Questions tagged [compactifications]
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110
questions
36
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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 ...
35
votes
2
answers
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"Transitivity" of the Stone-Cech compactification
Let $\beta \mathbb{N}$ be the Stone-Cech compactification of the natural numbers $\mathbb{N}$, and let $x, y \in \beta \mathbb{N} \setminus \mathbb{N}$ be two non-principal elements of this ...
34
votes
6
answers
3k
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Compactification theorem for differentiable manifolds ?
Just parallelling this question, that seemed not to admit an easy answer at all, let's "soft down" the category and ask the same thing in the case of $\mathcal{C}^{\infty}$-differentiable manifolds [...
20
votes
1
answer
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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 ...
13
votes
2
answers
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Direct construction of the Stone-Čech compactification using ultrafilters?
If $X$ is a set (regarded as a discrete space), its Stone-Čech compactification can be identified with the set of ultrafilters on $X$ with its natural (Stone) topology. If $X$ is a general ...
12
votes
1
answer
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Reference Request for Drinfeld and Laumon Compactifications
Background
Let $X$ denote a smooth projective curve over $\mathbb{C}$ and let $G$ denote a semi-simple simply connected algebraic group over $\mathbb{C},$ which has associated flag variety $G/B.$
...
12
votes
1
answer
566
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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 ...
11
votes
1
answer
407
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A product on the square roots of unit matrix
There is a strange product that takes two square roots of unit matrix, say $A$ and $B$, $A^2=I$, $B^2=I$ to a square root again,
$$ A\star B=(A+B)^{-1}(A-B+2I), \qquad (A\star B)^2=I$$
Could anybody ...
10
votes
2
answers
569
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Non-bimeromorphic compactifications
Let $X$ be a (smooth, non-compact) complex space. By a compactification of $X$ we mean a compact complex space $\overline X$ which contains a dense open subset biholomorphic to $X$ (we shall identify ...
10
votes
1
answer
319
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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^{\...
10
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2
answers
431
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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 ...
10
votes
1
answer
450
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Monograph or rich survey on infinite-dimensional Riemann manifolds
I'm working with the space of smooth curves $\mathcal{C}$ in a smooth manifold $M$, having (different, pre-determined) fixed endpoints. I'd like to endow it with a Riemann structure (I already have a ...
10
votes
0
answers
530
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When is the one-point compactification well-pointed?
This is a follow up to my previous
question.
Question:
Is there a reasonably natural set of conditions which guarantee that the one-point
compactification $X^+$ of a locally compact Hausdorff ...
9
votes
2
answers
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Stone-Čech compactification of $\mathbb R$
Let $\beta X$ - is a Stone-Čech compactification of $X$. $I=(-1,1)$ - is an interval of the real line. Is it true that $\beta \mathbb R\setminus I = \beta(\mathbb R\setminus I)$? In other words, it ...
9
votes
1
answer
698
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Compactifications of varieties with small complement
Let $X$ be a smooth variety over an algebraically closed field $k$. If it makes things easier, $X$ may be assumed to be quasi-projective. By Nagata (or quasi-projectivity) there exists a proper ...
9
votes
0
answers
366
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Embedding $\beta\mathbb{N}$ into a product of Cantor sets
Let us consider $\beta\mathbb{N}$, the Stone-Čech compactification of the natural numbers (where we do not take $0$ to be a natural number, so the only idempotent elements are nonprincipal ...
8
votes
1
answer
258
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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 ...
8
votes
2
answers
2k
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End point compactification for metric spaces
Freundenthal introduced ends of topological spaces and the end point compactification of locally compact topological spaces adding one point for each end of the topological space (see here).
For ...
8
votes
1
answer
1k
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Wonderful compactification
Suppose $G$ is a semi-simple group of adjoint type over an algebraic closed field, and $X$ its wonderful compactification a la De Concini and Procesi. Let $P=MU$ be a parabolic subgroup in $G$, and ...
8
votes
1
answer
648
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Sheaf (Gieseker) compactification of moduli space of vector bundles
I am given to understand that the moduli space $M_k^G$ of $G$ vector bundles with second Chern class $c_2=k$ over an algebraic curve/variety (for me a Riemann surface is enough/projective space for ...
8
votes
1
answer
417
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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 ...
8
votes
0
answers
500
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Deligne-Mumford moduli spaces and compactification of symmetric matrices
The real Deligne-Mumford moduli space $\bar M_{0,n+1}(\mathbb R)$ of stable genus
zero curves with $n+1$ marked points is a compactification of the space of
configurations of $n$ distinct ordered ...
7
votes
2
answers
704
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Existence of a compactification of $\mathbb{R}$ with $\aleph_0$ remainder
We know that the space $\mathbb{R}$ has compactifications with one point remainder, and two point remainder. but there is no compactification of $\mathbb{R}$ with three point remainder and the same ...
7
votes
1
answer
436
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Are countable FC-groups maximally almost periodic?
An FC-group is a group in which every element has a finite conjugacy class. A group G is said to be maximally almost periodic if there is an injective homomorphism from G into a compact Hausdorff ...
