All Questions
9,056 questions
12
votes
1
answer
316
views
Are the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ always nontrivial in the bordism rings for $n>2$?
In my recent research, I need to know if the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ are always nontrivial in the unoriented and oriented bordism rings for $n>2$. (For the ...
- 587
3
votes
0
answers
144
views
Is there a simple explicit expression of the Pontryagin square in terms of the cup product on a spin 4-manifold?
$A$ a finite abelian group, and denote $\Gamma(A)$ its universal quadratic group. The Pontryagin square $\mathfrak{P}\in H^4(B^2A,\Gamma(A))\cong \text{Hom}(\Gamma(A),\Gamma(A))$ is the element ...
- 813
1
vote
1
answer
249
views
Name for extension of the symplectic group
Let $S_g$ denote an ortientable surface of genus $g$. Let $\operatorname{Diff}(S_g)$ denote the group of diffeomorphism (that need not fix the orientation). Is there a name for the image of $\...
- 965
15
votes
1
answer
2k
views
Is Mazur's analogy between arithmetic and topology formal, in any sense?
I preface my question by admitting I know no algebraic geometry nor algebraic number theory. I do know some algebraic topology. I'm a student.
Recently I learned about sheaf cohomology. Then a little ...
- 609
4
votes
2
answers
330
views
Quillen pairs / $\infty$-adjunctions / adjunctions of homotopy categories
Some of the examples of $\infty$-categories are those arising from model categories. I would like to ask: what is the relationship between Quillen adjunctions between model categories and adjoint ...
- 1,514
13
votes
2
answers
1k
views
Is there a homotopy/homology-theory for probability spaces?
Please excuse that the following will be a somewhat soft question.
Let $(M,d)$ be a metric space and $X(\omega)$ a random variable on $M$ with distribution $\mu$.
Assume now that $M = \overline{B_1^n(...
- 549
6
votes
1
answer
466
views
Why, if the geometric realisation of a simplicial map $p$ is a (topological) covering map, must $p$ be a (simplicial) covering map?
I essentially am asking for an explanation of the comment under this post by Tom Goodwillie.
In the "Kerodon", Lurie defines a simplicial covering map as follows:
A map $p:E\to X$ of ...
- 1,020
5
votes
1
answer
333
views
Proof of homotopic essential simple close curves are isotopic
In the book by Benson Farb and Dan Margalit A primer on mapping class groups, Princeton Mathematical Series 49. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14794-9/hbk; 978-1-400-83904-9/...
- 119
4
votes
1
answer
239
views
On the sparsity of the descent spectral sequence computing homotopy groups of the K(n)-local sphere
There is a descent spectral sequence computing $\pi_*L_{K(n)}S^0$ with $E_2$-term
$$E_2^{s,t}\cong H^s_c(\mathbb{G}_n,(E_n)_t)$$
It is mentioned in Barthel-Beaudry (in the description of Figure 3.30) ...
- 155
4
votes
2
answers
200
views
Is $\operatorname{dim}_{K(h)_\ast} K(h)_\ast X$ increasing in $h$?
Let $X$ be a finite $p$-local spectrum. For each $h \in \mathbb{N} \cup \{\infty\}$, let $K(h)$ be Morava $K$-theory of height $h$. Recall that the coefficients $K(h)_\ast$ are a graded field, and $K(...
- 64k
24
votes
6
answers
2k
views
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 ...
- 2,327
1
vote
1
answer
282
views
Equivalent statement for Borsuk-Ulam theorem
I was going through this paper by Tanaka. In the introduction he says the following
"The classical Borsuk–Ulam theorem can be
restated as the point space is I-trivial."
I am not sure how to ...
176
votes
7
answers
19k
views
Proofs of Bott periodicity
K-theory sits in an intersection of a whole bunch of different fields, which has resulted in a huge variety of proof techniques for its basic results. For instance, here's a scattering of proofs of ...
- 6,308
14
votes
2
answers
1k
views
Derived topological stacks?
I apologize for the vagueness of the following.
Informally, in the site of commutative rings, one roughly get the notion of a derived stack by swapping out the commmutative rings with its subcategory ...
- 928
40
votes
16
answers
11k
views
"Homotopy-first" courses in algebraic topology
A first course in algebraic topology, at least the ones I'm familiar with, generally gets students to a point where they can calculate homology right away. Building the theory behind it is generally ...
35
votes
2
answers
5k
views
Why should have Peter May worked with CGWH instead of CGH in "The Geometry of Iterated Loop Space"?
This is a follow-up to Dan Ramras' answer of this question.
