Questions tagged [compactifications]
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129 questions
2 votes
0 answers
217 views
Are there spaces without a Fan-Gottesman compactification?
Def 1. Call $\mathcal{B}$ a Fan-Gottesman base on $X$ if it's an open base such that $\emptyset, X\in\mathcal{B}$ $U\in\mathcal{B}\implies X\setminus \overline{U}\in \mathcal{B}$ $U, V\in\mathcal{B}\...
- 2,307
5 votes
0 answers
111 views
Fadell-Neuwirth fibration for simplicial variant of Fulton-MacPherson compactification
Let $M$ be a smooth manifold, and let $C_n(M)$ denotes the space of ordered configurations of $n$ distinct points in $M$. Given an embedding of $M$ into $\mathbb{R}^m$, one can define the Fulton-...
- 151
1 vote
0 answers
89 views
Natural renormalizations of K theory classes on open varieties
I have a question about natural assignments of $K$-theory classes to open varieties. The question has turned out rather long, for which I apologize. All varieties I consider are smooth over $k=\mathbf{...
- 241
3 votes
0 answers
65 views
Toroidal Compactifications and Equivariance
Let $H \trianglelefteq G \subset \operatorname{O}(2,n)$ be arithmetic subgroups with $H$ normal in $G$ and let $Q=G/H$, which is assumed to be finite. If $D$ is the symmetric space associated with $\...
- 181
3 votes
1 answer
159 views
$SL(2,\mathbb R)$ and group structures on the compact solid $2$-torus
I am curious about the following, rather crude questions. If they are easily seen to be trivial, I apologise in advance. It is well-known that $SL(2,\mathbb R)$ is diffeomorphic to the open solid $2$-...
- 2,174
4 votes
1 answer
202 views
Is a space homeomorphic to $[0,1)$ a closed retract of this metric space?
An arc is a space $B$ homeomorphic to $[0,1]$. If $\phi:[0,1]\to B$ is the homeomorphism, we will denote the arc as $[a,b]$ if $\phi(0)=a$ and $\phi(1)=b$. A chain of arcs in a topological space $X$ ...
- 1,903
3 votes
1 answer
123 views
Separation in perfect compactifications
I refer to the book Isbell, Uniform spaces, 1970. p.97. Definition. A set $C$ separates $A$ and $B$ in a space $X$ if $X\setminus C=M\cup N$ where $M,N$ are separated sets containing $A,B$. And $M,N$ ...
- 73
3 votes
1 answer
212 views
Tychonoff spaces which are sequentially closed in their Stone–Čech compactifications
The sequential closure $\operatorname{sqcl}(A)$ of $A\subseteq Y$ is the set of all $x\in Y$ such that there is a sequence $x_n\in A$ with $x_n\to x$. If the space $X$ is nice enough, then $\...
- 2,307
14 votes
1 answer
1k views
Mistake on article about Bohr compactification?
$\DeclareMathOperator\b{b}\newcommand\B{{\operatorname B}}$I wish to get help understanding the content of two theorems of [Iva] that seem mutually contradictory. First some context. Let $\b(\mathbb{R}...
- 323
8 votes
1 answer
194 views
About path-connected components of the Bohr compactification of $\mathbb{R}^d$
Let ${\rm b}(\mathbb{R}^d)$ denote the Bohr compactification of $\mathbb{R}^d$, with $d\in\mathbb{N}$. This is the Pontryagin dual of the group $\mathbb{R}^d_d$, corresponding to $\mathbb{R}^d$ with ...
- 323
0 votes
1 answer
186 views
A question about G-Hewitt spaces
In the paper linked below, S. A. Antonyan gives the following proposition without proof (in fact all results are given without proof). I need a proof of this theorem. If anyone has information on this ...
- 1,661
1 vote
2 answers
296 views
Spaces $X$ with every compactification $0$-dimensional with $\beta X\setminus X$ not locally compact
Previously, in this post I've shown the following characterization of spaces with only zero-dimensional compactifications: Theorem. Let $X$ be strongly zero-dimensional and $\beta X\setminus X$ ...