7
votes
2
answers
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A question about the Stone–Čech compactification of discrete spaces
Let $D(\kappa)$ be the discrete space of cardinality $\kappa$, and $\beta D(\kappa)$ its Stone–Čech compactification.
Is there, for every infinite cardinal $\kappa$, a subset $Y \in [\beta D(\kappa)]^...
7
votes
1
answer
200
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An explicit description of neighborhoods of the rank 2 boundary in the Satake Compactification of $\mathbf{A}_2$
My Motivation: I'm having a hard time following the description of the topology in the Satake Compactification of locally symmetric spaces. The group theory is something I'm finding a bit tricky to ...
7
votes
1
answer
564
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Flatly compactifiable morphisms
Let $f:U \to S$ be a flat morphism. Let us say that $f$ is flatly compactifiable if there exists a proper morphism $\bar{f}:X \to S$ and a closed subscheme $Z \subset X$ such that
1) $U = X \...
7
votes
0
answers
492
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Compactification of the moduli space of Kähler manifolds with negative constant scalar curvatures
Moishezon compactification is very important in the study of the moduli space of varieties which admit canonical metrics. Moishezon showed that any non-projective Moishezon manifold $X$, after a ...
7
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0
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456
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Alterations of regular varieties
Let $X$ be a regular quasi-projective variety over a perfect field $k$. The existence of a "good compactification" of $X$, i.e. a regular projective variety $\bar{X}$ with an embedding $X\...
6
votes
2
answers
818
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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 ...
6
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2
answers
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A question about some special compactifications of $\mathbb{R}$
We Know that the topological space $Y$ is a compactification of the topological space $X$, if the space $Y$ is compact and hausdorff and $X$ is dense in $Y$. If for a positive integer $n$ we have a ...
6
votes
2
answers
991
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Bohr compactification as a topological compactification
Let $G$ be a locally compact Hausdorff group. Denote its Bohr compactification by $bG$.
Despite group structure, $G$ has several (Hausdorff) compactifications that, in a sense, the smallest one is ...
6
votes
1
answer
334
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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 ...
6
votes
1
answer
171
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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.
...
6
votes
1
answer
295
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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 ...
6
votes
1
answer
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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 ...
6
votes
0
answers
152
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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 ...
6
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0
answers
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A compactification of the non-negative rationals with the discrete topology
Let $S$ be the set of non-negative rational numbers. (If it makes any difference, feel free to take the non-negative dyadic rationals instead.) Let $B=\ell_\infty(S)$; as a ${\rm C}^*$-algebra this is ...
5
votes
2
answers
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Ends of topological spaces. Why independent of choice of ascending sequence of compact subsets?
Quoting from http://en.wikipedia.org/wiki/End_(topology):
"Let X be a topological space, and suppose that
K1 ⊂ K2 ⊂ K3 ⊂ · · ·
is an ascending sequence of compact ...
5
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1
answer
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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 ...
5
votes
1
answer
284
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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\...
5
votes
3
answers
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Are Hausdorff compactifications of a Tychonoff space $X$ in one-to-one correspondence with completely regular subalgebras of $BC(X)$?
Let $X$ be a completely regular (Tychonoff) topological space. It is known that if $\mathscr F\subseteq C(X,[0,1])$ separates points and closed sets (that is, for every closed set $E\subseteq X$ and $...
5
votes
1
answer
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Reference for homeomorphism between "analytic" compactification of $M_{g,n}$ and Deligne-Mumford compactification
There are several natural ways to endow the compactification of the space of
marked Riemann surfaces $M_{g,n}$ ($2g+n\geq 3$), with a topology, which is
defined using "differential geometric or ...
5
votes
0
answers
262
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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 ...
5
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0
answers
270
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On toroidal compactifications of Hilbert Kuga-Sato varieties
Let $F$ be a totally real field of degree d. There are Hilbert modular varieties over $\mathbb{Q}$ that paramatrize abelian varieties of dimension d with an action of $\mathcal{O}_F$ the ring of ...
4
votes
2
answers
838
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Non-uniqueness of smooth compactification
Let $U$ be a smooth quasi-projective variety. Does there always exist a smooth compactification of $U$? If not always when can we have smooth compactification?
In particular, suppose $X$ is a ...
4
votes
3
answers
938
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How is a MacNeille completion "universal" like a beta-compactification is "universal"?
The beta-compactification of a topological space is characterized as the largest space such that every mapping from the original space to another (range) space can be extended through to a mapping ...
4
votes
1
answer
513
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Uniqueness of smooth compactification upto a smooth morphism
By a $k$-variety, we will mean a separated scheme of finte type over a field $k$. Let $k$ be of characteristic 0. Given a smooth quasi-projective $k$-variety $X$, there is a projective $k$-variety $\...
4
votes
1
answer
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What is the Stone–Čech compactification of a dense set of $\beta N \setminus N$?
Is the Stone–Čech compactification of a dense $G_\delta$-set $X \subset\beta N \setminus N$ homeomorphic to $\beta N \setminus N$? Here, $\beta N \setminus N$ is the complement of $N$ in the Stone–...
4
votes
2
answers
532
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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 ...