The manifolds tag has no usage guidance.

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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299 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) = ...

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### 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 ...

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### 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 $(T_k^rM)_x$?

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

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### 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 ...

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### 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 ...

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### 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 ...

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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 ...

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### 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 $M$ ...

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### 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 ...

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### 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 ...

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### 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:
...

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### 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 ...

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### 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. ...

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### 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?

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### 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 ...

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### 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 $N$ ...

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### Good overview of singularity theory

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

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### 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

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### 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 ...

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### 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 $X$, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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) ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 = ...

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### 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 ...

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### 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 $x_1$ ...

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### 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 $M$ ...

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### 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 ...

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### Background to learn about manifolds

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, ...

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### uniqueness of regular/tubular neighborhood with equivariant boundary

Let $N$ and $N'$ be regular neighborhoods of a subpolyhedron $P$ in a closed PL manifold $M$, and suppose that $t$ is a free PL involution on $M$ such that each of $\partial N$, $\partial N'$ is ...

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### Numbers associated with boundaries of manifolds

I don't know what name if any is attached to the numbers I'm about to describe.
For a line segment, [a,b]
the number is 1 if for any k in (a,b)
and 2 if k=a or k=b.
For a square, [a,b] ...

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### a “homological dimension” for embedding of manifolds

Let $A\to B$ be a surjective map of commutative $k$-algebras, and suppose $C\to B$ is a free resolution of $B$ as an $A$-algebra, meaning that $C$ is a free non-negatively graded commutative ...

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### Fragmenting a homeomorphism of a compact manifold

Let $M$ be a compact manifold and let $f : M \rightarrow M$ be a homeomorphism which is isotopic to the identity. We will say that $f$ can be fragmented if it satisfies the following property. Let ...

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### Riemannian metrics on non-paracompact manifolds

After proving the existence of Riemannian metrics on manifolds, one of the students asked if the "paracompactness" is necessary. Of course the standard proof with the partition of unity
uses this ...