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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3
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1answer
83 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 ...
2
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0answers
74 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 ...
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2answers
135 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 ...
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0answers
40 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 $\...
2
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1answer
108 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 ...
3
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1answer
188 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 ...
3
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1answer
251 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 ...
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68 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 ...
6
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184 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 ...
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1answer
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 ...
6
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1answer
247 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 ...
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228 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.
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256 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 (...
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290 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, ...
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84 views

Probability density on Riemannian manifold

Let $\mathcal{M}$ be a $d-$dimensional (Riemannian) manifold and $X$ be a random variable taking values in $\mathcal{M}$. Assume that $X=f(Z)$ where $Z\sim \pi(z)$, and $f:\mathcal{Z}\subseteq \mathbb{...
5
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1answer
257 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 ...
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0answers
65 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?
10
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1answer
247 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 ...
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22 views

On the dimension of the projected rank $r$ matrix space

In $d$-dimensional complex number space $\mathbb{C}^{ d}$, we can define the rank at most $r$ matrix space $$ S=\{A|\ \mathrm{rank}(A)\leq r\}\subseteq \mathbb{C}^{d\times d}. $$ The dimension of ...
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1answer
971 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 ...
8
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1answer
161 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 ...
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1answer
257 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. ...
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141 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 ...
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81 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 ...
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17 views

Derivative $\left.\frac{\rm d}{{\rm d}t}\nu_{\partial T_t(\partial M)}\right|_{t=0}$ of outer normal field on a transformed boundary $T_t(\partial M)$

Let $d\in\mathbb N$, $v\in C_c^1(\mathbb R^d,\mathbb R^d)$, $X^x\in C^0([0,\infty),\mathbb R^d)$ denote the solution of $$T_t(x):=X^x(t)=x+\int_0^tv(X^x(s))\:{\rm d}s\;\;\;\text{for all }t\ge0\tag1$$ ...
3
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2answers
197 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 ...
2
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1answer
111 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 ...
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45 views

How can we calculate this tangential differential?

Let $\tau>0$, $d\in\mathbb N$ and $T_t$ be a $C^1$-diffeomorphism on $\mathbb R^d$ for $t\in[0,\tau]$ with $T_0=\operatorname{id}_{\mathbb R^d}$. Assume $$T_t(x)=x+\int_0^tv(s,T_s(x))\:{\rm d}s\;\;\...
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0answers
151 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-...
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0answers
102 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 $...
2
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0answers
51 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 ...
3
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1answer
162 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 "$\...
1
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1answer
111 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 ...
9
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1answer
209 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 ...
2
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0answers
102 views

Combinatorial condition for orientability on simplicial complexes

Let $K$ be a simplicial complex whose geometric realization is a topological or smooth manifold. Is it possible to restate the condition of orientability of $M$ exclusively in (combinatorial) terms of ...
13
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1answer
498 views

Is there an orientable prime manifold covered by a non-prime manifold?

A manifold is called prime if whenever it is homeomorphic to a connected sum, one of the two factors is homeomorphic to a sphere. Is there an example of a finite covering $\pi : N \to M$ of closed ...
6
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1answer
98 views

Subdivision of closed homology manifold reference request

I am interested in the barycentric subdivision of closed homology manifolds. Definition A (finite) simplicial complex $K$ is a closed homology manifold of dimension $n$ if for every $k$-simplex, its ...
5
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1answer
467 views

To what extent is a vector bundle on a manifold with boundary determined by its restriction to the interior?

Let $M$ be a manifold with boundary $\partial M$ and interior $M_0$. Let $E\rightarrow M_0$ be a fixed vector bundle. How many extensions of $E$ to a vector bundle $E'\rightarrow M$ are there, up to ...
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1answer
94 views

Topological connected eccentrics, not homeomorphic to commutative Lie groups

An eccentric is a universal algebra $\ (X\ \sigma\ \lambda\ \rho)\ $ such that operations $\ \sigma\ \lambda\ \rho\,:\,X\times X\to X\ $ satisfy: $\quad \forall_{x\ y\,\in X}\quad \lambda(\sigma(x\ y)...
3
votes
1answer
110 views

Tangent bundle with Sasaki metric is Kähler iff $M$ is locally flat

I'm having a hard time proving the following: If $M$ is an $n$-dimensional indefinite Riemannian manifold whose metric $g$ has index $s$, then the metric of Sasaki $g^{D}$ is an indefinite metric ...
1
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1answer
230 views

Homology of topological manifolds

Let $X$ be a topological manifold of dimension $n$ (assuming perhaps that there is a countable basis of open sets). Do NOT assume that $X$ is compact, or oriented, or triangulable (so do not assume it ...
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0answers
37 views

prove a bondle is an indefinite Hermitian manifold which is Kahler if and only if the manifold is locally flat

Let $M(J,g)$ be an indefinite Kahler manifold, then $% TM(J^{H},g^{D})$ is an indefinite Hermitian manifold which is Kahler if and only if $M$ is locally flat. Here $J^{H}$ denotes the horizontallift ...
3
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0answers
61 views

References on integration on non-compact manifolds

I am looking for references on integration on non-compact Riemannian manifolds, specially on the change of variables theorem. In particular I have non-compact manifold $M$ and I have an integral (in ...
2
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0answers
30 views

Calculate the Jacobian of a particular diffeomorphism of parallelizable manifold onto itself

Let $M$ be $d$-dimensional parallelizable manifold. Let $e_k(x)$ for $k=1, \dots , d$ be the smooth vector fields forming orthonormal basis in tangent bundle of 𝑀, these vector fields exist because ...
8
votes
1answer
205 views

Generators and relations for the 2-dimensional unoriented cobordism category

It is very well known in the field of TQFT that the 2-dimensional oriented cobordism category is generated by the disk and the pair of pants (each going in both directions), subject to a finite set of ...
7
votes
1answer
318 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 ...
6
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0answers
184 views

Signature of a non-compact manifold

Let $v_0,\dots,v_n\in\mathbb{Z}^2$ be integer vectors which satisfy the condition $\det\begin{pmatrix}v_{k-1}&v_k\end{pmatrix}=(-1)^k$, whose relevance will become apparent in a moment. We may ...
0
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2answers
76 views

Is this $a(p)=\lim_{r\to \infty} \frac{VolS(p,r)}{e^{h r}}$ exists and applied for manifolds with positive curvature?

In $1969$, Margulis proved, for suitable constant $h>0$and $r$ is a positive constant that : $a(p)=\lim_{r\to \infty} \frac{VolS(p,r)}{e^{h r}}$ with ($(S(p,r)$ is geodesic spheres), exists at ...
7
votes
1answer
302 views

Map which is null-homotopic on compacts

This is the missing ingredient towards answering my previous question. Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds). ...
10
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2answers
558 views

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 $\...

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