All Questions
1,240 questions
394
votes
115
answers
110k
views
Not especially famous, long-open problems which anyone can understand
Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please.
Motivation: I plan to use this list in ...
82
votes
12
answers
15k
views
Compelling evidence that two basepoints are better than one
This question is inspired by an answer of Tim Porter.
Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his ...
- 22.1k
149
votes
7
answers
23k
views
Homotopy groups of Lie groups
Several times I've heard the claim that any Lie group $G$ has trivial second fundamental group $\pi_2(G)$, but I have never actually come across a proof of this fact. Is there a nice argument, ...
- 4,014
35
votes
5
answers
11k
views
What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?
What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?
- 921
11
votes
1
answer
1k
views
Necessary conditions for cofibrancy in global projective model structure on simplicial presheaves
Consider the global projective model category
of simplicial presheaves on some category
(the category of smooth manifolds is particularly interesting to me).
In Section 9.1 of Dugger's paper “...
- 37.8k
70
votes
6
answers
8k
views
third stable homotopy group of spheres via geometry?
It is ''well-known'' that the third stable homotopy group of spheres is cyclic of order $24$. It is also ''well-known'' that the quaternionic Hopf map $\nu:S^7 \to S^4$, an $S^3$-bundle, suspends to a ...
- 20.9k
23
votes
2
answers
2k
views
Uniqueness of compactification of an end of a manifold
Let $M$ be an $n$-dimensional manifold (smooth or topological). I call $\bar{M}$ a compactification of $M$ if it is an $n$-dimensional compact manifold with boundary $\partial \bar{M}$, an $(n-1)$-...
- 21.6k
38
votes
3
answers
8k
views
The error in Petrovski and Landis' proof of the 16th Hilbert problem
What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis?
Please see this related post and also the following post.. For Mathematical development ...
- 366
80
votes
7
answers
12k
views
Cubical vs. simplicial singular homology
Singular homology is usually defined via singular simplices, but Serre in his thesis uses singular cubes, which he claims are better adapted to the study of fibre spaces. This young man (25 years old ...
- 47.5k
62
votes
9
answers
9k
views
Fundamental groups of noncompact surfaces
I got fantastic answers to my previous question (about modern references for the fact that surfaces can be triangulated), so I thought I'd ask a related question. A basic fact about surface topology ...
- 44.8k
38
votes
8
answers
6k
views
Why do we need model categories?
I cannot give a good answer to this question. And
2) Why this definition of model category is the right way to give a philosophy of homotopy theory? Why didn't we use any other definition?
3) Has ...
- 1,060
24
votes
3
answers
5k
views
Euler characteristic of a manifold and self-intersection
This is probably quite easy, but how do you show that the Euler characteristic of a manifold M (defined for example as the alternating sum of the dimensions of integral cohomology groups) is equal to ...
- 5,530
112
votes
6
answers
10k
views
Counterexamples in algebraic topology?
In this thread
Books you would like to read (if somebody would just write them...),
I expressed my desire for a book with the title "(Counter)examples in Algebraic Topology".
My reason for doing so ...
61
votes
2
answers
9k
views
Are spectra really the same as cohomology theories?
Let $E \to F$ be a morphism of cohomology theories defined on finite CW complexes. Then by Brown representability, $E, F$ are represented by spectra, and the map $E \to F$ comes from a map of spectra. ...
- 25.6k
57
votes
2
answers
7k
views
What arithmetic information is contained in the algebraic K-theory of the integers
I'm always looking for applications of homotopy theory to other fields, mostly as a way to make my talks more interesting or to motivate the field to non-specialists. It seems like most talks about ...
- 30.3k
54
votes
7
answers
15k
views
Why are local systems and representations of the fundamental group equivalent
My question: Let X be a sufficiently 'nice' topological space. Then there is an equivalence between representations of the fundamental group of X and local systems on X, i.e. sheaves on X locally ...
- 1,197
50
votes
0
answers
12k
views
Atiyah's paper on complex structures on $S^6$
M. Atiyah has posted a preprint on arXiv on the non-existence of complex structure on the sphere $S^6$.
https://arxiv.org/abs/1610.09366
It relies on the topological $K$-theory $KR$ and in ...
- 9,870
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
43
votes
4
answers
6k
views
Why the Dold-Thom theorem?
Dold-Thom Theorem: $$\pi_i(SP(X))\cong\tilde{H}_i(X)$$
It's pretty miraculous, no? I've seen its proof, where you show that the composition of the functors on the left-side satisfies the axioms of a ...
