All Questions
13,925 questions
21
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5
answers
4k
views
Isomorphisms of Banach Spaces
Suppose $X$ and $Y$ are Banach spaces whose dual spaces are isometrically isomorphic. It is certainly true that $X$ and $Y$ need not be isometrically isomorphic, but must it be true that there is a ...
- 629
8
votes
2
answers
592
views
Base change for category objects in topological spaces
I was prompted by this question, but the motivation is different.
Suppose we have an internal category object in topological spaces, i.e. an object space X and a morphism space Y, together with ...
- 52.6k
8
votes
3
answers
698
views
L_p norm balls for 1<p<2 - is it always similar to an L_q norm ball for some q>2?
The L_1 ball in 2D is shaped like a diamond (L_1 is also known as the Manhattan norm). The L_∞ ball is shaped like a square (L_∞ is also known as the supremum norm). They are similar, i.e. have same ...
- 101
1
vote
4
answers
5k
views
11
votes
1
answer
336
views
cardinality of final coalgebras in Top
Let P be a polynomial functor from Top to Top, by which I mean a functor of the form P(X) = ∐i ≥ 0 Si × Xi where the Si are finite sets, all but finitely many of which are empty. ...
- 25.2k
7
votes
6
answers
2k
views
How to partition R^3 into pairwise non-parallel lines?
Problem. How to partition R^3 into pairwise non-parallel lines?
A possible solution is to stack infinitely many ``concentric'' hyperboloids, by increasing radius and decreasing slope. And don't forget ...
- 1,110
23
votes
6
answers
2k
views
Is there a topological description of combinatorial Euler characteristic?
There are a collection of definitions of "combinatorial Euler characteristic", which is different from the "homotopy Euler characteristic". I will describe a few of them and give some references, and ...
- 54.6k
56
votes
17
answers
13k
views
Atiyah-Singer index theorem
Every year or so I make an attempt to "really" learn the Atiyah-Singer index theorem. I always find that I give up because my analysis background is too weak -- most of the sources spend a lot of ...
- 44.8k
17
votes
8
answers
3k
views
Smooth classifying spaces?
Take G to be a group. I care about discrete groups, but the answer in general would be welcome too. There are the various ways to construct the classifying space of G, bar construction, cellular ...
- 1,422
10
votes
6
answers
2k
views
What is an example of a topological space that is not homotopy equivalent to a CW-complex?
It would also be nice if someone can explain this comment appearing on the Wikipedia page on CW-complexes:
"The homotopy category of CW complexes is, in the opinion of some experts, the best if not ...
- 775
4
votes
2
answers
439
views
Legendrian homotopy of curves in a contact structure?
I'm aware of the great body of work on Legendrian knot theory in contact geometry, but suppose I'm curious just about homotopy and not isotopy. How does one understand the space of Legendrian loops ...
- 13.6k
2
votes
8
answers
3k
views
The core question of topology
As I see it, the core question of topology is to figure out whether a homeomorphism exists between two topological spaces.
To answer this question, one defines various properties of a space such as ...
- 159
2
votes
1
answer
493
views
Convergence of Affine Transformations
Hi,
I was wondering if anyone could point me to any sources regarding the convergence of iterated affine transformation, i.e. sequences where {a_n} is a set of affine transforms and the sequence:
...
- 690
76
votes
9
answers
15k
views
understanding Steenrod squares
There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called Steenrod squaring: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from ...
- 6,069
12
votes
4
answers
2k
views
Elements of infinite order in a profinite group
Say G is a profinite group with elements of arbitrarily large order. Do elements of infinite order exist (A) if we assume G is abelian? (B) in general?
A start for (A): we can ask the same question ...
- 11.3k
17
votes
10
answers
3k
views
References for homotopy colimit
(1) What are some good references for homotopy colimits?
(2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will ...
- 12.8k
66
votes
5
answers
8k
views
Does homology have a coproduct?
Standard algebraic topology defines the cup product which defines a ring structure on the cohomology of a topological space. This ring structure arises because cohomology is a contravariant functor ...
- 661
22
votes
3
answers
2k
views
What is a TMF in topology?
What is a topological modular form? How are they related to 'normal' (number-theoretic) modular forms?
- 15.1k
10
votes
2
answers
924
views
Has anyone tabulated 2-knots? Would anyone like to try?
I'd love to have a list of 'small' $2$-knots, for some sense of small. It's not clear what one should filter by, but there are two obvious candidates
Write a movie presentation, and count the frames.
...
- 7,800
14
votes
5
answers
4k
views
Is the long line paracompact?
A manifold is usually defined as a second-countable hausdorff topological space which is locally homeomorphic to Rn. My understanding is that the reason "second-countable" is part of the definition is ...
44
votes
7
answers
22k
views
How do you show that $S^{\infty}$ is contractible?
Here I mean the version with all but finitely many components zero.
- 10.5k
5
votes
3
answers
4k
views
5
votes
3
answers
1k
views
Does the "continuous locus" of a function have any nice properties?
Suppose $f:\mathbf{R}\to\mathbf{R}$ is a function. Let $S=\{x\in \mathbf{R}|f\text{ is continuous at }x\}$. Does $S$ have any nice properties?
Here are some observations about what $S$ could be:
$S$ ...
22
votes
4
answers
6k
views
Can you explicitly write $\mathbb{R}^2$ as a disjoint union of two totally path disconnected sets?
An anonymous question from the 20-questions seminar:
Can you explicitly write $\mathbb{R}^2$ as a disjoint union of two totally path disconnected sets?
- 1,059
12
votes
4
answers
877
views
Can you describe the image of the exponential map $B(H)\to B(H)$?
James Tener asks at the 20-questions seminar:
The exponential map $\exp:B(H)\to B(H)$ is just defined by its Taylor series. Can you describe its image?
- 1,059