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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Does there exist a manifold with finitely generated homology groups that is not homotopy equivalent to a compact manifold with boundary?

Does there exist a manifold with finitely generated homology groups that is not homotopy equivalent to a compact manifold with boundary? I am also interested in several variations of this question. ...
Arshak Aivazian's user avatar
2 votes
1 answer
197 views

A Riemann surface is automatically paracompact

[A question I remember from many years ago.] Definition A Riemann surface is a connected complex manifold $X$ of complex dimension one. This means that $X$ is a connected Hausdorff space that is ...
Gerald Edgar's user avatar
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3 votes
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non-smooth manifold with circle action (with fixed points)

I am interested to know if there a non-smooth manifold (i.e. a closed topological manifold admitting no smooth structure) $M$, having a continuous action $M \times S^1 \rightarrow M$, and the number ...
Nick L's user avatar
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3 votes
1 answer
175 views

Open manifolds which have stable $\pi_1$ at infinity but are not inward tame

Let $M$ be a $1$-ended open manifold. An important result of Siebenmann states that (in dimension $\geq6$) if $M$ is $(i)$ inward tame, i.e. for every closed neighborhood of infinity there exists a ...
Bargabbiati's user avatar
11 votes
2 answers
664 views

Can we embed a closed manifold into a homotopy equivalent CW complex?

Suppose $X$ is a CW complex and $M$ is a closed manifold and suppose further that there exists a homotopy equivalence $X \simeq M$. Does there exists an embedding of $M$ into $X$ (i.e. an injective (...
ThorbenK's user avatar
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4 votes
0 answers
175 views

Classifying singularities of the Ricci flow

Context: A solution $(M^n, g(t))$ of the Ricci flow is said to encounter a Type III Singularity if $g(t)$ is defined for all $t \geq 0$ and: $$ \sup _{\mathcal{M}^{n} \times[0, \infty)} \|\...
Matheus Andrade's user avatar
13 votes
1 answer
582 views

Are there examples of Einstein manifolds with unbounded curvature?

Are there any known examples of Einstein manifolds $(M, g)$ such that $$\sup_{x \in M} \|\text{Rm}(x) \| = \infty$$ I'm looking for these examples because they might provide a counter-example to a ...
Matheus Andrade's user avatar
12 votes
1 answer
611 views

Roadmap for L-Theory

Background: I spent sometime reading about algebraic K-theory and started reading research papers on the subject with relative facility at least I do understand constructions, statements of the ...
cellular's user avatar
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7 votes
0 answers
154 views

Homotopy equivalent cartesian product of closed manifold

I'm little bit lost with the following question: I have four connected closed orientable manifolds $M,N,S,S'$ such that $S$ and $S'$ are homotopy equivalent and $M\times S$ is homotopy equivalent to $...
Paris's user avatar
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23 votes
0 answers
652 views

Do most manifolds have symmetries? or not?

Let us say that a (closed, connected) manifold has a symmetry if it admits a non-trivial action by a finite group. Note that I am not asking the action to be free. So for example rotating the 2-sphere ...
Chris Schommer-Pries's user avatar
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
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9 votes
2 answers
407 views

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
10 votes
2 answers
649 views

Do colimits of manifolds coincide with underlying colimits as topological spaces?

Categories of manifolds (possibly with extra structure) tend not to have all colimits. Other questions have addressed when colimits of manifolds exist. I'd like to know what we can say in general ...
Alastair Grant-Stuart's user avatar
8 votes
1 answer
592 views

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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2 votes
0 answers
112 views

Birth of chaos due to nonautonomous perturbation

Let $\sigma, b>0$. I want to study the dynamics of the map $$ T \colon \mathbb{N} \times \mathbb{S}^1 \times \mathbb{R} \to \mathbb{S}^1 \times \mathbb{R}$$ such that $$T_{\sigma,b}(n,\theta,y) = (\...
Giuseppe Tenaglia's user avatar
3 votes
2 answers
181 views

Embedded submanifold in a cylinder

Let $M^n$ be an $n$-dimensional topological closed manifold. Suppose there exists an embedding $i:M \to M \times [0,1]$ such that $i(M)$ is contained in the interior of $M \times [0,1]$ and separates $...
Zhiqiang's user avatar
  • 881
17 votes
2 answers
1k views

Suspension of a topological space

Let $X$ be a topological space such that its suspension is a topological manifold. Can we prove that $X$ itself is a topological manifold?
Totoro's user avatar
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3 votes
0 answers
159 views

How do you compute the $w_2$ of Freedman's E8 manifold?

The Wikipedia page for Rokhlin's Theorem says "Michael Freedman's E8 manifold is a simply connected compact topological manifold with vanishing $w_{2}(M)$ and intersection form $E_{8}$ of ...
Stella Dubois's user avatar
14 votes
2 answers
2k views

What is the top cohomology group of a non-compact, non-orientable manifold?