- 17.5k
41
votes
5
answers
11k
views
Mathematically mature way to think about Mayer–Vietoris
This question is short but to the point: what is the "right" abstract framework where Mayer-Vietoris is just a trivial consequence?
- 6,348
33
votes
1
answer
1k
views
Nilpotence of the stable Hopf map via framed cobordism
The Pontryagin-Thom construction shows that the stable homotopy groups of spheres are the same as the groups of stably framed manifolds up to cobordism. Specifically the Hopf map corresponds to the ...
- 28.1k
19
votes
3
answers
5k
views
What determines a model structure?
It is easy to prove that a model structure is determined by the following classes of maps (determined = two model structures with the mentioned classes in common are equal).
cofibrations and weak ...
- 2,782
15
votes
6
answers
3k
views
Giving $\mathit{Top}(X,Y)$ an appropriate topology
$\DeclareMathOperator\Top{\mathit{Top}}$I am not sure if its OK to ask this question here.
Let $\Top$ be the category of topological spaces. Let $X,Y$ be objects in $\Top$.
Let $F:\mathbb{I}\...
- 1,117
9
votes
3
answers
2k
views
Reference for homotopy (co)limits of (co)chain complexes via totalization of double complexes
It seems to be a well-known fact that homotopy (co)limits
of (co)simplicial diagrams of nonnegatively graded
(co)chain complexes in (Grothendieck) abelian categories
can be computed by using the Dold-...
- 37.8k
8
votes
1
answer
732
views
Compositional inversion and generating functions in algebraic geometry
The exponential generating function of the graded dimension of the cohomology ring of the moduli space of n-pointed curves of genus zero satisfying the associativity equations of physics (the WDVV ...
- 10.5k
263
votes
29
answers
89k
views
Mathematical games interesting to both you and a 5+-year-old child
Background: My daughter is 6 years old now, once I wanted to think on some math (about some Young diagrams), but she wanted to play with me...
How to make both of us to do what they want ? I guess ...
104
votes
10
answers
24k
views
Motivation for algebraic K-theory?
I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like ...
98
votes
10
answers
14k
views
equivalence of Grothendieck-style versus Cech-style sheaf cohomology
Given a topological space $X$, we can define the sheaf cohomology of $X$ in
I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$)
or
II. the Čech ...
- 1,871
93
votes
20
answers
10k
views
Short papers for undergraduate course on reading scholarly math
(I know this is perhaps only tangentially related to mathematics research, but I'm hoping it is worthy of consideration as a community wiki question.)
Today, I was reminded of the existence of this ...
71
votes
10
answers
25k
views
Nice proof of the Jordan curve theorem?
As a student, I was taught that the Jordan curve theorem is a great example of an intuitively clear statement which has no simple proof.
What is the simplest known proof today?
Is there an intuitive ...
- 1,843
68
votes
12
answers
29k
views
Algebraic topology beyond the basics: any texts bridging the gap?
Peter May said famously that algebraic topology is a subject poorly served by its textbooks. Sadly, I have to agree. Although we have a freightcar full of excellent first-year algebraic topology texts ...
66
votes
8
answers
10k
views
What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$
I would like to know if there is a known necessary and sufficient
property on an open subset of $\mathbb{R}^n$ to be diffeomorphic to $\mathbb{R}^n$ :
For example :
Are all open star-shaped subsets ...
- 677
60
votes
6
answers
7k
views
Torsion in homology or fundamental group of subsets of Euclidean 3-space
Here's a problem I've found entertaining.
Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups is not ...
- 44.4k
58
votes
2
answers
2k
views
Can a subset of the plane have nontrivial $H_2$ or $\pi_2$?
This is a question that occurred to me years ago when I was first learning algebraic topology. I've since learned that it's a somewhat aesthetically displeasing question, but I'm still curious about ...
- 82.7k
51
votes
3
answers
12k
views
Spaces with same homotopy and homology groups that are not homotopy equivalent?
A common caution about Whitehead's theorem is that you need the map between the spaces; it's easy to give examples of spaces with isomorphic homotopy groups that are not homotopy equivalent. (See Are ...
- 10.1k
47
votes
1
answer
2k
views
Brown representability for non-connected spaces
In many places (on MO, elsewhere on the Internet, and perhaps even in some textbooks) one finds a statement of the classical Brown representability theorem that looks something like this:
If $F$ is ...
