All Questions
8,725 questions
14
votes
4
answers
2k
views
The ten most fundamental topics in geometric group theory
What are the ten most fundamental topics in geometric group theory?
This is a pedagogical question prompted by the fact that I am teaching geometric group theory to undergraduates. They are expected ...
3
votes
0
answers
99
views
+100
Can I get a spherical coordinate from a real cocycle?
The Setting
I am currently working on a project in Topological Data Analysis (TDA), where I aim to construct a density-robust spherical coordinate associated with a dataset $X$, sampled from a ...
- 71
4
votes
0
answers
193
views
Cell structure on the function space $\operatorname{Hom}(X,Y)$
By the Theorem of Milnor in his paper "On spaces having the homotopy type of a CW-complex", the function space $\operatorname{Hom}(X,Y)$ (with the compact-open topology) is homotopy ...
- 150
47
votes
10
answers
6k
views
Algebraic theorems with no known algebraic proofs
What are some good examples of algebraic theorems that have no known algebraic proofs?
A few I know concern classifications of (not necessarily associative) division algebras over $\mathbb{R}$: the ...
8
votes
0
answers
256
views
+300
Maps with small fibers between manifolds of equal dimension
The following question is an attempt to revise this one into what I intended.
Important revisions are shown in bold.
Are there any known examples of a compact Riemannian manifold $M$ with (possibly ...
- 501
5
votes
1
answer
287
views
Codimension zero embeddings and maps with small fibers
Edit: as explained in my comment on alesia's answer, I mistakenly did not ask below the question I intended (due to my misguided efforts to simplify it). Thus, I revised and reposted my question here.
...
- 501
2
votes
0
answers
117
views
Induced homology map zero implies zero in cobordism?
I had asked this in math stackexchange, but got no reply. Hence, I'm asking here.
[I'm no expert in (co)bordism theory, and I've been struggling with it for the past few weeks. Any good references on ...
- 21
8
votes
1
answer
401
views
Reduction of structure group and classifying spaces
Let $H, G$ be topological groups and $\phi : H \to G$ a group homomorphism. Let $M$ be a paracompact topological space.
For any principal $G$-bundle $P \to M$, a reduction (or sometimes 'lift') of its ...
- 113
7
votes
1
answer
300
views
What does Robert Stong mean when he says $H^*(MO(k))$ is a free Steenrod algebra in dimension less than $2k$?
$\newcommand{\Z}{\mathbb Z}\newcommand{\a}{\mathfrak a}\newcommand\widetildeH{\smash{\widetilde H}}$In Robert Stong's notes on Cobordism Theory, on page 95 he asserts the following:
$\widetildeH^* (...
- 391
10
votes
1
answer
666
views
Are there any tests for knowing whether a topological space admits a CW structure?
We know that for n $\ge$ 5, a manifold admits a piecewise linear structure if and only if its Kirby-Siebenmann class vanish and Galewski and Stern showed the existence of a similar invariant to test ...
- 168
4
votes
1
answer
284
views
Do the two orientations on an orientable manifold $M$ uniquely witness lifts of $\tau_M: M \to B\text{O($n$)}$ to $B\text{SO($n$)}$?
For an orientable $n$-manifold $M$ and its (orthonormal) frame bundle classifying map $\tau_M : M \to BO(n)$, we have a lift diagram of the following sort:
There are two orientations on $M$. Is it ...
- 113
5
votes
0
answers
240
views
Classification of principal $\mathrm{SO}(3)$-bundles on a 4-manifold via characteristic classes
I am interested in a reference with a detailed (as simple and topological as possible) proof of the following fact:
Theorem. A principal $\mathrm{SO}(3)$-bundle on a compact oriented 4-manifold are ...
- 6,962
13
votes
2
answers
394
views
What does $\mathrm{Conf}_n(M)^{h S_n}$ look like?
$\DeclareMathOperator\Conf{Conf}$Let $M$ be a manifold, and $\Conf_n M$ the ordered configuration space of n points on $M$. The symmetric group $S_n$ acts by permuting the points.
Is there a simple ...
- 275
2
votes
1
answer
147
views
An attempt at an alternative calculation of the rank of $\pi_n(MO)$
$\newcommand{\a}{\mathfrak a}\newcommand{\Z}{\mathbb Z}$Let $MO$ be the Thom Spectrum, then I am trying to come up with an alternative calculation that the rank of $\pi_n(MO)$ as a $\Z_2$ vector space ...
- 391
3
votes
2
answers
344
views
Cohomology version of Moore space
I asked this question on MSE a few days back but could not get any helpful response. So I am rewriting the post.
It is known to me that given a simply connected finite dimensional (which is also level-...
- 1,177
3
votes
2
answers
384
views
Is this true of the frame bundle $\operatorname{Fr}(M)$?
On an orientable (Riemannian) $n$-manifold $M$, with orthonormal frame bundle $\operatorname{Fr}(M)$, we have that the tangent bundle classifying map $\tau_M : M \to B{\operatorname O(n)}$ lifts to $B{...
- 113
1
vote
1
answer
111
views
The double of the genus two handlebody minus three tori [closed]
I am exploring the properties of the manifold $M$ defined as follows. Start with the handlebody $H$ of genus two, whose boundary surface is $\Sigma$. Let $P$ be a pants decomposition of $\Sigma$, ...
- 31
16
votes
1
answer
978
views
Pedagogically intuitive reformulation of Zorn's Lemma for functional analysis
While teaching an applied functional analysis class, I’ve noticed that students often struggle to develop an intuitive understanding of Zorn’s lemma. It’s relatively straightforward to explain why ...
- 5,824
2
votes
0
answers
120
views
Analogs of Plücker relations in Clifford algebras, and Bott periodicity (?)
Classical Plücker relations can be viewed as conditions on coefficients of an element $x=\sum_Sc_Se_S$, $S=(i_1,...,i_k)$, $i_1<\cdots<i_k$, $\{i_1,...,i_k\}\subset\{1,...,n\}$ of an exterior ...
- 18.4k
3
votes
0
answers
171
views
Cellular structure of $F_4$
Is there the cellular structure of the Exceptional Lie group $F_4$?
Is there a reference to it?
Thanks
4
votes
0
answers
101
views
Full subcategories of stable $\infty$-categories closed under all shifts
Is there a name for an $\infty$-category C that admits a zero object and suspensions such that the suspension functor $C \to C$ is an equivalence but which does not necessarily admit cofibers and ...
- 1,361
2
votes
1
answer
130
views
Question about maps on cofibers being zero
I have the following problem, I am skeptical that the solution is as easy as the one I wrote so would be grateful if some expert can enlighten me.
Let $G\colon \mathcal{C}\to \mathcal{D}$ be a functor ...
- 309
2
votes
0
answers
120
views
Is the fixed point index bounded?
I am working with the notion of fixed point index presented in the book "The Lefschetz fixed point theorem" of Robert Brown (MR283793, Zbl 0216.19601) and I would like to know if given any ...
- 21
2
votes
1
answer
386
views
Is it always possible to connect the endpoints of a smooth injective path, so the resulting path is smooth, closed and simple?
Motivation
The motivation for this question arises from the following problem: Is there a closed, simple, and infinitely differentiable path on the (complex) plane such that every straight line has a ...
5
votes
0
answers
181
views
Deformations of cotangent bundles
Let $M$ be a smooth variety of even dimension over $\mathbb C$. I am interested in necessary or sufficient conditions such that $X$ is a deformation of a family of cotangent bundles.
In other words, ...
- 6,622
4
votes
1
answer
238
views
When does a cofibrantly generated model category have this factorization property?
Let $\mathcal{C}$ be a cofibrantly generated model category, which is generated by $I$ and $J$. According to the small object argument (Hovey Theorem 2.1.14) of cofibrantly generated model categories, ...
- 143
3
votes
1
answer
168
views
Express fundamental group of $\mathcal H/\Gamma$ by $\Gamma$
Suppose $\mathcal H$ is the upper half plane, and $\Gamma$ is an arithmetic subgroup of $\operatorname{PSL}_2(\mathbb Z)$, I want to ask can we interpret the fundamental group of $\mathcal H/\Gamma$ ...
- 785
2
votes
1
answer
216
views
Compute the singular homology group modulo barycentric subdivision
Let $X$ be a topological space, and let $C(X)$ denote its singular chain complex with boundary operator $\partial$ and $n$-th chain group $C_n$. We know there exists a barycentric subdivision operator ...
- 807
7
votes
1
answer
310
views
Homotopy between posets
This is entirely a new area for me and I apologise in advance if the questions are silly.
In Quillen's paper "Homotopy properties of the posets of non-trivial $p$-subgroup of a group" (see ...
- 139
7
votes
2
answers
383
views
Connectivity of fibers under fibration replacement
Assume all the spaces mentioned below are simply connected CW complexes. Let $ f: X \to Y $ be a continuous surjctive map between CW complexes, where $ f $ is not necessarily a fibration. Assume that ...
- 1,177
7
votes
2
answers
323
views
Formula for compositions of Steenrod squares that produce a form in the top degree
On a smooth $d$ dimensional compact connected manifold $M$, for an $\mathbb{Z}_2$-valued $(d-j)$-cocycle $x_{d-j}$ we have the formula ${\text{Sq}}^{j} (x_{d-j}) = u_{j} \cup x_{d-j}$. Here $u_j \in ...
- 283
3
votes
1
answer
246
views
Model structure on simply-connected topological spaces in which the weak equivalences are the rational homotopy equivalences
I recently started learning rational homotopy theory, and found the claim on page 7 of this survey that there is a suitable model category structure in which the weak equivalences are the rational ...
- 33
4
votes
2
answers
258
views
Waldhausen S-construction for exact categories
Let $\mathcal{C}$ be an exact category. Then, we can consider $\mathcal{C}$ as a Waldhausen category, where the cofibrations are admissible monomorphisms. By Waldhausen $S$-construction we know that $...