- 2,307
11 votes
2 answers
411 views
Spaces with every compactification $0$-dimensional which aren't locally compact
Recently I've proven the following theorem Theorem. Let $X$ be a zero-dimensional locally compact Hausdorff space. Then the following are equivalent: Every compactification of $X$ is zero-dimensional....
- 2,307
13 votes
1 answer
404 views
Is there a metric compactification that doesn't create new paths?
Every separable metric space $A$ has a metrizable compactification, i.e. a compact metrizable space $X$ for which $A$ embeds topologically as a dense subspace of $X$. There are many approaches to ...
- 7,710
0 votes
0 answers
232 views
Behavior of subtree of $\mathbb{Z}^2$ embedded in $\mathbb{C}$ under compactification of the latter to the riemann sphere
I consider a countable subtree $T$ of the integer lattice isomorphic to $\mathbb{Z}^2$ with directed edges. It shall be embedded in $\mathbb{C}$ where the edge $(u,v)$ points from $u$ to $v$ if and ...
- 329
-2 votes
1 answer
168 views
Mismatch between equivalent definitions of the Bohr compactification of the reals
I feel I'm overlooking something very silly. The Bohr compactification of $\mathbb R$ has two equivalent definitions. The set of (possibly discontinuous) homomorphisms $\mathbb R \to \mathbb T$ under ...
- 2,085
10 votes
1 answer
813 views
Is there an explicit construction of the Bohr Compactification of the Integers?
Is it possible to explicitly describe the Bohr compactification of $\mathbb Z$? This is equivalent to describing all the group homomorphisms $\mathbb R/\mathbb Z \to \mathbb R/\mathbb Z$ including ...
- 2,085
1 vote
0 answers
142 views
When is the "Gelfand Remainder" compact?
Suppose we have a noncompact Hausdorff space $S$ and a Banach algebra $A \subset C^*(S,\mathbb R)$ of the space of real-valued bounded functions on $S$. For niceness let's assume $A$ separates the ...
- 2,085
3 votes
0 answers
159 views
Fulton-MacPherson compactifications (and wonderful compactifications) as relative Proj
Let $X$ be a (smooth complex projective) variety. The Fulton-MacPherson compactification $X[n]$ is obtained from $X^n$ by blowing up the diagonals in a certain order. Is it possible to write down a (...
- 350
1 vote
0 answers
167 views
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 $\Delta_n$, ...
- 1,834
1 vote
1 answer
223 views
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 ...
2 votes
0 answers
153 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 ...
- 343
4 votes
2 answers
742 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 ...
- 339
0 votes
1 answer
261 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}...
- 1,524
1 vote
0 answers
297 views
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 \...
- 1,524
1 vote
0 answers
118 views
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 ...
- 93
2 votes
0 answers
196 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}...
- 41
2 votes
1 answer
162 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-...
- 21
4 votes
1 answer
295 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 ...
- 2,086
7 votes
2 answers
891 views
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 ...
- 3,691
2 votes
0 answers
252 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 ...
- 41
5 votes
0 answers
249 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 ...
- 947
3 votes
0 answers
92 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 ...
1 vote
0 answers
181 views
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 ...
1 vote
1 answer
266 views
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-...
- 10.4k
4 votes
0 answers
178 views
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 ...
- 1,044
0 votes
0 answers
196 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 $G$ ...
- 59
19 votes
1 answer
1k views
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 ...
- 51.4k
6 votes
1 answer
401 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 ...
- 7,710
3 votes
0 answers
275 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 $\mathbb C^n$. ...
- 2,709
3 votes
0 answers
294 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 ...
- 483
3 votes
1 answer
193 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 ...
- 223
8 votes
1 answer
372 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 ...
- 1,697
1 vote
1 answer
262 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)^+\...
- 151
6 votes
0 answers
186 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 ...
- 81
11 votes
1 answer
731 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^{\...
- 347
9 votes
1 answer
492 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 ...
user158636
5 votes
0 answers
343 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 ...
- 3,454
0 votes
0 answers
203 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 ...
- 44.3k
4 votes
1 answer
329 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 ...
- 3,691