Questions tagged [manifolds]
A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.
534
questions
16
votes
2
answers
918
views
Homotopy equivalent but non-homeomorphic high-dimensional manifolds
I have a question motivated by the classification theory of simply-connected closed $4$-manifolds.
Questions: Given any $n\geq 5$, is it possible to find two simply-connected closed $n$-manifolds $M$ ...
- 6,933
0
votes
3
answers
223
views
Extending $\mathbb{R}$ to a higher dimensional manifold [closed]
If a topological space $X$ is Hausdorff, connected, second countable, homogeneous (i.e. it has transitive homeomorphism group) and embeds the real line $\mathbb{R}$, does it follow that $X$ is a ...
- 11.4k
5
votes
3
answers
809
views
Naturality of Lie bracket — alternate proof
Let $M$ and $N$ be smooth manifolds, and let $F: M \to N$ be a smooth map. Let $X$ and $Y$ be vector fields on $M$, and let $\tilde{X}$ and $\tilde{Y}$ be vector fields on $N$. We say that $X$ and $\...
- 11.4k
9
votes
0
answers
215
views
Existence of $1$-separated and $(1-\varepsilon)$-dense set in metric spaces
Is it know which metric spaces $M$ do have the following property: there is $\varepsilon>0$ and a maximal $1$-separated set which is $(1-\varepsilon)$-dense?
In other words, when does at set $S\...
6
votes
1
answer
238
views
Why does Bott's obstruction theorem imply the vanishing of some cohomology classes of $B\Gamma_q$?
Recall that Bott's obstruction for integrability [Bott70] asserts that:
Given a smooth (=$C^\infty$) $m$-manifold $M$ and a completely integrable vector subbundle $E\subset TM$ of rank $m-q$, every ...
- 1,803
2
votes
0
answers
161
views
When is the closure of a manifold a manifold with boundary?
I'm looking for conditions that ensure the closure of an embedded manifold is a manifold with boundary.
The specific case I'm interested is as follows. Consider an oriented $C^1$ surface ${\cal S}$ in ...
- 5,632
2
votes
1
answer
239
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 ...
- 43.1k
17
votes
3
answers
2k
views
Is symmetric power of a manifold a manifold?
A Hausdorff, second-countable space $M$ is called a topological manifold if $M$ is locally Euclidean. Let $SP^n(M): = \left(M \times M \times \cdots \times M \right)/ \Sigma_m$, where product is done $...
- 11.9k
1
vote
0
answers
96
views
criticality for nonlinear wave equations on manifolds
On $\mathbb{R}^{1+n}$, the initial value problem for the homogeneous wave equation
$$ \Box \phi = \partial_t^2 \phi - \Delta_{\mathbb{R}^n} \phi = 0, \\ (\phi, \partial_t \phi)|_{t=0} = (\phi_0, \...
- 894
0
votes
0
answers
81
views
Example of a metrizable space that is not an ANR
I have been looking for an example of a metrizable space that is not an absolute neighborhood retract (ANR).
Recall that a metrizable space $X$ is called an ANR if there exists an open set $U$ in a ...
- 359
1
vote
0
answers
187
views
Local to global complexity of triangulations
Alright 3rd time's the charm - editing again to put all my cards on the table.
Consider a PL $n$-manifold $M$. Define the complexity $c(M)$ of $M$ to be the minimum number of $n$-simplices needed to ...
- 159
23
votes
2
answers
2k
views
Uniqueness of compactification of an end of a manifold
Let $M$ be an $n$-dimensional manifold (smooth or topological). I call $\bar{M}$ a compactification of $M$ if it is an $n$-dimensional compact manifold with boundary $\partial \bar{M}$, an $(n-1)$-...
- 1,446
6
votes
2
answers
1k
views
Manifolds with negative dimension – Definition, References
Does the concept of differential manifold with negative dimension make sense, in differential geometry?
If yes, how is it defined? Do you have any reference to recommend?
My problem was born in ...
- 36.3k
3
votes
2
answers
413
views
A question on the manifold $ \{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\} $
Consider a manifold $ N $ defined as follows
$$
N=\{n\otimes n-m\otimes m:n,m\in S^2,\quad(n,m)=0\}\subset M^{3\times 3},
$$
where $ S^2 $ denotes the two dimensional sphere, $ (\cdot,\cdot) $ ...
- 9,388
0
votes
0
answers
70
views
Topological transversality by dimension
We know that to achieve transverality in the topological category, for example to make a continuous map into a manifold transverse to a topological submanifold, we need the existence of micro normal ...
- 971
2
votes
0
answers
66
views
Is it known whether a homeomorphism close to the identity of a compact manifold with nonzero Euler characteristic necessarily has a fixed point?
I recently saw in Kirby's list of open problems that it isn't known if two commuting homeomorphisms of a compact manifold close to the identity necessarily share a common fixed point, when the ...
- 21
0
votes
1
answer
111
views
A sufficient condition for a collection of open sets of a manifold to contain all open sets
Question
Let $k\geq 0$ be an integer and let $M$ be a topological $n$-manifold. Let $\mathcal{U}$ be a set of open sets of $M$ which satisfies the following closure properties:
(1). Let $U\subset M$ ...
- 1,803
4
votes
1
answer
426
views
Detecting a "bad map" in Fintushel-Stern knot surgery
Background
Let $X$ be a simply-connected smooth 4-manifold which contains a smoothly embedded torus $T$ with trivial normal bundle (in other words, $T^2\times D^2\subset X$). Let $K$ be a knot in $S^3$...
- 10.5k
6
votes
2
answers
624
views
Removing a submanifold from a closed manifold
Let $M$ be a simply-connected closed manifold. Can we find a closed submanifold $N \subsetneq M$ such that $M\backslash N$ is simply-connected and has finite second homotopy group?
- 43.1k
5
votes
1
answer
330
views
To what extent differentiable mappings of an affine line into a manifold determine its differentiable structure? What about mappings of a plane?
If $M$ is a (real) differentiable manifold, its differentiable structure is completely determined if it is known which mappings $M\to\mathbf{R}$ are differentiable.
How much can be said about the ...
- 36.3k
6
votes
1
answer
295
views
Sobolev embedding theorems on manifolds
I had asked the following question on math.stackexchange but did not get any response:
I'm looking for a reference which states the Sobolev embedding theorems on Riemann manifolds for fractional ...
- 11
8
votes
1
answer
353
views
Given an embedded disk in $\mathbb{R}^n$, is there always another disk which intersects it nontrivially in a disk?
We call an open subset $D\subset X$ of a manifold $X$ an embedded disk, if there exists a homeomorphism $D\cong \mathbb{R}^n$.
The precise formulation of the question in the title is as follows:
Let $...
- 9,749
1
vote
0
answers
81
views
Is there a standard name for the following class of functions on non-Hausdorff manifolds?
Let $M$ be a (not necessarily Hausdorff) smooth manifold. Given an open chart $U\subset M$ and a compactly-supported smooth function $f:U\to\mathbb{R}$ on $U$, define $\widetilde{f}:M\to\mathbb{R}$ by ...
- 5,169
3
votes
1
answer
405
views
Cohomology of finite symmetric products of manifolds
Let $M$ be a closed, orientable manifold of dimension $k$. I am looking for results to determine explicitly the (co)homology groups and/or cohomology ring structure (in integer or rational ...
- 66.8k
143
votes
20
answers
23k
views
Are there examples of non-orientable manifolds in nature?
Whilst browsing through Marcel Berger's book "A Panoramic View of Riemannian Geometry" and thinking about the Klein bottle, I came across the sentence:
"The unorientable surfaces are never discussed ...
10
votes
2
answers
276
views
Reference request - Fibrations between spaces of embeddings
This is a cross-post of this question from MSE.
Given topological manifolds $M$ and $N$ of the same dimension, let $\operatorname{Emb}(M,N)$ denote the subspace of $\operatorname{Map}(M,N)$ ...
- 7,933
0
votes
0
answers
66
views
Is $\left|\frac{\det(A_{\mu\nu})}{\det(B_{\mu\nu})}\right|$ an invariant for two tensors $A_{\mu\nu}$ and $B_{\mu\nu}$ in a manifold?
I was doing some math around the determinant of 2nd-order covariant tensors. In a general $n$-dimensional manifold, I deduced that the determinant of a tensor $A_{\mu\nu}$ can be defined as
$$
\det(A_{...
- 176
0
votes
0
answers
161
views
Homeomorphism groups on manifolds and topological properties
Let $M$ be a compact $n$-dimensional manifold let $H(M)$ denote the homeomorphism group of $M$.
If $n=2$ then $H(M)$ enjoys nice properties such as being an ANR, is locally contractible, separable. ...
- 60.2k
3
votes
1
answer
147
views
Quotient by freely acting group on Banach manifold
I have a Banach manifold $\mathcal{M}$ and I have a Lie group $G$, that is finite dimensional, such that $G$ acts freely on $\mathcal{M}$. I would like to know if $\mathcal{M} / G$ is a Banach ...
- 359
3
votes
1
answer
292
views
A detail in Brown's proof of the generalized Schoenflies theorem
Consider a homeomorphic embedding $h:S^{n-1}\times [0,1]\rightarrow S^n$ and denote
$$S^{n-1}_t=h(S^{n-1}\times \{t\}).$$
The generalized Schoenflies theorem states the closure of each connected ...
- 1,175
8
votes
3
answers
2k
views
Inverse function theorem for manifolds with boundary as the domain
I wonder that whether there exists a version of the inverse function theorem for smooth maps from a smooth manifolds with boundary to a smooth manifold without boundary? More precisely, whether the ...
- 323
11
votes
0
answers
247
views
Detecting topology change of tubular neighbourhoods via smoothness of volume function
Let $M$ be an embedded closed manifold in $\mathbb R^n$, define $M_r=\{x\in\mathbb R^n:d(x,M)<r\}$.
Define $r\in\mathcal S_M$ iff $M_r\subset M_{r+\epsilon}$ is not a homotopy equivalence for all ...
- 111
0
votes
1
answer
162
views
Mappings of reducible 3 manifolds with boundary
In section 3 of his paper "Mappings of reducible 3 manifolds" McCullough, proves that every self-homeomorphism of a reducible 3 manifold can up to isotopy be written as a composition of ...
- 26.3k
-2
votes
1
answer
110
views
Is this limit a tangent vector? [closed]
Let $M$ be a smooth compact sub-manifold of $\mathbb R^d$. Let $p\in M$ and $x_n,y_n \in M$ be sequences such that $x_n,y_n\rightarrow p$. Does the following hold when passing to a convergent sub-...
- 5,632
5
votes
2
answers
338
views
Does every triangulable manifold have a vertex-transitive triangulation?
Does every triangulable manifold have a vertex-transitive triangulation?
When I talk about a vertex-transitive triangulation of a manifold, I mean in the sense of realizing a manifold homeomorphically ...
- 31k
11
votes
1
answer
339
views
Lower bounds for Betti numbers of a manifold given its boundary?
Let $B$ be some compact, path connected $n$-manifold without boundary such that its cobordism class is trivial, so that there exists some other $n+1$ manifold $M$ with $\partial M= B$. While there is ...
- 10.5k
1
vote
0
answers
136
views
Is $\pi_m(M) = 0$ if $\pi_m(M-X) = 0$ for a low-dimensional subset $X$?
I am doing a problem where I am stuck at this point.
Let $M$ be a connected smooth manifold of dimension $n$ and let $X$ be any subset of $M$. Assume that there is a positive integer $m$ such that $n&...
- 9,421
1
vote
1
answer
67
views
When a support of an isotopy is disjoint from a subset
Let $M$ be a compact connected manifold, $X\subset M$ a closed subset, and $f:M \times [0;1] \to M$ an isotopy such that each $f_t:M \to M$ is fixed on some open neighborhood $N_t$ of $X$, but there ...
- 8,076
0
votes
0
answers
79
views
Determinant of SU(N) elements, and radius of associated manifold
I'm wondering if the fact $SU(2)$ group elements have $det = 1$ is connected with the radius of the unitary $S^{3}$ manifold associated.
The context is demonstration of dU being an Haar invariant ...
- 1
2
votes
0
answers
76
views
Question about stable manifold theorem and Frobenius integrability theorem
I have a question about Anosov diffeomorphism (Wikipedia: Anosov diffeomorphisms)
For hyperbolic fixed point $p$, $W^{s}(p)$ is a smooth manifold and its tangent space has the same dimension as the ...
- 60.2k
18
votes
4
answers
3k
views
(Very) High dimensional manifolds
Usually one regards manifolds up to dimension 4 as a part of low dimensional topology. There are plenty of various results which work only in low dimensional topology; especially in dimension 4. ...
- 5,384
43
votes
4
answers
3k
views
Do rings of smooth functions differ from rings of continuous functions?
Let $M$, $N$ be connected nondiscrete compact smooth manifolds. Can the ring of continuous functions on $M$ be isomorphic to the ring of smooth functions on $N$?
CommunityBot
- 1
2
votes
0
answers
74
views
Can vector fields in manifolds with corner and sharp edges still satisfy Poincare-Hopf theorem?
We know that the sum of singularity index of the vector fields on a sphere equal to Euler characteristics of sphere, satisfying the Poincare-Hopf theorem. But how about situations of the geometry with ...
- 60.2k
10
votes
4
answers
2k
views
Elliptic regularity on compact manifold without boundary
Let $(M,g)$ be a Riemannian compact manifold without boundary, and $\Delta$ is the Laplace-Beltrami operator on $M$. Is there any result on the elliptic regularity like this:
For any $u\in H^1(M)$, ...
- 729
3
votes
0
answers
83
views
(When) can you embed a closed map with finite discrete fibers into a (branched) cover?
Assume all spaces are topological manifolds. A branched cover is a continuous open map with discrete fibers. A finite branched cover is one with finite fibers.
Questions. Given closed map $X\to S$ ...
- 10.3k
2
votes
1
answer
279
views
If there exists a function on a Riemannian manifold such that its Hessian matrix is the identity matrix?
In Euclidean space $\mathbb{R}^n$, $n\geq 2$, the Hessian matrix of the function $\frac{|x|^2}{2}$ is the identity matrix. While on a smooth manifold $(M^n, g)$, do there exists a function on $(M^n, g)...
- 5,632
5
votes
1
answer
284
views
Stable torus that is not a torus [duplicate]
Let $M$ be a closed manifold such that $M\times \mathbb{S}^1$ is a torus.
Is it true that $M$ is homeomorphic to a torus?
- 18.8k
7
votes
1
answer
431
views
Why does the tangent classifier classify the tangent (micro)bundle?
Let $\mathcal{M}\mathrm{fld}_n$ denote the $\infty$-category of topological manifolds (without boundary) and embeddings; more precisely, it is the homotopy coherent nerve of the simplicial category ...
- 5,201
7
votes
1
answer
255
views
Lens space bounding a topological, simply-connected 4-manifold with $b_2=1$
The following is written in section 1.6 (p.7) of this paper: https://arxiv.org/pdf/1010.6257.pdf.
($\cdots$) Which lens spaces bound a smooth, simply-connected 4-manifold $W$ with $b_2(W)=1$? ($\cdots$...
- 19.2k
8
votes
1
answer
275
views
Non-compact three-manifolds with the same proper homotopy type are homeomorphic?
I am looking for some literature with some (counter) examples of the following fact (though I don't know if the fact is true or not):
Let $M, M'$ be two non-compact connected $3$-manifolds with the ...
- 106