All Questions
5,184 questions
0
votes
1
answer
360
views
Triviality of finite fiber bundles [closed]
Hello,
I suspect the following is true and easy but I am unable to prove. Suppose (E, B, π, F) is a fiber bundle, where E,B are compact and F is finite, prove that E is a trivial fiber bundle. Any ...
6
votes
1
answer
1k
views
Some questions on Nicolai Reshetikhin's lectures on quantization of gauge theories.
This in reference to this fascinating lecture by Nicolai Reshetikhin-
http://staff.science.uva.nl/~nresheti/Holb-Quant-Gauge.pdf
Given what is said on page 13 in section 4.1 its not clear to me why ...
- 3,541
2
votes
1
answer
218
views
Shrinkable maps and universal weak equivalences
Recall that a morphism $f:X \to B$ is called shrinkable is there exists a section $s:B \to X$ together with a homotopy $$H:I \times X \to X$$ from $sf$ to $id_X$ over $B,$ i.e. for all $t$, the map $$...
- 15.5k
5
votes
2
answers
524
views
space of homotopy equivalences of $S^1$
Does the space of homotopy equivalences of $S^1$ deformation retract onto the space of homeomorphisms of $S^1$? If so, does anyone have a reference?
I found that Kneser proved that $Homeo(S^1)$ ...
- 53
8
votes
2
answers
689
views
What does the space induced by this unusual metric(?) on R/Z look like?
The motivation for this question comes from music theory. Dmitri
Tymoczko models "good" voice leading as minimizing distance between
pitches in successive chords. While this theory works well for ...
- 3,093
2
votes
0
answers
203
views
Faithful actions of finite groups on topological spaces
Suppose that $G$ is a finite group acting faithfully on a topological space $X$. In the smooth setting, one can deduce that for each $x$ in $M$, the induced map $$G_x \to Diff_x\left(M\right)$$ from ...
- 15.5k
3
votes
1
answer
695
views
Hausdorff-dimension of connected closed subsets of R^2
Let $A \subseteq \mathbb{R}^2$ be closed and connected, and for any $c \in \mathbb{R}$, let $A_c \subseteq A$ be a closed and connected subset of $A$, s.t. for $c \neq d$ we have $A_c \cap A_d = \...
- 4,727
2
votes
1
answer
526
views
Meaning of "Compact" in 1932 Paper by van der Waerden "Continuity Theorem for Semisimple Lie Groups".
I am putting together an exposition on Lie theory; maths research is not my day job, let alone real maths history, so apologies in advance for any ignorance shown by these questions.
I am attempting ...
- 1,161
0
votes
1
answer
280
views
"Skein" equations sets that can reduce any graph
Consider for example the approach to the Kuperberg G2 or the Yamada polynomial. I don't know whether relations of graph (in contrast to knot) theory are also usually called "skein" but
I simply carry ...
- 4,829
6
votes
3
answers
750
views
Characterization of Tychonoff spaces in terms of open sets
Metrizability and complete regularity are topological properties that are, in a sense, different from the Hausdorff condition because they are not defined purely in the terms of the open sets, but ...
- 656
6
votes
2
answers
554
views
Automorphism groups and etale topological stacks
Recall that an etale topological stack is a stack $\mathscr{X}$ over the category of topological spaces (and open covers) which admits a representable local homeomorphism $X \to \mathscr{X}$ from a ...
- 15.5k
25
votes
6
answers
5k
views
Is there a classification of open subsets of euclidean space up to homeomorphism?
I hope this question is reasonable enough to have a well known answer. i.e either there is a simple invariant (like the homotopy groups) that characterizes the homeomorphism type of such set among ...
- 2,027
8
votes
3
answers
846
views
A compactness property for Borel sets
Is the following generalised compactness property of Borel sets in a Polish space consistent with ZFC?
($*$) Let $\mathcal{B}$ be a family of $\aleph_1$-many Borel sets. If $\bigcap \mathcal{B} = \...
- 866
3
votes
1
answer
292
views
Can a closed trefoil appear as a space-time "cut" of an open trefoil?
An open trefoil is a trefoil tied in an infinitely long line. An open trefoil that is at rest in (flat) space, in space-time is a knotted hypersurface.
Different observers in space-time have ...
- 75
2
votes
2
answers
1k
views
How many lanes has a freeway? (Crossing free Kuperberg G2, that is.) EDITED
For referencing, I keep the original title and post and ask
only about the simplest case (and forget the freeway with crossings
for now).
Consider a trivalent graph, e.g. the dodecahedron or cube net....
- 4,829
1
vote
1
answer
1k
views
Representations of regular maps (four color theorem)
For the scope of the four color problem and without lack of generality, maps can be represented in different ways. This is generally done to have a different perspective on the problem.
For example, ...
