The manifolds tag has no usage guidance.

**27**

votes

**2**answers

785 views

### good covers of manifolds

It is well-known and easy to prove (see for instance this post) that every smooth manifold admits a "good cover", i.e. a locally finite cover by open balls such that all nonempty intersections of the ...

**2**

votes

**1**answer

221 views

### homology of punctured manifolds

Let $M$ be an $n$-dimensional closed manifold. Choose $x \in M$. Using long exact sequence of pairs $(M,M - x)$, we have
$$H_k(M - x, \mathbb{Z}) \cong H_k(M, \mathbb{Z})$$
for $k<n-1$. For ...

**1**

vote

**0**answers

180 views

### Does there exist any subspace of R^n, homeomorhic to a manifold but not a C^0 submanifold of R^n?

At first I thought that if a subspace of $\mathbb{R}^n$ is homeomorphic to a manifold, then it is a $C^0$ submanifold of $\mathbb{R}^n$. But I found an asterisked exercise in the book Differential ...

**13**

votes

**1**answer

746 views

### How can gauge theory techniques be useful to study when topological manifolds can be triangulated?

I was reading a review article arXiv:1310.7644 and it was explained there that in the last few years it was proven that there are topological manifolds of dimension greater than four that cannot be ...

**1**

vote

**0**answers

68 views

### Tubular neighbourhood which is nowhere piecewise linear

I recently asked this question.
I think, if the following were true, then I would solve my problem.
Let $E\subset\{(x_1,\dots,x_n)\in\mathbb R^n\;|\;x_i\geq 0\, \&\, \sum_ix_i=1\}$ be a convex ...

**15**

votes

**1**answer

604 views

### Are there geometrically formal manifolds, which are not rationally elliptic?

Formality of a space is meant in the sense of Sullivan, i.e. a space $X$ is called formal, if it's commutative differential graded algebra of piecewise linear differential forms $(A_{PL}(X),d)$ is ...

**5**

votes

**0**answers

152 views

### A question about something like “shelling” in a PL manifold

If $P$ and $Q$ are compact codimension zero submanifolds of a PL manifold, say that they meet nicely if $P\cap Q$ is a codimension zero submanifold of both $\partial P$ and $\partial Q$. In ...

**2**

votes

**0**answers

323 views

### What kinds of manifolds admit non-vanishing vector fields defining convergent congruences?

One of the corollaries of the Poincaré–Hopf index theorem is that a closed, connected manifold $M$ admits non-vanishing vector fields iff its Euler characteristic is zero; i.e. $\chi(M) = 0$.
I am ...

**6**

votes

**1**answer

308 views

### Can the monoidal structure on manifolds be strictified?

I'm asking this question purely out of curiosity.
Let $\{M_\alpha\}_{\alpha\in A}$ be a collection of closed smooth manifolds, with exactly one in every diffeomorphism class of closed smooth ...

**1**

vote

**0**answers

80 views

### PL or projective PL map on the links of a PL manifold

Let $M$ be a PL manifold and $f: M\rightarrow M$ be a PL homeomorphism. Suppose that $f(x)=x$ for some vertex $x$. Is the restriction map of $f$ on the links of $x$ also PL? Someone claims that this ...

**9**

votes

**2**answers

643 views

### Reference Request: Compact manifolds with boundary have the homotopy type of a CW-complex

Let $M$ be a compact manifold (possibly non-smooth) manifold with boundary $\partial M$.
Is the inclusion $\partial M\hookrightarrow M$ homotopy equivalent to the inclusion of a subcomplex into a ...

**2**

votes

**0**answers

243 views

### When is the realisation of a simplicial set a manifold?

It is known that a simplicial complex is homeomorphic to a manifold if the link of every vertex is a simplicial sphere, for which there exists a definition. (I know that for high dimension, the ...

**9**

votes

**3**answers

350 views

### Characteristic polynomials of trees and E8

In thinking about constructing manifolds via surgery or plumbing, the following combinatorial problem comes up:
If T is a tree with adjacency matrix A and I is the identity matrix of the same order, ...

**5**

votes

**3**answers

277 views

### Constructing a vector field with given zeros on a torus

By the Hopf-Poincaré theorem, the sum of the indices of the zeros of a vector field on the d-dimensional torus must equal zero. Given an even number of points $x_i$ on a d-dimensional torus, and ...

**7**

votes

**0**answers

228 views

### Reference Request: Topological h-cobordism theorem in higher dimensions

I think this question on math.stackexchange is more appropriate on mathoverflow. Correct me, if you don't think so.
The h-cobordism theorem is true in the topological and in the smooth category in ...

**7**

votes

**1**answer

234 views

### High dimensional generalized Poincare hypothesis without the h-cobordism theorem?

The generalized PL Poincare hypothesis states that in dimension $n$ there is a unique PL manifold that has the homotopy-type of $S^n$. It's known to be true in all dimensions except perhaps $n=4$.
...

**17**

votes

**2**answers

609 views

### Distinct manifolds with the same configuration spaces?

For a space $X$, let $C_k X$ denote the space of configurations of $k$ distinct unordered points in $X$.
What is an example of a pair of smooth manifolds $M$ and $N$ that are not homeomorphic but ...