Let $M$ be a connected, non-compact, non-orientable topological manifold of dimension $n$. Question: Is the top singular cohomology group $H^n(M,\mathbb Z)$ zero? This naïve question does not seem to ...
Georges Elencwajg's user avatar
3 votes
0 answers
115 views

When is the dual block decompositition a CW decomposition?

Given a triangulated homology manifold $X$, the dual block decomposition is defined by setting, for each simplex $\sigma$ of $X$, the block $\overline{D}(\sigma)$ to be the union of all simplices of $\...
Pedro's user avatar
  • 269
5 votes
1 answer
463 views

Invariance of morse homology, doubt in proof in book "Morse Theory and Floer homology"

I am reading the book "Morse theory and Floer Homology" by Michele Audin and Mihai Damian. Now I am reading the proof of the following theorem. Link to the statement of the theorem ...
Luis Carlos 's user avatar
5 votes
1 answer
241 views

What locales correspond to Manifolds?

I am studying the categorical equivalence between (sober) topological spaces and (spatial) locales with enough points. As the title implies, I am interested in finding localic analogues of both ...
Bumblebee's user avatar
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8 votes
2 answers
246 views

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
  • 265
2 votes
0 answers
159 views

Theory of mollifiers on the boundary of a $C^2$ domain

Let $D\subseteq\mathbb{R}^d$ be a nice but not smooth domain, somewhere between Lipschitz and $C^2$. I am looking for a reference on the theory of mollifiers and regularization for functions on $\...
ajr's user avatar
  • 171
2 votes
1 answer
447 views

Collar neighborhood theorem for manifold with corners

I was reading this wonderful sequence of posts: nlab: manifold with boundary and nlab: collar neighbourhood theorem and I couldn't help but wonder. Is there an extension of the Collar neighborhood ...
ABIM's user avatar
  • 5,019
4 votes
1 answer
473 views

$H^1(X, GL(n, \mathcal{O}_X))$ and Vector Bundles

Let $X$ be a nice space: a manifold, a variety over a field, or so on. We know that line bundles on $X$ up to isomorphisms are given by $H^1(X,\mathcal{O}_X^*)$. Can it be generalized to higher rankal ...
P. Grabowski's user avatar
3 votes
1 answer
366 views

Closed manifolds are not absolute retracts?

A fundamental result in topology is that the $n$-sphere is not a retract of the $n+1$-ball. It implies that the $n$-sphere is not an absolute retract. Is there a generalization from the sphere to ...
mathieu's user avatar
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1 vote
0 answers
124 views

Codimension one submanifold gives cofibrant pair

Let $M$ be a smooth manifold, and $N$ be an embedded smooth submanifold of $M$ with $\partial M=\varnothing=\partial N$. Suppose, $\dim M-\dim N=1$, and $N$ is a closed subset of $M$. Does the ...
Someone's user avatar
  • 265
6 votes
0 answers
207 views

If $X_d$ is a non-triangulable manifold, can $X_d \times T^k$, $X_d \times I^k$, or $X_d \times \mathbb{R}^k$ be a triangulable manifold?

If $X_d$ is a non-triangulable manifold, can $X_d \times T^k$, $X_d \times I^k$, or $X_d \times \mathbb{R}^k$ always be a triangulable manifold? Let $X_d$ be a $d$-manifold which is NOT a ...
wonderich's user avatar
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25 votes
1 answer
1k views

On a curious map from the complex projective plane into $S^5$

I have heavily edited the post (including the title), based on a comment by @GregoryArone that my map $f$ is not injective. In an earlier version of this post, I had thought to have constructed a ...
Malkoun's user avatar
  • 4,991
6 votes
1 answer
268 views

Characteristic class that cannot be represented by disjoint tori

Is there a simply-connected smooth closed 4-manifold with a characteristic class $x \in H_2(X; \mathbb{Z})$ such that $x$ can not be represented by a disjoint union of tori in $X$? I would not know ...
user101010's user avatar
  • 5,319
2 votes
0 answers
343 views

Is $U(n)$ a Kahler manifold?

I am wondering if it is known whether the unitary group $U(n)$ is a Kahler manifold, and, if so, what is a reference for this.
Mo Behzad Kang's user avatar
7 votes
0 answers
599 views

How much differs the category of real-analytic manifolds from $C^\infty$ ones?

I was thinking about the difference between the concept of real-analytic function (for any point the Taylor-series of $f$ converge to the function in a neighborhood of the point) and complex analytic (...
John117's user avatar
  • 395
13 votes
0 answers
439 views

Structures between PL and smooth

Let $X$ be a topological manifold of dimension at least five. The Kirby-Siebenmann invariant $ks(X)\in H^4(X,\mathbb{Z}_2)$ is an obstruction to the existence of a PL structure on $X$. If it vanishes, ...
Philip Engel's user avatar
  • 1,493
6 votes
1 answer
373 views

What are some "good" examples of Kan simplicial manifolds?

According to the definition 1.1 of the paper Kan Replacement of simplicial manifolds by Chenchang Zhu https://arxiv.org/pdf/0812.4150.pdf, A Kan simplicial manifold is a simplicial manifold $X$ such ...
Adittya Chaudhuri's user avatar
0 votes
0 answers
124 views

Einstein submanifold of Einstein manifold - References

Is there any work in which the conditions under which an Einstein manifold admits an Einstein submanifold are studied? If yes, can you give me the references?
MathDG's user avatar
  • 242
11 votes
1 answer
371 views

Existence of normal microbundles

In the same paper where Milnor introduced the concept of microbundles, he gave the following definition. $M$ has a microbundle neighborhood in $N$ if there is a neighborhood $U$ of $M$ in $N$ and a ...
user101010's user avatar
  • 5,319
18 votes
1 answer
1k views

The group of isometries of a manifold is a Lie group, isn't it?

Let $M$ be a connected finite dimensional topological manifold and $g$ be any metric on it that induces the topology of $M$ ($g$ is not a Riemannian metric). How to prove that the group of isometries ...
aglearner's user avatar
  • 14k
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
10 votes
1 answer
357 views

A piecewise-linear or topological Fulton-MacPherson compactification

The Fulton-MacPherson compactifications of configuration spaces are smooth manifolds with corners which have the ordered configuration spaces of distinct points in a smooth manifold as their interior. ...
skupers's user avatar
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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
0 votes
0 answers
205 views

Intersection of zero sets of continuous functions

Let the zero sets $F=\{x \in \mathbb{R}^n: f(x) = 0\}$, $G = \{x \in \mathbb{R}^n : g(x) = 0\}$, where $f$ and $g$ are $m$-dimensional real, analytic, continuous, and nonlinear vector functions. Under ...
Chris's user avatar
  • 1
3 votes
2 answers
276 views

minimal embedding space of a manifold in smooth and PL case

Given a manifold $M$, we can always embed it in some Euclidian space (general position theorem). Hence we can define the minimal embedding space of $M$ to be the smallest euclidean space that we can ...
Steve's user avatar
  • 494
2 votes
1 answer
204 views

Quotient space by discrete group and $L^2(\Gamma\backslash \mathcal{H})$

I'm reading Daniel Bump's "Automorphic forms and representations" chapter 2, and in the book they define an integral over $\Gamma\backslash\mathcal{H}$ (here, $\Gamma$ is a discontinuous ...
hong's user avatar
  • 21
0 votes
0 answers
340 views

What is the physical meaning of torsion

The torsion tensor in 4 dimensions $S_{ab}^{\hphantom0\hphantom0 c}$ has 24 components and it can be split into a vector part $\hphantom0^{V}S_{ab}^{\hphantom0\hphantom0 c}=\frac{1}{3}(S_a\delta^c_b-...
Eris's user avatar
  • 1
0 votes
0 answers
160 views

Problem of Thickening an Arc in a Topological $ 2 $-Manifold

Let $ M $ be a topological $ 2 $-manifold (possibly with boundary), $ C $ an arc in the interior of $ M $ (i.e., an injective continuous function from $ [- 1,1] $ into $ \operatorname{Int}(M) $), and $...
Transcendental's user avatar
3 votes
0 answers
69 views

On isospectral planar domains (and a paper by Buser, Conway, Doyle and Semmler)

I have never seen a short, elegant way (from the viewpoint of a non-topologist) which constructs isospectral planar domains from Sunada group triples, although essentially those triples live at the ...
THC's user avatar
  • 4,313
4 votes
1 answer
187 views

The $\operatorname{spin}^c$ index for manifolds

$\DeclareMathOperator\spin{spin}\DeclareMathOperator\ch{ch}\DeclareMathOperator\ind{ind}$In the paper Čadek, Crabb, and Vanžura - Obstruction theory on 8-manifolds, the authors discussed the "$\...
Xing Gu's user avatar
  • 935
1 vote
1 answer
361 views

Number of connected components of the set of invertible matrices over the reals when some of the matrix entries are fixed

Let $\mathcal{M} = \text{GL}(n, \mathbb{R})$ denote the set of $n \times n$ invertible matrices over $\mathbb{R}$. We know that $\text{GL}(n, \mathbb{R})$ has exactly 2 connected components. I have ...
Rahul Sarkar's user avatar
10 votes
1 answer
577 views

Is every open topological $d$-manifold homotopy equivalent to a CW-complex of dimension $\leq d-1$?

Let $M$ be a connected open topological $d$-manifold (without boundary). Whitehead showed that if $M$ has a PL structure, there exists a subcomplex of dimension $\leq d-1$ onto which $M$ deformation ...
Cihan's user avatar
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