- 351
7
votes
1
answer
2k
views
Universally measurable sets and weak topology
After I posted this question, a couple of months ago, and got from MO-users several
good hints, I think i'm ready, after some study, to ask another related question (or rather, to focus on the main ...
user11618
8
votes
1
answer
1k
views
Ring of continuous functions, reference request.
I am looking for a reference for the following facts in functional analysis and topology. (If these "facts" are not true, I suppose I'm looking for the closest approximation which is true.)
Let $X$ ...
- 13.3k
6
votes
2
answers
492
views
Distinct, non-homeomorphic, profinite topologies on a given abstract group ?
Just a silly little question which arose in connection with infinite Galois groups and their Krull topology:- can a given abstract group be endowed with distinct, non-homeomorphic, profinite ...
13
votes
3
answers
8k
views
$fgf = f$, $gfg = g$, $fg$ not necessarily identity, what is this called?
A very simple question, I just totally forgot how it was called, and Google is not helping.
There's a pair of functions $f:X\to Y$, $g:Y\to X$.
$fgf = f$, $gfg = g$, but $fg$ and $gf$ don't need to ...
- 241
3
votes
1
answer
1k
views
$\Delta_{2}^{1}$-hard set?
Hello everybody!
I'm interested in $\Delta_{2}^{1}$ subsets of Polish spaces, i.e. those sets that are both $\Pi_{2}^{1}$ and $\Sigma_{2}^{1}$ in the boldface hierarchy of Polish spaces.
There is a ...
user11618
3
votes
4
answers
1k
views
Topologically split extensions of topological groups
Let $1 \to N \to G \to H \to 1$ be a short exact sequence of topological groups. Such an exact sequence is said to be topologically split if $G$ is $N \times H$ as a
topological space.
Can someone ...
- 49
2
votes
0
answers
254
views
Simple terminology question about the Dubrovnik (Kauffman) polynomial
In my S matrix classification attempts I encounter a lot of
Dubrovnik polynomials of the form D(z-1/z,z^n) and D(-z+1/z,z^n).
[Second variable is for writhe, n is an integer; for the first I don't
...
- 4,829
5
votes
2
answers
587
views
Is there a formula to count how many different topological regular maps can be created with n faces (on a sphere)?
Notes:
For "regular" I intend maps in which the boundaries form a 3-regular planar graph
For "different" I intend maps that cannot be topologically transformed one into another (faces have ...
4
votes
1
answer
1k
views
Cantor set and Hilbert cube, or anything else?
I have recently rediscovered (after several years) the wonder of the Cantor set (so rich and so beautiful!). I have two questions that are unrelated, but they are both about Cantor sets.
Let $K$ be a ...
- 2,829
0
votes
2
answers
796
views
Extending Continuous Sublinear maps on dense subsets of a Banach space
Suppose X' and Y are Banach spaces and X is a linear subspace dense in X'. Let T be a continuous map of X to Y satisfying:
(1) ||T(x+y)|| is less than or equal to ||T(x)||+||T(y)||.
Please prove ...
- 11
4
votes
1
answer
1k
views
Are there two non-homeomorphic finite regular CW complexes with isomorphic face posets?
Hello again! More of the same bumbling down the road of algebraic topology. This time, I am trying to figure out exactly how much information the face poset of a CW complex encodes. It has often ...
- 2,392
5
votes
3
answers
1k
views
On closed totally disconnected subgroups of connected real Lie groups
So the following statement seems to be obvious but I don't see how to prove it:
Q: How does one prove that a closed totally disconnected subgroup of a connected real Lie group is discrete?
Note that ...
- 7,609
2
votes
1
answer
259
views
Nuclearity of certain semigroup crossed product C*-algebras
This question is related to this question link.
Suppose we have an (abelian) semigroup $S$ acting by endomorphisms on a $C^*$-algebra A giving rise to a semigroup crossed product $B = A\rtimes S$. ...
- 2,029
3
votes
1
answer
392
views
Can cones (toric monoids) be built as colimits of their faces?
Suppose $L$ is a lattice (free abelian group) and $\sigma$ is a (pointed) spanning rational cone in $L\otimes\mathbb Q$. Then $M=L\cap \sigma$ is a monoid with $M^{gp}=L$. A monoid of this form is ...
- 24.1k
5
votes
1
answer
401
views
Topological space associated to a real or complex scheme
Hi, consider a scheme $X$ of finite type over $\mathbb{R}$ (or $\mathbb{C}$). In Hartshorne's appendix B on 'transcendental methods' it is shortly mentioned how to assign a reasonable topological ...
- 103
3
votes
1
answer
404
views
Picking a representative in a continuous way
I'm hoping for some ideas/pointers here. I'm experimenting with a Livschitz theorem for functions on a locally compact Abelian group, where the periodic orbit sums take values in a closed subgroup.
...