**3**

votes

**1**answer

500 views

### How is the notion of a Lipschitz structure on a manifold defined?

According to wikipedia, there is such a definition. $\:$ The candidate that I can come up with is
"an equivalence class of metrics that induce the topology and make the space locally bi-Lipschitz
to ...

**-2**

votes

**1**answer

225 views

### A diffeomorphism between complex manifolds which is not a holomorphic map [closed]

Can someone give an example or a reference on this?

**1**

vote

**0**answers

90 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 ...

**1**

vote

**0**answers

224 views

### On tangent spaces of Stiefel Manifolds

I was trying to read Edelman et al.'s 1998 paper "The Geometry of Algorithms with Orthogonality Constraints" and, since I don't have any differential geometry or much linear algebra background, I am ...

**2**

votes

**2**answers

180 views

### Preimage of $1 \in H^n(M^n)$ under Chern character

Let $M$ be a closed, oriented manifold of dimension $n$. We know that the Chern character induces an isomorphism $K^\ast(M) \otimes \mathbb{Q} \cong H^\ast(M; \mathbb{Q})$ and now I was wondering how ...

**4**

votes

**0**answers

161 views

### Haken manifolds and characterising sutured manifold hierarchies

In Gabai's paper (Knot Theory and Manifolds Lecture Notes in Mathematics Volume 1144, 1985, pp 14-17 An internal hierarchy for 3-manifolds) he considers sutured manifold decompositions of Haken ...

**1**

vote

**1**answer

215 views

### On the canonical neighborhoods

Suppose $M$ is a 3-dimensional manifold, John W. Morgan and Frederick Tsz-Ho Fong in their "Ricci Flow
and Geometrization
of 3-Manifolds" book as a definition of canonical neighborhoods have ...

**3**

votes

**2**answers

980 views

### Visualization of the real projective plane [closed]

Consider a closed (compact and without boundary) and non-orientable 2-manifold $M$. By Whitney embedding theorem, one can embed $M$ in $\mathbb{R}^4$. $M$ cannot be embeded in $\mathbb{R}^3$ and just ...

**5**

votes

**0**answers

230 views

### Embedding tower in low codimension

If $F$ is a suitably nice functor from manifolds to spaces, it has a degree $k$ "polynomial" approximation $T_k F$ in the sense of embedding calculus. We set $T_\infty F := \mathrm{holim} T_k F$.
The ...

**14**

votes

**3**answers

877 views

### Does a *topological* manifold have an exhaustion by compact submanifolds with boundary?

If $M$ is a connected smooth manifold, then it is easy to show that there is a sequence of connected compact smooth submanifolds with boundary $M_1\subseteq M_2\subseteq\cdots$ such that ...

**11**

votes

**4**answers

2k views

### geometric interpretation of Lie bracket

On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written:
We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to
which the integral ...

**0**

votes

**1**answer

157 views

### Does some type of curvature require the space be an embedded manifold in a higher-dimensional space? [closed]

Assuming an appropriate definition of 'curvature', is there a theorem that says:
"At least n+1 dimensions are necessary for a particular curvature to exist in an n-dimensional space, and the n-space ...

**5**

votes

**1**answer

708 views

### About Sectional Curvature [closed]

In a paper by Yann Ollivier:
Let $x$ be a point in $X$, $v$ a small tangent vector at $x$, $y$ the endpoint
of $v$, $w_x$ a small tangent vector at $x$, and $w_y$ the parallel transport of $w_x$ from ...

**3**

votes

**1**answer

118 views

### A k-form is thought of as measuring the flux through an infinitesimal k-parallelepiped

On the wikipedia has written "A $k$-form is thought of as measuring the flux through an infinitesimal $k$-parallelepiped." How does a $k$-form do this? if this sentence is right, then the flux of ...

**0**

votes

**0**answers

229 views

### A question from Hamilton's Ricci Flow book by bennett chow

On page 3 of the book before exercise 1.2, is written: "torsion free is a compatibility condition with the differentiable structure". I correctly do not understand how torsion-free condition results ...

**7**

votes

**1**answer

549 views

### Cancellation law for $M^n\times \mathbb R= N^n\times \mathbb R$.

Assume $M^n$ and $N^n$ are null bordant, i.e. each can be realized as boundary of an $n+1$ dimensional manifold. Suppose $M^n \times \mathbb R$ is homeomorphic to $N^n\times \mathbb R$. Is there any ...

**14**

votes

**2**answers

487 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 ...

**12**

votes

**3**answers

844 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 ...

**1**

vote

**1**answer

227 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

**2**answers

606 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**

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**0**answers

67 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 ...

**33**

votes

**3**answers

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 ...

**12**

votes

**2**answers

1k 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 ...

**13**

votes

**1**answer

504 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

**1**answer

698 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

**1**answer

238 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

**2**answers

634 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 ...

**9**

votes

**1**answer

450 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?

**18**

votes

**2**answers

911 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

**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, ...

**4**

votes

**1**answer

375 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

536 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**

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**1**answer

601 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 ...