The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here).
The weak Hausdorff rather ...
- 2,974
7
votes
1
answer
567
views
Long exact sequences for parametrized cohomology
I'm reading Michael Shulman's articles on cohomology in HoTT here and here, as well as Floris van Doorn's thesis here.
Given $E: Z \to \mathsf{Spectrum}$ a family of spectra over a homotopy type $Z$, ...
- 6,025
2
votes
1
answer
404
views
Reference request: a cousin to the log semiring
Let $f$ be strictly increasing on $\mathbb{R}$. Then $x \oplus y := f^{-1}(f(x)+f(y))$ gives rise to a strict symmetric monoidal ($\Rightarrow$ commutative monoid) structure on $(\mathbb{R},\ge)$ with ...
- 15.4k
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
11
votes
1
answer
593
views
Examples of 6-manifolds without an almost complex structure
Question: I am searching for examples for closed (hence orientable ), smooth $6$-manifolds without an almost complex structure.
Finding such an example is equivelant to finding a manifold where the ...
- 6,995
4
votes
1
answer
120
views
Characterization of complex line bundles induced from $\mathbb C P^1$
Let $\eta$ be a complex line bundle over some (good) space. Then it is induced from the canonical line bundle over $\mathbb C P^{\infty}$. It may happen that $\eta$ in fact is induced from $\mathbb C ...
- 41
5
votes
0
answers
107
views
Size of minimal generating set of ideal over Laurent polynomial ring
Recently in attacking a problem in algebraic topology relating to the construction of stably-free non-free modules over integral group rings I’ve noticed that it is often fairly easy to reduce to ...
- 187
2
votes
0
answers
55
views
Tangential normal invariant isomorphism
Recently, I was reading the paper "Finite Group Actions on Kervaire Manifold" by Crowley, Hambolton. But I am having problem understanding a definition. Here it is,
In page 15-16 they are ...
1
vote
2
answers
232
views
Abelian covering of link complement
I'm considering finite index abelian (regular) covering of link complement:
$$ X \rightarrow S^3\setminus L$$
where $L$ is a minimally twisted chain link.
I'm interested in covering space. Can we ...
- 11
2
votes
0
answers
109
views
Description of a point cloud being "undersampled" wrt persistent homology, confidence level?
I am completely new to topological data analysis, so I apologise if this is a well-known area of persistent homology, as well as for any imprecise language.
Suppose we know completely the topological ...
- 21
5
votes
1
answer
217
views
How many pairwise non-homeomorphic non-empty closed subsets of the Cantor set are there? [duplicate]
My question is more or less related to basic set theory. But I don't know even that. Apologies if I added the wrong tags.
Motivation: How many non-compact (planar) surfaces are there upto ...
- 1,097
45
votes
13
answers
9k
views
Motivating the de Rham theorem
In grad school I learned the isomorphism between de Rham cohomology and singular cohomology from a course that used Warner's book Foundations of Differentiable Manifolds and Lie Groups. One thing ...
- 82.7k
12
votes
2
answers
786
views
Is the Petersen graph a "Cayley graph" of some more general group-like structure?
The Petersen graph is the smallest vertex-transitive graph which is not a Cayley graph. Is it the "Cayley graph" of some slightly more general group-like structure?
- 1,947
7
votes
1
answer
260
views
Relation between cohomology operations and the Adams spectral sequence
$\newcommand{\Z}{\mathbb Z} \DeclareMathOperator{\Ext}{Ext}
\DeclareMathOperator{\Cone}{Cone}$
I'm trying to understand how higher order cohomology operations are related to the Adams spectral ...
- 71
12
votes
2
answers
721
views
Fields in monoidal categories
We can speak of rings in monoidal categories, including also the non-Cartesian case. What about fields?
Question 1: Definitions
What are some possible notions of a (skew or commutative) field in a ...
- 11.8k
96
votes
4
answers
10k
views
Which manifolds are homeomorphic to simplicial complexes?
This question is only motivated by curiosity; I don't know a lot about manifold topology.
Suppose $M$ is a compact topological manifold of dimension $n$. I'll assume $n$ is large, say $n\geq 4$. The ...
- 27.2k
5
votes
2
answers
217
views
Topology of a union of facets of a convex polytope
The following question arose from a survey paper I am writing on
combinatorial reciprocity. Let $\mathcal{P}$ be a $d$-dimensional
convex polytope. Let $\mathcal{Q}$ be a union of facets (codimension
...
