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 a manifold-with-boundary with given interior and non-empty boundary essentially unique?

Let $M$ be a compact connected manifold-with-boundary such that $\circ M \neq \emptyset$, where $\circ M$ is the boundary of $M$. Let $N$ be a compact connected manifold-with-boundary such that $\circ ...
kaba's user avatar
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6 votes
1 answer
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Characterisation of Sobolev Spaces on manifolds of bounded geometry via geodesic coordinates

I have a reference request concerning equivalent norms on Sobolev Spaces on manifolds of bounded geometry. This may be obvious to the experts but I am not working in the field and only want to use ...
Julia's user avatar
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1 vote
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Matrix trace minimization of quadratic and linear terms under orthogonal manifold constraints

How would one solve the following orthogonal manifold problem? $$\begin{array}{ll} \text{maximize} & \mbox{tr}(X^\top A X - X^\top B)\\ \text{subject to} & X^\top X = I\end{array}$$ where $A ...
John's user avatar
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-1 votes
1 answer
562 views

Metrics on derived smooth manifolds

Derived geometry explains how to remove the transversality condition and make sense out of a nontransversal intersection. For example, if $X$ and $Y$ are embedded submanifolds of a manifold (or ...
MathDG's user avatar
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7 votes
1 answer
514 views

Can a hyperbolic manifold be a product?

I was interested in whether a manifold which admits a metric of constant sectional curvature can be homotopy equivalent to a product of non-contractible manifolds. Of course, there are three cases: ...
Michael Albanese's user avatar
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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2 votes
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Triple link in a 5-sphere -- Proposal

In this post I would like to propose a triple link in a 5-sphere. Let us start with the following gluing into a 5-sphere: $$S^5=(D^2_{} \times T^3_{}) \cup_{T^4} ({S^5 \smallsetminus D^2 \times T^3})$...
wonderich's user avatar
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5 votes
1 answer
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harmonic coordinates on non-compact manifolds

Is it possible to show the existence of harmonic coordinates (e.g., on uniform-sized balls) on certain classes of non-compact Riemannian manifolds? For example, one may expect that such harmonic ...
Siran Victor Li's user avatar
2 votes
2 answers
721 views

Self-adjoint extensions for pseudo-differential operators

The class $\Sigma^1$ of symbols on $\mathbb R^{2n}$ is made with $C^\infty$ functions $a$ of $X=(x,\xi)\in \mathbb R^n\times\mathbb R^n$ such that $$ \vert\partial_X^\alpha a\vert\le C_\alpha(1+\vert ...
Bazin's user avatar
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2 votes
0 answers
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Vector bundle endomorphism diffeomorphism invariant?

Let's say we have a vector bundle $V$ on a compact Riemannian manifold $M$ (of dimension $m$, with metric $g$). Given a differential operator $P$, acting on the sections of $V$, of the form: $$P = \...
phydev's user avatar
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9 votes
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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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1 vote
0 answers
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Manifold with no closed components?

Let $M$ be a manifold with boundary. Reading some papers on $3$-manifolds I have come across some statements where they require that: ”$M$ has no closed components.” What does this mean? The ...
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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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16 votes
1 answer
887 views

Can one determine the dimension of a manifold given its 1-skeleton?

This may be an easy question, but I can't think of the answer at hand. Suppose that I have a triangulated $n$-manifold $M$ (satisfying any set of conditions that you feel like). Suppose that I give ...
Simon Rose's user avatar
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4 votes
0 answers
229 views

Reference for a proof of a Theorem by Joseph Wolf

We know that Lie Groups are parallelizable, I was looking for a version of the converse and came across this: https://books.google.com/books?id=w4bhBwAAQBAJ&pg=PA115 in Introduction to Smooth ...
Partha's user avatar
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11 votes
1 answer
839 views

Critical dimensions D for "smooth manifolds iff triangulable manifolds"

I am aware that at least for lower dimensions, "smooth manifolds iff triangulable manifolds" at least for dimensions below a certain critical dimensions D. My question is that for For ...
wonderich's user avatar
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14 votes
1 answer
476 views

3-fold of general type homeomorphic to rational 3-fold

Is there a smooth (complex projective) 3-fold of general type which is homeomorphic (in the complex topology) to a rational $3$-fold? I am aware of such examples in complex dimension $2$, for ...
Nick L's user avatar
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4 votes
0 answers
310 views

Topological approach to create a space between clouds

I have a dataset associated with labels. According to https://arxiv.org/pdf/1802.03426.pdf --> UMAP (Uniform Manifold Approximation and Projection) which is a novel manifold learning technique for ...
davegaut's user avatar
4 votes
1 answer
357 views