- 66.8k
44
votes
6
answers
4k
views
Does $\mathbb C\mathbb P^\infty$ have a group structure?
Does $\mathbb C\mathbb P^\infty$ have a (commutative) group structure? More specifically, is it homeomorphic to $FS^2$, (the connected component of) the free commutative group on $S^2$?
$\mathbb C\...
- 8,727
41
votes
1
answer
6k
views
Not all manifolds can be triangulated: In which dimensions?
I know that Ciprian Manolescu has settled the triangulation conjecture in the negative: Not all manifolds can be triangulated. I've only read secondary literature on this result, which did not detail ...
- 151k
40
votes
4
answers
7k
views
What is a simplicial commutative ring from the point of view of homotopy theory?
Let $k$ be a field. There are two natural categories to consider:
The category of simplicial commutative $k$-algebras.
The category of connective $E_\infty$ $k$-algebras (i.e., chain complexes of $...
- 25.6k
35
votes
3
answers
7k
views
Higher Topos Theory- what's the moral?
I've often seen Lurie's Higher Topos Theory praised as the next "great" mathematical book. As someone who isn't particularly up-to-date on the state of modern homotopy theory, the book seems ...
35
votes
5
answers
3k
views
Small simplicial complexes with torsion in their homology?
Fix a prime $p$. What is the smallest integer $n$ so that there is a simplicial complex on $n$ vertices with $p$-torsion in its homology?
For example, when $p=2$, there is a complex with 6 vertices (...
- 4,254
34
votes
4
answers
5k
views
The Jouanolou trick
In Une suite exacte de Mayer-Vietoris en K-théorie algébrique (1972) Jouanolou proves that for any quasi-projective variety $X$ there is an affine variety $Y$ which maps surjectively to $X$ with ...
- 23.5k
33
votes
2
answers
2k
views
What are the "correct" conventions for defining Clifford algebras?
I have three related questions about conventions for defining Clifford algebras.
1) Let $(V, q)$ be a quadratic vector space. Should the Clifford algebra $\text{Cliff}(V, q)$ have defining ...
- 118k
31
votes
1
answer
3k
views
What was the error in the proof of Roos' theorem?
Background: In 1961, Roos (who, sadly, apparently passed away just last month) purported to prove [1] that in an abelian category with exact countable products (AB4${}^\ast_\omega$), limits of inverse ...
- 64k
29
votes
2
answers
2k
views
Equivariant classifying spaces from classifying spaces
Given compact Lie groups $G$ and $\Pi$, there is a notion of "$G$-equivariant principal $\Pi$-bundle", and a corresponding notion of classifying space, often denoted $B_G\Pi$, so that $G$-equivariant ...
- 27.2k
29
votes
4
answers
3k
views
Geometric interpretation of the lower central series for the fundamental group?
For any group $G$ we can form the lower central series of normal subgroups by taking $G_0 = G$, $G_1 = [G,G]$, $G_{i+1} = [G,G_i]$. We can check this gives a normal chain
$$G_0 \ge G_1 \ge ... \ge G_i ...
- 573
29
votes
3
answers
5k
views
finite generated group realized as fundamental group of manifolds
This is discussed in the standard textbooks on algebraic topology.
Pick a presentation of the group $G = \langle g_1,g_2,...,g_n|r_1,r_2,...r_m \rangle$
where $g_i$ are generators and $r_j$ are ...
- 291
29
votes
5
answers
13k
views
(Co)homology of the Eilenberg-MacLane spaces K(G,n)
Let $(G, n)$ be a pair such that $n$ is a natural number, $G$ is a finite group which is abelian if $n \geq 1$. It is well-known that $\pi_n(K(G,n)) = G$ and $\pi_i (K(G,n)) = 0$ if $i \neq n$.
Also ...
- 3,699
23
votes
5
answers
4k
views
Fundamental groups of topological groups.
Let $G$ be a topological group, and $\pi_1(G,e)$ its fundamental group at the identity. If $G$ is the trivial group then $G \cong \pi_1(G,e)$ as abstract groups. My question is:
If $G$ is a non-...
- 273
21
votes
7
answers
4k
views
Whitehead for maps
I made the following claim over at the Secret Blogging Seminar, and now I'm not sure it's true:
Let $f: X \to Y$ and $g: X \to Y$ be two maps between finite CW complexes. If f and g induce the same ...
- 156k