- 23.2k
0
votes
0
answers
189
views
On Birman-Wenzlyfying the B2 spider
Prelude: First of all, let "S matrix" denote "an abstract 4D tensor satisfying
the usual isotypy rules (with no arrows!)". I'm busy trying to classify all possible
S matrices (paper pending) - it's ...
- 4,829
5
votes
1
answer
655
views
an example of a semigroup with solvable word problem but unsolvable power problem
We say that a semigroup $S$ has solvable power problem if there is an algorithm that takes as input an element $s \in S$ and decides whether or not there exist $m,n \in \mathbb{N}$ with $m \neq n$ and ...
- 549
3
votes
0
answers
637
views
Fixed point theorem for convex, closed multivalued mapping
There is well-known fixed point theorem theorem for multivalued l.s.c. maps, based on Michael selection theorem:
Suppose, that $X$ is compact, convex and metrizable in locally convex Hausdorff ...
- 696
0
votes
2
answers
643
views
Collinear vertices and definition of k-simplex
On page 120 of his Basic Topology, Armstrong defines the $k$-simplex in $\mathbb{E}^n$ with verices $v_0,\ldots,v_k$ to be complex hull of said vertices. (A similar definition is given on Wikipedia).
...
- 3,079
2
votes
1
answer
1k
views
Lebesgue covering dimension
Roughly from wikipedia: The covering dimension of a topological space $X$ is defined to be the minimum value of $n$ such that every finite open cover of $X$ has a finite open refinement in which no ...
- 6,034
9
votes
1
answer
1k
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A question concerning separate and joint continuity of bilinear maps
Suppose that $V$ is a locally convex topological vector space and $f:V^2 \to V$ is a bilinear map. Suppose that $C \subseteq V$ is compact and convex, $f$ maps $C^2$ into $C$ and
$f \restriction C^2$ ...
- 3,547
-8
votes
2
answers
1k
views
Special infinitary relations and ultrafilters
(This problem appeared in face of me trying to generalize my theory of (binary) funcoids to the theory of $n$-ary funcoids (I call them "multifuncoids") for arbitrary $n$.)
Let $I$ is some indexing ...
- 765
5
votes
2
answers
2k
views
Simple connectedness via closed curves or simple closed curves?
I've recently read some papers and books involving simply connected domains in Euclidean space (dimension at least 2), where domain is an open connected set. The usual definition is a (connected) set ...
34
votes
1
answer
2k
views
Square roots of $\mathbb R^{2n}$
Recently, Richard Dore asked us if $\mathbb R^3$ is the cartesian square of some space, and Tyler Lawson answered beautifully in the negative.
The even powers of $\mathbb R$ were left out in that ...
- 47.3k
231
votes
4
answers
16k
views
Is $\mathbb R^3$ the square of some topological space?
The other day, I was idly considering when a topological space has a square root. That is, what spaces are homeomorphic to $X \times X$ for some space $X$. $\mathbb{R}$ is not such a space: If $X \...
- 5,275
7
votes
1
answer
650
views
Cones, monoids, and the space of (very) ample divisors
An interesting and useful tool to study a projective variety is its ample cone. Understanding the structure of this cone reveals information about the variety, and it is an isomorphism-invariant so ...
- 1,871
8
votes
0
answers
833
views
Is there a generalization of Brouwer's fixed point theorem?
In essence, this is the same problem as in
“The generalization of Brouwer's fixed point theorem?”.
But now I am determined to be careful. The main question is
the following:
Is there any ...
- 6,891
1
vote
1
answer
515
views
Braid*Temperley-Lieb=?
I would be very astonished if this algebra isn't named.
You simply have the braid AND the Temperley-Lieb generator in
the algebra. Rules are the usual Reidemeister equivalents
plus the kink and ...
- 4,829
9
votes
1
answer
2k
views
The generalization of Brouwer's fixed point theorem?
Let $X$ be a contractible compact [edit: locally connected] topological space
(Hausdorff and second countable). Let $f\colon X\to X$
be a continuous map. Then (I suppose) $f$ has a fixed
point. ...
- 6,891
0
votes
0
answers
365
views
Finding paths in a path connected space
I'm looking for such literature as exists relevant to the following problem.
Problem Given a compact, path-connected region $E$ on the plane and a positive constant $r$. Find (if possible) a path ...
- 627
0
votes
2
answers
641
views
Looking for general approaches to show connectedness of topological groups
Let $G$ be a topological group. One general approach to show that $G$ is connected is the following:
For every subgroup $H\leq G$ (not necessarily closed) we have a projection map:
$$
\pi: G\...
- 7,609
5
votes
1
answer
523
views
Injections to binary sequences that preserve order
Suppose we have a countable set S with a total order. Can we give an injection from S to the set of finite binary sequences that end in all zeros that preserves the ordering? The order on binary ...
- 493
23
votes
13
answers
7k
views
What should be taught in a 1st course on smooth manifolds?
I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic ...