Questions tagged [compactifications]

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Homeomorphism between interiors of simplex and permutohedron

The $n$-dimensional permutohedron $P_n$ is a polytope whose facets (i.e.\ codimension $1$ faces) are in 1-to-1 correspondence with all faces (of codimension${}\geq 1$) of the $n$-simplex $\Delta_n$, ...
Xin Nie's user avatar
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1 answer
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Points in the Stone Cech compactification are intersection of open sets

Let $\beta \mathbb{N}$ be the Stone Cech compactification of the natural numbers and let $ x\in \beta \mathbb{N}$. Is it true that there exists a sequence of open sets $\{U_n\}_{n=1}^\infty$ in $\beta ...
Serge the Toaster's user avatar
1 vote
0 answers
82 views

What is the meaning of universal family of Fulton Macpherson configuration space?

Fulton and Macpherson suggests the way to compactify the set of $n$-labelled distinct point on variety in their paper, "A Compactification of Configuration Spaces" In this paper, the process ...
ChoMedit's user avatar
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4 votes
2 answers
535 views

Compactification of a product of manifolds

Let $M$ be a smooth manifold. We make the assumption that $M$ can be viewed as the interior of a compact manifold with boundary $\overline{M}$. In practice, for an explicit manifold, any ...
zarathustra's user avatar
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1 answer
197 views

Are degrees and ramification degrees preserved upon passing to the smooth compactification?

Let $\phi :C_1\to C_2$ be morphism of projective singular curve. Let $\tilde{C}_1$ and $\tilde{C}_2$ be their smooth compactification. Then $\phi$ extends to $\tilde{\phi} : \tilde{C}_1\to \tilde{C}...
Duality's user avatar
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1 vote
0 answers
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Implicit function theorem and compactification of algebraic curve

Let $C$ be a singular curve defined over a local field $K$. Let $\tilde{C}$ be its smooth compactification(maybe this is not normalization). Why $\tilde{C}(K)\neq \emptyset$ implies ${C}(K)\neq \...
Duality's user avatar
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Are the irreducible components appearing in the resolution of singularities of a Hilbert modular surface defined over $\mathbb{Q}$?

It seems to me that this is claimed in van der Geer's "Hilbert modular surfaces" on p. 245 at the beginning of XI.2 (without justification). My current state of belief/knowledge: The ...
dfn's user avatar
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2 votes
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139 views

Line bundles on toric varieties associated to Weyl chamber

I am interested in studying toric varieties associated to the fan of Weyl chambers. General information would be best but I am also interested in the specific case of the Weyl chamber of $\mathfrak{sl}...
Merrick Cai's user avatar
2 votes
1 answer
129 views

On the zero-dimensional strata of the Fulton-MacPherson conpactification

Let $\operatorname{Conf}_n(\mathbb{R})$ be the configuration space of $n$ marked points on the real line. What is the difference between $\operatorname{Conf}_n(\mathbb{R})$ and the locus of zero-...
Banana23's user avatar
4 votes
1 answer
220 views

What is the Freudenthal compactification of a wildly punctured n-sphere?

Let $C$ be a compact and totally-disconnected subspace of the $n$-sphere $\mathbb{S}^n$, where $n\geq 2$. Question: Must the Freudenthal compactification of $\mathbb{S}^n \setminus C$ be homeomorphic ...
Agelos's user avatar
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6 votes
2 answers
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Do all homogeneous spaces have homogeneous compactifications?

Let $X$ be a separable metric space which is homogeneous, i.e. for every two points $x,y\in X$ there is a homeomorphism $h$ of $X$ onto itself such that $h(x)=y$. A compactification of $X$ is a ...
D.S. Lipham's user avatar
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2 votes
0 answers
199 views

Borel-Weil-Bott theorem for wonderful compactification in characteristic p

Are there any known results for a Borel-Weil-Bott theorem for the wonderful compactifications over characteristic $p$ (i.e., theorems that classify the cohomologies of all line bundles on a wonderful ...
Merrick Cai's user avatar
4 votes
0 answers
153 views

Ends of a metric space?

I'm looking for a definition of “ends” of a metric space that is well-defined even for non geodesic or locally finite metric spaces, invariant under quasi-isometries (or more generally coarse ...
user148575's user avatar
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0 answers
58 views

Algebraic characterisation of the end space of a proper geodesic space in terms of non-continuous functions

$\DeclareMathOperator\Bf{B_\mathrm{f}}\DeclareMathOperator\Bc{B_\mathrm{c}}\DeclareMathOperator\Cf{C_\mathrm{f}}\DeclareMathOperator\Cd{C_\mathrm{d}}\DeclareMathOperator\Cc{C_\mathrm{c}}$Based on a ...
Carlos Adrián's user avatar
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End space of non-compact 2-manifolds described with proper rays

I am wondering if the classification of noncompact surfaces given by Ian Richards can be stated with proper rays instead of nested sequences of connected open subsets with compact boundary. I asked ...
Carlos Adrián's user avatar
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1 answer
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Representing split-complex numbers as intervals and related compactification

Since there is an isomorphism between split-complex numbers and $\mathbb{R}^2$ with element-wise operations, of the following form $a + bj \leftrightarrow (a - b, a + b)$, one can think about a split-...
Anixx's user avatar
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Explicit toroidal compactification of Hilbert modular varieties

Hirzebruch's construction of toroidal compactification of Hilbert modular surfaces is explicit, namely one can explicitly choose rational polyhedral cone decomposition in a sort of optimal way using ...
GTA's user avatar
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0 answers
137 views

Ergodic action on product spaces

Let $(X_1 \times X_2,d\mu)$ be a measure space with $X_2$ compact. Suppose that we have a continuous (diagonal) action of a topological group $G$ on $X=X_1 \times X_2$. I know that the action of $G$ ...
Osheaga's user avatar
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Is the one-point compactification of $\mathbb{N}$ computably countable?

The one-point compactification $\mathbb{N}_\infty$ of $\mathbb{N}$ is obtained from the discrete space $\mathbb{N}$ by adjoining a limit point $\infty$. It may be identified with the subspace of ...
Andrej Bauer's user avatar
6 votes
1 answer
296 views

How complicated can the path component of a compact metric space be?

Let $X$ be a compact metric space and $P$ be a path component of $X$. Since we are not assuming $X$ is locally path connected, $P$ must need not be open nor closed. Certainly, $P$ must be separable ...
Jeremy Brazas's user avatar
3 votes
0 answers
177 views

Smooth toric compactification of $\mathbb C^n$

By a compactification $(X,Y)$ of $\mathbb C^n$, we mean an irreducible compact complex space $X$ and a closed analytic subspace $Y\subset X$ such that $X\setminus Y$ is biholomorphic to $\mathbb C^n$. ...
Hang's user avatar
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3 votes
0 answers
239 views

Functoriality for compactifications of locally symmetric spaces

Let $X$ be a symmetric space associated to an algebraic group $G$ defined over $\mathbb{Q}$ and $G(\mathbb{R})$ acts on $X$ from the left. Let $\Gamma \subset G(\mathbb{Q})$ be an arithmetic subgroup ...
random123's user avatar
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2 votes
1 answer
148 views

Inducing maps between Martin boundaries

This is a reworking of a question I asked on math.se. Given two countable discrete metric spaces $X_{1}$ and $X_{2}$, each equipped with a (irreducible and transient)* random walk given by transition ...
Robert Thingum's user avatar
8 votes
1 answer
258 views

Characterization of pretty compact spaces

This is a cross post from MSE. I believe that the following problem have already been considered by some sophisticated topologist. Definition 1. A non-compact Hausdorff topological space $X$ is called ...
Norbert's user avatar
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1 vote
1 answer
212 views

Fixed points of one-point-compactification

Let $M$ be a locally compact (Hausdorff) space, and $g:M\to M$ an isomorphism (think of an action of a finite cyclic group). By some generalities one can show that the "obvious" map $(M^g)^+\...
Leonard's user avatar
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6 votes
0 answers
152 views

Completion/Compactification of a Kähler metric on $\mathbb C^2$

Consider $\mathbb{C}^{2}$ equipped with the Kähler form $$ \omega_{\mu}=\frac{i}{2} \partial \bar{\partial} \log \left(\left(1+|z|^{2}\right)^{\mu}+|w|^{2}\right), $$ where $\mu$ is a positive real ...
Robbixmaths's user avatar
10 votes
1 answer
319 views

Possible cardinalities of the remainders of compactifications of $\Bbb R$

With the usual topology on $\Bbb R$, a compactification $\mathrm{id}_{\Bbb R}:\Bbb R\to v\Bbb R$ can have a remainder $v\Bbb R \setminus \Bbb R$ of cardinality $1,2, 2^{\aleph_0}=\mathfrak c,$ or $2^{\...
DanielWainfleet's user avatar
8 votes
1 answer
417 views

Does a flat compactification always exist?

Let $\pi:X\to S$ be a separated flat morphism of finite type of Noetherian schemes. Does $\pi$ necessarily factor as an open immersion followed by a proper flat morphism? The analogue of this question ...
user avatar
5 votes
0 answers
263 views

Intuition for the McGerty-Nevins compactification of quiver varieties

In Section 4 of the paper Kirwan surjectivity for quiver varieties (Inventiones Math. 2018) McGerty and Nevins define a compactification of the moduli space of representations of the preprojective ...
Yellow Pig's user avatar
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0 votes
0 answers
156 views

A ``1-soft'' improvement of the Parovichenko theorem

This is a ``1-soft'' modification of this problem. We start with the necessary definitions. Definition 1. A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called 1-soft if for any ...
Taras Banakh's user avatar
  • 40.9k
4 votes
1 answer
263 views

Nowhere compact subsets of the plane

Suppose $X\subseteq \mathbb R^2$ is nowhere compact ($X$ has no compact neighborhood) and non-empty. Can $X$ be densely embedded into the plane? In other words, is there a dense set $X'\subseteq ...
D.S. Lipham's user avatar
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1 vote
1 answer
206 views

The Stone-Čech compactification of a inverse system

Is the Stone-Čech compactification of the inverse limit of an inverse system $\left\{ X_{i},f_{ij},I\right\} $ of Tychonoff spaces equal to the limit of the inverse system $\left\{ \beta X_{i},\beta ...
Mehmet Onat's user avatar
  • 1,161
36 votes
1 answer
3k views

Is there a general theory of "compactification"?

In various branches of mathematics one finds diverse notions of compactification, used for diverse purposes. Certainly one does not expect all instances of "compactification" to be specializations of ...
Tim Campion's user avatar
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1 vote
0 answers
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Subspaces of compact spaces and quotients of Hausdorff spaces

Let $\operatorname{Top}$ be the class of topological spaces. Furthermore, let $\mathcal{U}\subset\operatorname{Top}$ and $\mathcal{V}\subset\operatorname{Top}$ classes satisfying the following ...
cl4y70n____'s user avatar
0 votes
1 answer
88 views

Does surjective map induce surjective map on Hewitt real compactifications?

Let $\beta X$ be the Stone-Čech compactification and $\upsilon X$ be the Hewitt real compactification of a completely regular space $X$. It is well known that any continuous surjective map $f:X\...
Mehmet Onat's user avatar
  • 1,161
3 votes
1 answer
186 views

Research in compactifications of locally compact spaces

I would like to know how is it going the research in compactifications of locally compact Hausdorff spaces. Are there people doing this? Are there relevant conjectures on it?
Lucas Henrique's user avatar
3 votes
0 answers
190 views

Universal closure of schemes à la Nagata

Nagata compactification theorem is the following fundamental result: Let $S$ be a qcqs scheme. Let $X$ be a separated $S$-scheme of finite type. Then there exists a proper $S$-scheme $\overline{X}$ ...
user avatar
6 votes
1 answer
173 views

Is the conformal compactification of $M \setminus \{ p \}$ unique?

Let $(M,c)$ be a compact conformal manifold and $p \in M$. $M$ is a conformal compactification of $M \setminus \{ p \}$, because the embedding $M \setminus \{p\} \hookrightarrow M$ is an isometry. ...
user143031's user avatar
10 votes
2 answers
435 views

Which points in the Samuel compactification of a metric space $X$ are limits of uniformly discrete subsets of $X$?

Given a metric space $(X.d)$ the Samuel compactification of $X$, written $sX$, is the unique compactification with the property that if $Y$ is an arbitrary compact Hausdorff space and $f:X\rightarrow ...
James Hanson's user avatar
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2 votes
1 answer
73 views

Are compactifications of completely $T_{4}$ spaces completely $T_{4}$?

The title is the question. Given a locally compact completely $T_{4}$ space $X$ (every subspace is $T_{4}$) and a (Hausdorff) compactification $\overline{X}$ of $X$, is $\overline{X}$ also completely ...
Robert Thingum's user avatar
2 votes
1 answer
122 views

Naive compactification of $\mathbb{C}^*$-fibrations

Let $\pi:X \to Y$ be a $\mathbb{C}^*$-fibration between complex manifolds in the sense that there exists a fixed integer $a$ such that for every $y \in Y$, $\pi^{-1}(y)=(\mathbb{C}^*)^a$. Suppose ...
Jana's user avatar
  • 2,022
2 votes
1 answer
473 views

Stone-Cech Compactification of the real line

I have a question in $\beta\mathbb{R}$, the Stone-Cech compactification of the real line $\mathbb{R}$. My question is: is $\beta\mathbb{R}$ a $\mathrm{F}$-space, i.e., the closure of two disjoint open ...
user132068's user avatar
1 vote
0 answers
370 views

The Deligne-Mumford Compactification for Closed Surfaces

I am reading this note on super-Riemann surfaces. In the second paragraph of section 7.4.1 (page 87), there is a statement that I am trying to understand: The compactified moduli space of closed ...
QGravity's user avatar
  • 969
6 votes
1 answer
335 views

Is each Parovichenko compact space homeomorphic to the remainder of a soft compactification of $\mathbb N$?

Definition 1. A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called soft if for any disjoint sets $A,B\subset\mathbb N$ with $\bar A\cap\bar B\ne\emptyset$ there exists a ...
Taras Banakh's user avatar
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5 votes
1 answer
284 views

Is each compactification of $\mathbb N$ soft?

Definition. A compactification $c\mathbb N$ of the countable discrete space $\mathbb N$ is defined to be soft if for any disjoint sets $A,B\subset\mathbb N\subset c\mathbb N$ with $\bar A\cap\bar B\ne\...
Taras Banakh's user avatar
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12 votes
1 answer
570 views

Compactification of 6d (2, 0) SCFT on 4-manifolds

This question is about the 6d (2, 0) superconformal field theory (also called 'theory X' by some people). This SCFT, which can be considered as a relative quantum field theory (see here for a ...
user avatar
3 votes
2 answers
520 views

What are the components of the Stone-Cech Remainder?

Suppose $X = \displaystyle\bigsqcup_{i \in I} X_i$ is the disjoint union of infinitely many continua. The components of the Stone-Cech remainder $X^*$ can be described as follows. Treat $I$ as a ...
Daron's user avatar
  • 1,761
5 votes
1 answer
1k views

One point compactification of $(\mathbb{C}^{\ast})^n$

I would like to know if there is a closed form formula for the homotopy type of $\widehat{(\mathbb{C^{\ast}})^n}$? For example, it is not difficult to see that $\widehat{\mathbb{C^{\ast}}}$ has the ...
Priyavrat Deshpande's user avatar
1 vote
0 answers
41 views

Do maximally almost periodic groups embed homeomorphically into their Bohr compactifications? [duplicate]

If $G$ is a Hausdorff topological group and $bG$ is its Bohr compactification, Wikipedia says that $G$ is called maximally almost periodic (MAP) if and only if the natural map $i : G \to bG$ is ...
Alex M.'s user avatar
  • 5,217
6 votes
1 answer
289 views

Is there a compactification with nontrivial connected remainder?

Question: Let $X$ be a continuum and $p \in X$. Under what conditions does there exist a compactification $\gamma (X-p)$ with $\gamma (X-p) - (X-p)$ connected and nondegenerate? Throughout, $X$ is a ...
Daron's user avatar
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