What is the topological/smooth analogue of Nagata compactification

A celebrated theorem of Nagata and subsequent refinements to schemes and algebraic spaces say that over a not-completely-monstrous base scheme, any separated morphism can be openly immersed in a ...
Arrow's user avatar
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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 $u\in H^1(M)$, ...
S. Maths's user avatar
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9 votes
1 answer
420 views

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
  • 1,175
1 vote
1 answer
753 views

Intersection of hyperspheres

Suppose we have $n$ hyperspheres in $\mathbb{R}^m$, $m\geq n$, of centers $x_1,\ldots x_n$, $x_i\neq x_j\,\forall i,j$, and radii $r_1,\ldots ,r_n$. Suppose that, for every $i,j$, the quantities $r_i$,...
Capublanca's user avatar
5 votes
0 answers
107 views

Induced new structures on Poincare dual manifolds

"R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows Given a spin structure on $M^3$, the submanifold $\text{PD}(a)$ can be given a natural induced $\text{Pin}^-$...
wonderich's user avatar
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7 votes
0 answers
227 views

GSO (Gliozzi-Scherk-Olive) projection and its Mathematics?

GSO (Gliozzi-Scherk-Olive) projection is an ingredient used in constructing a consistent model in superstring theory. The projection is a selection of a subset of possible vertex operators in the ...
wonderich's user avatar
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3 votes
0 answers
92 views

Relation between the orientation sheaves of the interior and the boundary of a topological manifold

Let $(M, \partial M)$ be an $n$-dimensional topological manifold with boundary. Let $\mathcal{O}_{M \setminus \partial M}$ and $\mathcal{O}_{\partial M}$ denote the orientation sheaves of $M \setminus ...
Rahmpilz's user avatar
  • 165
5 votes
1 answer
189 views

Relating bordism groups of different dimensions

Let $M_d$ be a $d$-manifold generator of a subgroup of bordism group $$ \Omega_d^{G}, $$ or further generalization $$ \Omega_d^{G}(K(\mathcal{G},n+1)), $$ which $G$ is the given structure ...
wonderich's user avatar
  • 10.3k
10 votes
3 answers
1k views

Is there a closed non-smoothable 4-manifold with zero Euler characteristic?

I will just repeat the title: Is there a closed non-smoothable 4-manifold with zero Euler characteristic? I am guessing yes simply based on other existence theorems I have seen for 4-manifolds.
Cihan's user avatar
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1 vote
0 answers
98 views

Upper bounds on $\epsilon$-covers of arbitrary compact manifolds

Let $M \subset \mathbb{R}^d$ be a compact $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$-cover $P$ of $M$, that is for every ...
user122374's user avatar
4 votes
0 answers
131 views

Lower bound on $\epsilon$-covers of arbitrary manifolds

Let $M \subset \mathbb{R}^d$ be a $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$-cover $P$ of $M$, that is for every point $...
user122374's user avatar
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
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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 ...
MathDG's user avatar
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5 votes
1 answer
485 views

Every unorientable 4-manifold has a $Pin^c$, $Pin^{\tilde c+}$ or $Pin^{\tilde c-}$ Structure

The precise statement on J. W. Morgan's "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds (MN-44)" that 4-manifold $X$ admits a Spinc structure (Lemma 3.1.2) ...
wonderich's user avatar
  • 10.3k
4 votes
1 answer
151 views

The homological negligibility of certain subsets in compact manifolds

Let $n\ge 3$ and $X$ be a compact connected $n$-manifold (without boundary). I need a reference to the following facts (which I believe are true at least in dimension $n=3$): Fact 1. For every ...
Taras Banakh's user avatar
  • 40.8k
3 votes
1 answer
252 views

Poincaré dual of the generators of $H^d(\mathbb{RP}^5,\mathbb{Z}_2)$

We know $H^d(\mathbb{RP}^5,\mathbb{Z}_2)=\mathbb{Z}_2$. So there are two classes of $\mathbb{Z}_2$ generators, trivial and nontrivial, for $d=0,1,2,3,4,5$. Wha are the Poincaré dual $(5-d)$-...
wonderich's user avatar
  • 10.3k
4 votes
0 answers
66 views

Irreducible separators of compact manifolds

Definition. A closed subset $S$ of a topological space $X$ is called $\bullet$ a separator of $X$ if $X\setminus S$ is disconnected; $\bullet$ an irreducible separator if $S$ is a separator of $X$ ...
Taras Banakh's user avatar
  • 40.8k
15 votes
0 answers
331 views

Beyond smoothness-the clear picture about the notion of a differential form

In this paper N.Teleman constructs the signature operator on an arbitrary (closed, oriented) Lipschitz manifold with coefficients in a vector bundle $\xi$. In particular the notion of a differential $...
truebaran's user avatar
  • 9,140
4 votes
0 answers
235 views

