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.

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Is $\operatorname{Spin}(8)$ a direct product of $\operatorname{Spin}(7)$ and $S^7$?

Is $\textrm{Spin}(8)$ a direct product of $\textrm{Spin}(7)$ and $S^7$? I met this statement in the literature, but without a reference. If it is true, where is it explicitly written?
Andrei Smilga's user avatar
9 votes
2 answers
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Is limit of null-homotopic maps null-homotopic?

The question is motivated by my failed comment to this one. Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds). Let $\...
erz's user avatar
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9 votes
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Action of diffeomorphism group on non-vanishing vector fields

Let $M$ denote a closed manifold. Let $\Gamma(TM\setminus 0) $ denote the space of non-vanishing sections of $TM$. Note that the diffeomorphism group $\text{Diff} (M)$ acts on $\Gamma(TM\setminus 0)...
ThorbenK's user avatar
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9 votes
1 answer
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When does the diagonal cohomology class of a non-compact oriented manifold vanish?

Let $M$ be a non-compact, connected and oriented topological $d$-manifold without boundary. My understanding is that there are two (equivalent) ways of defining the diagonal class $\delta_M \in H^{d}(...
Cihan's user avatar
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9 votes
1 answer
390 views

Reference for the Brown-Gersten property for smooth manifolds

A classical result by Brown and Gersten says that to verify the homotopy descent property for the Zariski topology it suffices to verify it for Zariski squares and the empty cover of the empty scheme. ...
Dmitri Pavlov's user avatar
9 votes
0 answers
212 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\...
Christian's user avatar
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9 votes
0 answers
386 views

History of the definition of smooth manifold with boundary

I am trying to determine the earliest source for the definition of smooth ($C^\infty$) manifold with boundary. Milnor and Stasheff (1958) give a definition, but a scrutiny of that definition shows it ...
John Klein's user avatar
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$k$ times differentiable but not $C^k$ manifold

I asked the following question on Math Stack Exchange 3 months ago but got no answer. So maybe Math Overflow is a more suitable place for such a question: I cannot find the notion of $k$ times ...
fm12's user avatar
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9 votes
0 answers
368 views

Is it possible to glue together complex manifolds?

In the case of Riemannian manifolds, there are ways to take two manifolds and glue them together to get a new Riemannian manifold. For example, taking connected sums in local regions where the two ...
Kim's user avatar
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0 answers
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Almost Poincaré duality

Let $M^n$ be a connected, closed manifold. It has Poincaré duality with $\mathbb{Z}/2$ coefficients $H^k(M;\mathbb{Z}/2)\cong H_{n-k}(M;\mathbb{Z}/2)$, induced by cap product with the fundamental ...
Mark Grant's user avatar
8 votes
2 answers
911 views

Pin$^+$ and Pin$^−$ structure for manifolds in any dimensions

For an oriented $d$-manifold $M$, we can ask whether the manifold admits a Spin structure, say, if the transition functions for the tangent bundle, which take values in $SO(d)$, can be lifted to $\...
wonderich's user avatar
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8 votes
3 answers
662 views

Maximal exotic $\mathbb{R}^4$

Article Exotic $\mathbb{R}^4$ on Wikipedia says that there is at least one maximal smooth structure on $\mathbb{R}^4$, that is such an atlas on $\mathbb{R}^4$ that any other smooth $\mathbb{R}^4$ can ...
Konstantin Slutsky's user avatar
8 votes
2 answers
684 views

Number of critical points of smooth functions on $S^1$

Let $u$ be a smooth function on the unit circle $S^1$ such that $\int_{S^1}ux_j=0$, for $j=1,2$. Is the number of critical points of $u$ strictly bigger than 2?
Matchmaticians's user avatar
8 votes
1 answer
859 views

On the Euler characteristic of a Poincaré duality space

Background. Suppose that $M$ is an oriented, connected, closed manifold of dimension $d$ with fundamental class $\mu \in H_d(M;\Bbb Z)$. Let $\Delta : M \to M \times M$ be the diagonal map. Then the ...
John Klein's user avatar
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8 votes
2 answers
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Degree one self-map of $\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$ not homotopic to any self-homotopy equivalence

Consider the surface $\Sigma=\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$. Does there exist a proper map $f\colon \Sigma\to \Sigma$ of degree $1$ and not homotopic to any self-homotopy ...
Someone's user avatar
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8 votes
2 answers
314 views

Vanishing of Aronhold S-invariant on the cubic forms on $H^2(X, \mathbb Q)$

I am considering several examples of compact complex threefolds $X$ such that $rk H^2(X)=3$. Note that we have a cubic form on $H^2(X, \mathbb Q)$ which comes from the cup product. I calculated the ...
user104109's user avatar
8 votes
2 answers
459 views