- 50.8k
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
7
votes
1
answer
715
views
Complex vector bundles on compact complex manifolds
The complex vector bundles on complex projective space $\mathbb{CP}^n$ are explicitly classified for low dimensions. When $n\leq 3$, they are exactly the holomorphic vector bundles; when $n\geq 4$ we ...
user482036
35
votes
3
answers
1k
views
Incorrect information in an old article about the Kervaire invariant
In the Soviet times there was a famous Encyclopedia of Mathematics. I think it is still familiar to every Russian mathematician maybe except very young ones, and yours truly is in possession ...
- 6,891
9
votes
1
answer
758
views
Two vague questions about TFT
Question 1. Take a smooth projective Calabi-Yau $X$. Then $D^b(X)$ is a fully-dualizable category and there's an associated 2d TFT. This the usual 2d B-model with target $X$.
But $D^b(X)$ is actually ...
- 460
31
votes
1
answer
1k
views
What results about the topology of manifolds depend on the dimension mod 3?
There are a lot of interesting results about the topology of manifolds that depend on the dimension of the manifold mod 2, mod 4, or mod 8. The simplest ones involve the cup product
$$ \smile \colon ...
- 22.3k
5
votes
1
answer
251
views
Monoid associated to $>2$-player Hackenbush
There is some literature on multiplayer combinatorial game theory, but as far as I can tell none of it follows the line of attack below. I'd love a pointer to a similar approach taken in the ...
- 20.5k
1
vote
0
answers
198
views
topological functor of tor functor
The framework of Quillen's model categories gives us a very general way of defining things as derived functors. For instance, in this way one can realise the singular homology as Andre-Quillen ...
- 449
9
votes
1
answer
756
views
Does there exist a GRR-like generalization of the AS Index Theorem?
The Hirzebruch Riemann-Roch Theorem (HRR) expresses an analytic/algebraic invariant, namely the Euler-Poincaré characteristic of a vector bundle $V$ over a compact complex/algebraic manifold $X$, as ...
- 1,020
15
votes
1
answer
2k
views
How to motivate constructible sheaves
I'm writing some notes for some students which just finished a first course in scheme theory. There I would like to talk about constructible sheaves, but I found it hard to give a compelling ...
- 771
14
votes
1
answer
919
views
What is known about exotic spheres up to stable diffeomorphism?
In even dimensions $n=2k$ we can define two smooth manifolds $M$ and $N$ to be stably diffeomorphic if they become diffeomorphic after the connect sum with $r$ many copies of $S^k \times S^k$ for some ...
- 27.5k
4
votes
0
answers
196
views
Valuations and (semi)norms on ring spectra
Valuations and seminorms on rings play a big role in number theory and analytic geometry, with seminorms being heavily used in Berkovich geometry and valuations featuring heavily in adic geometry.
Let'...
- 11.8k
2
votes
1
answer
104
views
DK equivalences are Reedy equivalences for complete Segal spaces
$\require{AMScd}$
Dear all,
I have a question concerning Charles Rezk's paper "A model for the homotopy theory of homotopy theory
", precisely Proposition 7.6 in this paper. It is proven ...
- 1,759
6
votes
1
answer
417
views
Do $h$-cobordism groups arise from a 'Thom-like' spectrum?
Not thinking about $h$-cobordism, one usually defines a cobordism between manifolds, realizes it is an equivalence relation, chooses an appropriate class of structured manifolds (framed, unoriented, ...
- 609
9
votes
1
answer
235
views
Links and non-orientable surfaces
Let $\Sigma \subset \mathbb{R}^3$ be a compact embedded surface with boundary $\partial \Sigma$ and $i:\Sigma\setminus \partial\Sigma \to \mathbb{R}^3 \setminus \partial\Sigma$ the inclusion.
Is the ...
- 443
1
vote
0
answers
137
views
Is $\pi_m(M) = 0$ if $\pi_m(M-X) = 0$ for a low-dimensional subset $X$?
I am doing a problem where I am stuck at this point.
Let $M$ be a connected smooth manifold of dimension $n$ and let $X$ be any subset of $M$. Assume that there is a positive integer $m$ such that $n&...
13
votes
1
answer
624
views
Ultracategories with one object
Historically, the theory of ultracategories was invented by Makkai to prove a strong conceptual completeness theorem for first-order logic, roughly: if $T$ and $S$ are two first-order theories such ...
- 133
26
votes
5
answers
2k
views
Surprising properties of closed planar curves
In https://arxiv.org/abs/2002.05422 I proved with elementary topological methods that a smooth planar curve with total turning number a non-zero integer multiple of $2\pi$ (the tangent fully turns a ...
- 405
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