 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 ...

**32**

votes

**3**answers

2k views

### When is a submanifold of $\mathbf R^n$ given by global equations?

Let $M \subset \mathbf R^n$ be a (smooth) submanifold of dimension $d$. Under which conditions does there exist global equations defining $M$? By global equations I mean : does there exist a smooth ...

**1**

vote

**2**answers

267 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) ...

**6**

votes

**4**answers

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 ...

**1**

vote

**3**answers

468 views

### symplectic form with partition on unity

Assume $M$ is a $2n-$dimensional differentiable manifold. Let $(U_{i})$ be a open covering of $M$. With respect to this covering let $\rho_{i}$ be a partition of unity. Assume that on each $U_{i}$ we ...

**9**

votes

**2**answers

584 views

### Cardinality of connected manifolds

Consider the assertion:
Every connected, but not necessarily paracompact, n-manifold is of cardinality
$2^{\aleph_0}$ (at least assuming the axiom of choice).
For n=1 this may be proved via ...

**24**

votes

**3**answers

2k views

### A “meta-mathematical principle” of MacPherson

In an appendix to his notes on intersection homology and perverse sheaves, MacPherson writes
Why do we want to consider only spaces $V$ that admit a decomposition into manifolds? The intuitive ...

**11**

votes

**2**answers

802 views

### How is the differential in complex cobordism defined?

This is my first MO question...hopefully it's not a bad one...
Background: As a stable homotopy theorist, I like to think of complex cobordism $MU$ as a ring spectrum. If I needed to get my hands ...

**7**

votes

**3**answers

399 views

### Question concerning h-cobordisms

Suppose we have a cobordism $W$ of manifolds $M_0$ and $M_1$ and suppose the inclusion of $M_0$ into $W$ is a homotopy equivalence. Is the same true for the inclusion of $M_1$ (ie. is $W$ already an ...

**2**

votes

**2**answers

536 views

### cayley transform for non-square matrices

Hi,
I am optimizing a function over a matrix $U$, where $U \in \mathbb{R}^{m \times n}$ and $U^TU = I$. I do not want to run a constrained maximization program, since employing the constraint $U^TU = ...

**6**

votes

**1**answer

587 views

### fundamental domain of universal covering

Let $M$ be a connected compact manifold without boundary, $\pi:\widetilde{M}\to M$ be the universal covering map. A fundamental domain of $(\pi,\widetilde{M}, M)$ is a compact subset $D\subset ...

**7**

votes

**0**answers

529 views

### Homometric $\Rightarrow$ isometric?

Suppose you know that there is a mapping between
two Riemmanian manifolds $M_1$ and $M_2$ such that,
for each $x_1 \in M_1$, the (codimension-1) measure of the set of points
at distance $d$ from $x_1$ ...

**9**

votes

**1**answer

319 views

### Can an action of a compact Lie group be nontrivial if it is trivial on the boundary?

Let $G$ be a compact Lie group acting on a connected topological manifold $M$ with boundary. Suppose the action on one boundary component is trivial. Does it follow that the action on the whole of $M$ ...

**1**

vote

**1**answer

337 views

### Do bistellar flips preserve shellability?

I notice there is a strong connection between shellability of simplicial complexes and bistellar flips on these complexes; in particular, adding in a new facet of a shelling induces a bistellar flip ...

**1**

vote

**5**answers

3k views

### Background to learn about manifolds [closed]

Greetings
As a necessity to go forward with physics, I find myself in the need to learn about manifolds. Being an engineering student, I don't have the chance to study topology in all its glory.
So, ...