- 167
12
votes
0
answers
257
views
When do (or don't) residue fields generate the derived category of a ring?
Let $R$ be a commutative ring, and $D(R)$ its derived category of unbounded chain complexes. I'm interested in when the residue fields $k(\mathfrak{p}) = \mathrm{Frac}(R/\frak p)$ for $\mathfrak p \in ...
- 3,784
3
votes
0
answers
133
views
Grothendieck spectral sequence (cohomology version) for posets with functor coefficient
In this paper, Quillen mentioned a spectral sequence as follows.
Let $f:X\to Y$ is a poset map and $\mathcal{F}:X\to Ab$, where $Ab$ is the category of abelian groups, a functor which is contravariant ...
- 139
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 ...
17
votes
1
answer
414
views
Is $MU/I_\infty$ an $E_\infty$ ring?
Fix a prime $p$, and suppose that $p>2$ for simplicity, although many things should also work for $p=2$. Let $F$ be the usual formal group law defined over $MU_*$, and let $I_\infty$ be the ideal ...
- 56.9k
3
votes
0
answers
246
views
Fundamental group of degree 4 del Pezzo surface minus 16 (-1)-curves [Reference request]
Let $S$ be a degree $4$ del Pezzo surface (over $\mathbb{C}$).
That is, $5$ points blow-up of $\mathbb{P}^2$, or $4$ points blow-up of $\mathbb{P}^1 \times \mathbb{P}^1$.
The classical fact is that $...
- 111
27
votes
8
answers
3k
views
Object of proven finiteness, yet with no algorithm discovered?
I explain my title by two examples in number theory:
The rational points on elliptic curve over number fields forms a finitely generated abelian group, so its rank is an integer, but so far we do not ...
- 1,053
5
votes
1
answer
379
views
Why is this Brieskorn manifold a rational homology sphere?
In Némethi's book "Normal surface singularities", Example 5.1.17, there is a formula to find the Seifert invariants of a Brieskorn complete intersection $\Sigma(a_1,...,a_n)$. I am ...
- 197
7
votes
1
answer
297
views
Unoriented cobordism of oriented manifold
We can regard an oriented manifold as an unoriented manifold by forgetting the orientation. This gives a homomorphism from the oriented cobordism group to the unoriented cobordism group. What is the ...
- 73
6
votes
2
answers
523
views
Is the logic of the ($\infty$-?)topos of simplicial sets "contradictory up to homotopy"?
In a sense this is a followup to my earlier question Does the (1-)topos structure on simplicial sets have any homotopy-theoretic significance?.
In the topos of simplicial sets, the subobject ...
- 18.4k
2
votes
1
answer
300
views
G-equivariant homotopy between G-spaces
I apologize for asking too many questions in a single post. I am not very conversant with equivariant homotopy theory. While discussing with some faculty I was told that certain fact is true. All ...
- 139
4
votes
0
answers
178
views
Basis of topology on space of properly embedded smooth manifolds
In A Short Exposition of the Madsen-Weiss Theorem, Hatcher discusses (starting at p.6) a basis for a topology on the space $\mathcal{C}^n$, the space of all smooth oriented $d$-dimensional ...
- 141
25
votes
1
answer
583
views
Does every oriented $3$-dimensional submanifold of $\mathbb{R}^6$ bound an oriented $4$-dimensional submanifold?
In my recent research, I encountered the following problem about embeddings. Let $M^3$ be a closed compact oriented smooth $3$-dimensional submanifold of $\mathbb{R}^6$. Does there exist a compact ...
- 587
3
votes
0
answers
120
views
Signature vs commensurability
If a closed oriented $4n$-dimensional manifold $M$ has an orientation-reversing homeomorphism, then the signature $\sigma(M)$ of the intersection form vanishes. More generally, $\sigma(M) = 0$ if $M$ ...
- 41
8
votes
1
answer
224
views
Can increasing the winding number of a 2-cell make a CW complex embeddable?
Let $X$ be a 2-dimensional CW complex and let $c\subset X$ be a 2-cell attached along a simple closed loop $\gamma$ in the 1-skeleton $X^{(1)}$.
For a natural number $n\ge 2$ consider the operation of ...
- 13.6k
4
votes
1
answer
256
views
Third page differential in the Lyndon–Hochschild–Serre Spectral Sequence
I am trying to understand the description of the group cohomology of $Q_8$ from Adem–Milgram’s “Cohomology of finite groups”. The main result is the following:
Theorem 2.9. In the Lyndon–Hochschild–...
- 141
3
votes
1
answer
184
views
Freudenthal suspension homomorphism
I asked this question in MathStackExchange a couple of months ago, with little feedback, hence I try here.
The Hopf invariant $h(f)$ of a mapping $f:\mathbb S^{2m-1}\to \mathbb S^m$ is a homotopy ...
- 203
10
votes
2
answers
337
views
Finitely dominated universal spaces for the family of solvable subgroups
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Sz{Sz}$In short, I am interested in the question which finite groups $G$ admit a finitely dominated universal space with respect to the family of ...
- 231