All Questions
8,725 questions
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Tensor product of sheaves
Let $R$ be a ring and let $F$ and $G$ be sheaves of $R$-modules over a Hausdorff space $X$.
Define the tensor product $F\otimes G$ as the sheafification of $U\mapsto F(U)\otimes G(U)$. (All tensor ...
- 470
11
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0
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858
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Infinite-dimensional affine space in algebraic geometry and algebraic topology
In topology, there are of course many different infinite-dimensional topological vector spaces over $\mathbb R$ or $\mathbb C$. Luckily, in algebraic topology, one rarely needs to worry too much about ...
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3
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When does a CW-complex of dimension 2 embed in $\Bbb R^4$?
Let $X$ be a finite CW-complex of dimension two having just one 0-cell
(+ finitely many 1-cells + finitely many 2-cells).
Is it true that X can be embedded in $\Bbb R^4$?
If true, is it due to ...
- 13.6k
4
votes
1
answer
284
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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 ...
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0
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99
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+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
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5
answers
2k
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"Classical" consequences of Bezout's theorem in dimensions $>2$
By Classical I mean something that could have been found before 1900 (say).
A well known consequence of Bezout's theorem for plane curves is Pascal's theorem http://en.wikipedia.org/wiki/Pascal'...
- 5,359
14
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4
answers
2k
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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 ...
- 28.2k
18
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1
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1k
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Application of higher categories in algebra
Higher categories and derived algebraic geometry are relatively new areas and probably fewer people are working on them compared to the majority of topologists or geometers. I believe higher ...
CommunityBot
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256
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+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
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0
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193
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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 ...
- 12.9k
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6
answers
2k
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Multiplicative Structures on Moore Spectra
The motivation for this question is that I want "toy examples" of how to prove/disprove the existence of multiplicative structures on examples of spectra. The class of examples I am thinking of is the ...
- 11.2k
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2
answers
2k
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K-Theory and the Stack of Vector Bundles
I have some understanding that vector bundles provide a basic, familiar example of what I should call a stack. Namely, consider the functor $Vect$ that assigns to a space $X$ the set of isomorphism ...
- 35.5k
5
votes
2
answers
583
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Construction of the Mayer-Vietoris spectral sequence
Given subspaces $\{U_i\}_{i \in I}$ of a topological space $X$ with $X = \bigcup_i U_i$ satisfying some conditions, there is a Mayer-Vietoris spectral sequence converging to the homology of $X$. Here ...
- 12.9k
2
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0
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117
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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 ...
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1
answer
287
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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.
...
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7
votes
1
answer
816
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How to prove that topological Hochschild homology of a smooth proper stable k-linear infinity category is dualizable?
Let $k$ be a perfect field of characteristic $p$. I heard that the Topological Hochschild homology of a smooth proper stable infinity category (or dg-category) is dualizable as a THH(k)-module ...
- 15.9k
8
votes
1
answer
401
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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 ...
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votes
1
answer
233
views
Superlevel sets of a parametrized quadratic forms
Let $N$ be an odd integer, $n\in\mathbb{N}$, and $-\frac{2T}{NR^2}\leq a\leq0$ for given $R,T\in\mathbb{R}$ with $\frac{T}{NR^2}\leq\frac{\pi}{2}$.
Now consider the quadratic form $\Omega(a)=\sum_{l\...
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394
votes
115
answers
110k
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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 ...
- 24.2k
5
votes
0
answers
240
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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
2
votes
1
answer
216
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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 ...
CommunityBot
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1
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147
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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
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1
answer
246
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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 ...
- 37.8k
4
votes
1
answer
476
views
Equivariant K-theory for products of groups?
Let $X$ be a $(G \times H)$-space. What is known about the connection between the groups $K_G(X)$, $K_H(X)$ and $K_{G \times H}(X)$? The $G$ and $H$ action on $X$ come from the canonical inclusions $G ...
CommunityBot
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votes
1
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300
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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^* (...
- 12.9k
10
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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 ...
- 66.3k
9
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2
answers
2k
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The simplicial Nerve
The nerve functor $N:Cat\to SSet$ from the category of small categories to simplicial sets can be obtained as follows: The left Kan extension of the functor $F$ which sends $[n]$ to the category $\...
- 9,627
1
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1
answer
379
views
Creating an inverse system which "stratifies density"
Setting:
Let $X'$ be a dense subset of an infinite-dimensional Fréchet space $X$ and suppose that $(X_n')_{n \in \mathbb{N}}$ is a nested sequence of non-empty subsets of $X'$ satisfying
$$
\bigcup_{n ...
CommunityBot
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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 $\Sigma$, ...
- 28.2k
2
votes
1
answer
130
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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 ...
- 64k
4
votes
0
answers
101
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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
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0
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120
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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
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0
answers
171
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Cellular structure of $F_4$
Is there the cellular structure of the Exceptional Lie group $F_4$?
Is there a reference to it?
Thanks
3
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2
answers
384
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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
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2
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344
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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-...
- 10.8k
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0
answers
120
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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 ...
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181
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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, ...
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1
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386
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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 ...
- 91.8k
5
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1
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920
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Stacks and Maurer-Cartan elements
One can associate to any deformation problem a dg Lie or $L_{\infty}$-algebra $g$.
For instance, in algebraic deformation theory, let's say the deformation theory of algebras over a Koszul operad $P$, ...
CommunityBot
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6
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1
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380
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The optimal ranges for the integral homological stability of $\operatorname{GL}_n(F)$'s for a field $F$
$\DeclareMathOperator\GL{GL}$
$\DeclareMathOperator\co{H}$
$\DeclareMathOperator\ko{K}$
$\DeclareMathOperator\trd{tr-deg}$
$\DeclareMathOperator{\ch}{char}$Given a field $F$ and a homological degree $...
- 1,726
13
votes
2
answers
394
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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 ...
- 56.9k
47
votes
10
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6k
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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 ...
- 38.6k
3
votes
0
answers
133
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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 ...
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4
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1
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238
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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, ...
- 42.4k
3
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1
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168
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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 $\mathcal H/\Gamma$ ...
- 63.9k
7
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1
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310
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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 ...
- 3,948
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1
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978
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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 ...
- 13.1k
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257
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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
11
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1
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957
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Technology for various models of spectra
There are a couple different models for spectra, or constructions of the categories of spectra that have the desired properties (homotopically and otherwise). The construction of the Categories of $S$-...
CommunityBot
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1
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470
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Finite domination and compact ENRs
Edit: In the comments, Tyrone points out that West's positive answer to Borsuk's conjecture implies that every compact ENR is homotopy equivalent to a finite CW complex. It follows that the only ...
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