Non-spin 5-manifold and $2^2$-Bockstein homomorphism

The $2^2$-Bockstein is $\beta_4$ is associated to $$0\to\mathbb{Z}/2\to\mathbb{Z}/{8}\to\mathbb{Z}/{4}\to 0,$$ (The $2^n$-Bockstein homomorphism $$\beta_{2^n}:H^*(-,\mathbb{Z}/{2^n})\to H^{*+1}(-,\...
wonderich's user avatar
  • 10.3k
3 votes
0 answers
854 views

Quotient space, homogeneous space, and higher homotopy groups

Preparation and my input: For the quotient space $G/H$, knowing the homotopy groups of $G$ and $H$ one can determine homotopy groups from the long exact sequence $$ ... \to \pi_n(H) \to \pi_n(G) ...
wonderich's user avatar
  • 10.3k
1 vote
0 answers
203 views

Completing the proof of that the set of points where $f(x) = 0$ is a $k$-manifold [closed]

[I have asked this question with the previous versions of my answer in math.SE; however, I did not get any comment / answer, so I thought I might asked this in here with the improved version of my ...
Our's user avatar
  • 133
1 vote
1 answer
177 views

Can a continuous map on a Hilbert manifold be approximated by a map which has infinitely many critical points?

It is well-known that a continuous map $f:M\to\mathbb{R}^n$ from a Hilbert manifold can be closely approximated by a smooth map $g:M\to\mathbb{R}^n$ which has no critical points. But, can such a ...
Sergio Charles's user avatar
11 votes
1 answer
634 views

Poincare duality spaces vs. manifolds via lifting maps, the obstruction theory and the role of simply connectedness

Suppose that we are given a topological space $X$: assume for simplicity that $X$ is compact we want to adress the following question: Is it true that one can find a manifold $M$ which is homotopy ...
truebaran's user avatar
  • 9,140
14 votes
1 answer
565 views

Obstruction of spin-c structure and the generalized Wu manifods

Bockstein homomorphim and obstruction of spin-c structure: Let $w_2$ be the Stiefel Whintney class of manifold $M$. Let the Bockstein homomorphim $\beta$ be the $$ H^2(\mathbb{Z}_2,M) \to H^3(\mathbb{...
annie marie cœur's user avatar
10 votes
1 answer
586 views

Non-triangulable 4-manifold as a boundary of some 5 manifold

We know that there are non-triangulable 4-manifolds, such as the E$_8$ manifold. Can E$_8$ manifold be a boundary of some 5-manifold $M_5$? Can such a $M_5$ be triangulable or non-triangulable? What ...
wonderich's user avatar
  • 10.3k
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
14 votes
1 answer
628 views

Is there a PL, or topological, bordism hypothesis?

The bordism hypothesis says that the $(\infty, n)$-category of smooth, framed $n$-bordisms, $(n-1)$-dimensional boundaries, and corners down to points, is freely generated symmetric monoidal with ...
Manuel Bärenz's user avatar
14 votes
3 answers
1k views

Existence of non-null-homotopic map from $M^n$ to $S^{n-1}$

Let $M^n$ be compact, connected, oriented $n$-dimensional smooth manifold without boundary, the Hopf degree theorem states that the homotopy class of continuous maps from $M^n$ to $S^n$ is classified ...
sawdada's user avatar
  • 6,148
1 vote
0 answers
67 views

Does this "algebraic" method for the application of the constructive proof of the classification of closed & compact surfaces have any use? [closed]

Disclaimer: This question is cross-posted in here. I have never asked a question in mathoverflow before, so if the level of this question is not appropriate for this site, please just vote close it. ...
Our's user avatar
  • 133
5 votes
0 answers
160 views

Theoretical justification of time-series forecasting using Takens' embedding

This is a cross-posting where I couldn't get an answer. In the meantime I have tried to improve the original logic: As in Takens original paper about his embedding theorem, consider a compact $m$-...
Sarem Seitz's user avatar
12 votes
3 answers
608 views

Is there a discrete lattice analogue of conformal transformations?

There is a simple discrete combinatorial analogue of manifolds and homeomorphisms: Replace manifolds by simplicial complexes and homeomorphisms by Pachner moves. Equivalence classes of manifolds under ...
Andi Bauer's user avatar
  • 2,901
9 votes
3 answers
388 views

Are there invariants of cell complexes similar to the Euler characteristic?

The Euler characteristic is an invariant (under homeomorphism) of manifolds that can be computed from a cellulation by (weighted) counting of different kinds of objects, namely \begin{equation} \chi=\...
Andi Bauer's user avatar
  • 2,901

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