Inverse cohomological isomorphisms

Let $\ M'\ M''\ $ be simply-connected Hausdorff compact manifolds (possibly with boundary for another variant of the question). Let $\ f:M'\rightarrow M''\ $ be a continuous function which induces an ...
Włodzimierz Holsztyński's user avatar
8 votes
1 answer
378 views

Universal cover with one end

Suppose that $M$ is a non-compact manifold of finite topological type with one end which is the universal cover of some closed manifold $N$. Is $M $ necessarily homeomorphic to the total space of some ...
Nick L's user avatar
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8 votes
1 answer
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When is a triangulation of sphere two-colorable?

Let $T$ be a triangulation of sphere. We say that $T$ is $k$-colorable if the triangles of $T$ can be assigned with $k$ colors such that any two triangles with a common edge have different colors. I ...
Hailong Dao's user avatar
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8 votes
1 answer
578 views

A topological group which is also a (not necessarily smooth) manifold is orientable

I am trying to show that a topological group which is also a (not necessarily smooth) manifold is automatically orientable. I know of a proof involving transition functions for smooth manifolds, in ...
Doeke's user avatar
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8 votes
1 answer
207 views

If $M$ is contractible manifold and $X\subset \partial M$, does the cone over $X$ embed in $M$?

Let $M$ be a compact contractible manifold, $X\subset\partial M$ and $C_X$ the cone over $X$. Question: Is it true that $C_X$ embeds in $M$ with its boundary $\partial C_X$ mapped to $X\subset \...
M. Winter's user avatar
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8 votes
1 answer
720 views

Status of Hilbert-Smith conjecture and H-S conjecture for Hölder actions

The Hilbert-Smith conjecture states that If $G$ is a locally compact group which acts effectively on a connected manifold as a topological transformation group then is $G$ a Lie group. It was ...
Zarathustra's user avatar
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8 votes
1 answer
271 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 ...
Random's user avatar
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8 votes
1 answer
558 views

Fundamental groups of non-orientable closed four-manifolds

The fundamental group of a closed orientable manifold is finitely presented, and every finitely presented group arises as the fundamental group of a closed orientable four-manifold; see this question. ...
Michael Albanese's user avatar
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 ...
ProbLe's user avatar
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8 votes
1 answer
1k views

Cobordism Theory of Topological Manifolds

Unfortunately, due to my ignorance, my present knowledge is limited to the cobordism Theory of Differentiable Manifolds. Cobordism Theory for DIFF/Differentiable/smooth manifolds However, there are ...
wonderich's user avatar
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8 votes
2 answers
315 views

Poisson structures on non-smooth manifolds with singularities

It's very known how we can describe a Poisson structure on a manifold $M$, where $M$ is a smooth manifold, but what about a Non-smooth manifold with singularities? In section $(2)$ of the paper The ...
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8 votes
1 answer
179 views

Are all monotonically normal manifolds of dimension at least two metrizable?

Alan Dow and Frank Tall recently proved the consistency of the statement Every hereditarily normal manifold of dimension at least two is metrizable. See: Dow, Alan; Tall, Franklin D., Hereditarily ...
Santi Spadaro's user avatar
8 votes
1 answer
315 views

Tietze's extension theorem for contractible manifolds

I've read that the Tietze's extension theorem was still valid for continuous applications from a closed subspace of a normal topological space to a contractible topological manifold (understood as ...
ychemama's user avatar
  • 1,326
8 votes
1 answer
910 views

Can the graph Laplacian be well approximated by a Laplace-Beltrami operator?

It seems rather well known that given a Laplace-Beltrami operator $\mathcal{L}_{M}$ on a manifold $M$ we can approximate its spectrum by that of a graph Laplacian $L_{G}$ for some $G$ (where $G$ is ...
Felix Goldberg's user avatar
8 votes
1 answer
499 views

Universal covers of non-prime 3-manifolds

Let $M$ be a closed, connected, oriented 3-manifold. If $M$ is prime, then we know what the universal cover of $M$ looks like: it is either $S^3, \mathbb{R}^3$ or $S^2 \times \mathbb{R}$ depending on ...
Minkowski's user avatar
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8 votes
1 answer
1k views

An orientable surface that cannot be embedded into $\Bbb R^3$? [duplicate]

I previously asked this question on MSE, without success. By Whitney's embedding theorem, every 2-dimensional manifold (aka. a surface) can be embedded into $\Bbb R^4$. Now, Wikipedia states in this ...
M. Rumpy's user avatar
  • 263
8 votes
1 answer
347 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 $...
Tashi Walde's user avatar
8 votes
0 answers
193 views

A modified version of the converse to the Sard's Theorem

When I learned Sard's Theorem in differential topology by myself, I was thinking whether it would be possible to prove a converse version of the theorem. That is to say, can we somehow show that each (...
pureorapplied's user avatar
8 votes
0 answers
163 views

Is there a combinatorial representation of general topological manifolds similar to triangulations?

Piece-wise linear manifolds are combinatorially represented by simplicial complexes modulo Pachner moves. However, for dimensions greater than $3$, the notions of piece-wise linear and topological ...
Andi Bauer's user avatar
  • 2,901
8 votes
0 answers
213 views

Geometric argument for "easy" part of Jordan-Brouwer separation theorem without local flatness

Let $M^n \subset \mathbb{R}^{n+1}$ be an $n$-dimensional compact connected topological submanifold. The Jordan-Brouwer separation theorem says that $\mathbb{R}^{n+1} \setminus M^n$ contains two ...
Lisa's user avatar
  • 321
8 votes
0 answers
193 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 ...
johnwbarrett's user avatar
7 votes
2 answers
769 views

Transitive embedding of the projective plane $\Bbb R P^2$ into the $4$-sphere

Is there an embedding (i.e. injective continuous map) $$\phi:\Bbb R \Bbb P^2\hookrightarrow S^4\subseteq\Bbb R^5$$ of the projective plane $\Bbb R\Bbb P^2$ into the $4$-sphere, that is ...
M. Winter's user avatar
  • 12.5k
7 votes
2 answers
1k views

Classification of closed 3-manifolds with finite first homology group?

I am interested in a topological classification of connected closed 3-manifold $M$ that have finite homology group $H_1(M)$. Since $H_1(M)$ is the abelization of the fundamental group $\pi_1(M)$, ...
Taras Banakh's user avatar
  • 40.8k
7 votes
3 answers
447 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 h-...
HenrikRüping's user avatar
7 votes
1 answer
340 views

Exotic $C^k$ manifolds

How much is known about exotic $C^k$-manifolds? For example, Is it known whether there are $C^k$-differentiable manifolds that are homeomorphic, but not $C^k$ diffeomorphic? More generally, is it ...
ExoticCk's user avatar
7 votes
2 answers
1k views

Any 3-manifold can be realized as the boundary of a 4-manifold

We know "Any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$." See this post: Elegant proof that any closed, oriented 3-manifold is the boundary of some ...
wonderich's user avatar
  • 10.3k
7 votes
1 answer
923 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 ...
J. GE's user avatar
  • 2,593
7 votes
3 answers
518 views

Contractible set in a manifold

Let $M$ be an $n$-dimensional topological closed manifold. Suppose $K$ is a compact subset of $M$ which is contractible in the sense that there exists a continuous map $F:K \times [0,1] \to M$ with $F(...
Zhiqiang's user avatar
  • 881
7 votes
2 answers
438 views

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 $\...
Andy Putman's user avatar
  • 43.4k
7 votes
2 answers
1k views

G-spaces and manifolds

In his book "The geometry of geodesics" H. Busemann defines the notion of a G-space to be a space which satisfies the following axioms: The space is metric The space is finitely compact, i.e., a ...
Dror Atariah's user avatar
7 votes
1 answer
2k views

The relation between Hausdorff dimension of an $n$-manifold and $n$

It is known that for a topological space with different metrics, the Hausdorff dimensions may not be equal in general. For the case of manifolds, suppose $M$ is a $n$-manifold with a metric(distance)...
Lewis Zhang's user avatar
7 votes
3 answers
815 views

Is the bordism from disjoint union to connected sum universal for connected manifolds?

Let $M_1$ and $M_2$ be two oriented, connected, closed $n$-manifolds. It is known that the disjoint union $M_1 \sqcup M_2$ and the connected sum $M_1 \# M_2$ are cobordant, via a bordism $\Sigma_{M_1, ...
Manuel Bärenz's user avatar
7 votes
1 answer
440 views

Cohomology of the mapping class group of a non-orientable surface?

What is the low degree cohomology of the mapping class group of a non-orientable surface? More specifically, what is the universal central extension of the mapping class group of a non-orientable ...
Kevin Walker's user avatar
  • 12.3k
7 votes
1 answer
885 views

On eigenfunctions of the Laplace Beltrami operator [closed]

How can we generate the eigenspace for the Laplace Beltrami operator on SU(2)?
Lateef's user avatar
  • 91

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