All Questions
9,056 questions
39
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2
answers
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What parts of the theory of quasicategories have been simplified since the publication of HTT?
It has been almost ten years since Lurie published Higher Topos Theory, where (following Joyal and probably others) he set up foundations for higher category theory via quasicategories. My impression ...
- 5,958
39
votes
1
answer
5k
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Flatness in Algebraic Geometry vs. Fibration in Topology
I am currently trying to get my head around flatness in algebraic geometry. In particular, I'm trying to relate the notion of flatness in algebraic geometry to the notion of fibration in algebraic ...
- 21.3k
38
votes
7
answers
7k
views
What is DAG and what has it to do with the ideas of Voevodsky?
In Toen's and Vezzosi's article From HAG to DAG: derived moduli stacks a kind of definition of DAG is given. I am not an expert and can't see what's the relation between DAG and the motivic cohomology ...
- 1,085
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 ...
- 1,040
38
votes
4
answers
4k
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What manifolds are bounded by RP^odd?
Real projective spaces $\mathbb{R}P^n$ have $\mathbb{Z}/2$ cohomology rings $\mathbb{Z}/2[x]/(x^{n+1})$ and total Stiefel-Whitney class $(1+x)^{n+1}$ which is $1$ when $n$ is odd, so it follows that ...
- 576
38
votes
4
answers
8k
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Relative De Rham cohomologies
as far as I know, there are two main ways to have a relative version of De Rham Cohomology for a pair (M,N), where M and N are smooth manifolds and N is a closed (as a topological subspace) ...
- 830
38
votes
3
answers
8k
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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 ...
- 356
38
votes
3
answers
2k
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If $X$ and $Y$ are homotopy equivalent, then are $X \times \mathbb{R}^{\infty}$ and $Y \times \mathbb{R}^{\infty}$ homeomorphic?
Let $X$ and $Y$ be reasonable spaces. Since $\mathbb{R}^{\infty}$ is contractible,
$$
X \times \mathbb{R}^{\infty} \cong Y \times \mathbb{R}^{\infty} \;\;\; \implies \;\;\; X \simeq Y.
$$
Is the ...
- 5,898
38
votes
3
answers
6k
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What is so "spectral" about spectral sequences?
From recent mathematical conversations, I have heard that when Leray first defined spectral sequences, he never published an official explanation of his terminology, namely what is "spectral" about a ...
- 1,764
38
votes
3
answers
2k
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Is there a "simplification" functor in algebraic topology?
Recall that a space (=CW complex) is called simple if it is connected, the fundamental group is abelian, and the fundamental group acts trivially on all higher homotopy groups. Call Simp(X) a ...
- 28.1k
38
votes
2
answers
13k
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Explanation for the Thom-Pontryagin construction (and its generalisations)
In 1950, Pontryagin showed that the n-th framed cobordism group of smooth manifolds was equal to n-th stable homotopy group of spheres:
$$ \lim_{k \to \infty} \pi_{n+k}(S^k) \cong \Omega_n^{\text{...
- 5,530
38
votes
2
answers
2k
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What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?
This question concerns a set-theoretic aspect that I found interesting in the recent question asked by user Nick R., namely, Is
$\mathbb{R}^3\setminus\mathbb{Q}^3$ simply connected? He had asked ...
- 236k
38
votes
2
answers
2k
views
Finite complexes whose homotopy groups are not "finitely generated"
I'll say $K$ has "finitely generated" homotopy groups if there is a finite wedge of spheres $W = \bigvee S^{n_i}$ and a map $f: W\to K$ which induces a surjection on $\pi_*$.
It seems likely that ...
- 12.5k
37
votes
3
answers
5k
views
Topological Langlands?
In a workshop about the geometry of $\mathbb{F}_1$ I attended recently, it came up a question related to a mysterious but "not-so-secret-anymore" seminar about... an hypothetical Topological Langlands ...
- 2,992
37
votes
4
answers
3k
views
Why is it so hard to compute $\pi_n(S^n)$?
Of course it isn't really that hard - nowhere near as hard as $\pi_k(S^n)$ for $k>n$, for instance. The hardness that I'm referring to is based on the observation that apparently nobody knows how ...
- 29.2k
37
votes
5
answers
7k
views
Inference using Topological Data Analysis: Is it worth it for a regular statistician to learn TDA?
After having read Gunnar Carlsson's Topology and Data I feel enthusiastic to use some topological data analysis (TDA) methods in my current research, mostly in social sciences. We often handle huge ...
- 729
37
votes
3
answers
3k
views
Are there pairs of highly connected finite CW-complexes with the same homotopy groups?
Fix an integer n. Can you find two finite CW-complexes X and Y which
* are both n connected,
* are not homotopy equivalent, yet
* $\pi_q X \approx \pi_q Y$ for all $q$.
In Are there two non-...
- 27.2k
37
votes
2
answers
4k
views
How can we detect the existence of almost-complex structures?
Any smooth $k$-manifold $M$ comes with a well-defined map $f:M\rightarrow BGL_{k}(\mathbb{R})$ (up to homotopy) classifying its tangent bundle. Since $GL_{k}(\mathbb{R})$ deformation-retracts onto $...
- 6,069
37
votes
1
answer
1k
views
Does there exist a continuous 2-to-1 function from the sphere to itself?
I am interested in the following question:
Does there exist a continuous function $f:S^2\to S^2$ such that, for any $p\in S^2$, $|f^{-1}(\{p\})|=2$?
I suspect the answer is no, but I don't know ...
- 513
37
votes
1
answer
1k
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If $A$, $B$ are abelian groups such that $\mathrm{Hom}(A, G) \cong \mathrm{Hom}(B, G)$ for all abelian groups $G$, must $A$ and $B$ be isomorphic?
$\DeclareMathOperator\Hom{Hom}$The question is in the title. If the isomorphism $\Hom(A, G) \cong \Hom(B, G)$ is natural in $G$ then this is just the Yoneda Lemma. If $A$ and $B$ are finitely ...
- 1,141
37
votes
1
answer
3k
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Morava on Shafarevich conjecture
$\DeclareMathOperator\Q{\mathbf{Q}}$Jack Morava has some interesting ideas stemming from stable homotopy theory and geometric topology on the Shafarevich Conjecture.
The Shafarevich Conjecture states: ...
- 2,734
36
votes
3
answers
6k
views
In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?
I'm interested in whether one only needs to consider simple loops when proving results about simply connected spaces.
If it is true that:
In a Topological Space, if there exists a loop that cannot ...
- 4,862
36
votes
21
answers
6k
views
Generalizations of Planar Graphs
This is a follow up to Harrison's question: why planar graphs are so exceptional. I would like to ask about (and collect answers to) various notions, in graph theory and beyond graph theory (topology; ...
- 24.7k
36
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 ...
36
votes
5
answers
6k
views
What is the equivariant cohomology of a group acting on itself by conjugation?
This question makes sense for any topological group $G$, but I'd particularly like to know the answer for $G$ a compact, connected Lie group.
$G$ acts on itself by conjugation. One has the equivariant ...
- 13.2k
36
votes
2
answers
4k
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Maps which induce the same homomorphism on homotopy and homology groups are homotopic
I am interested in the following question. Are maps which induce the same homomorphism on homotopy and homology groups homotopic? I am sure the answer is no, however I cannot imagine how to construct ...
- 1,129
36
votes
9
answers
5k
views
Computing fundamental groups and singular cohomology of projective varieties
Are there any general methods for computing fundamental group or singular cohomology (including the ring structure, hopefully) of a projective variety (over C of course), if given the equations ...
- 21k
36
votes
4
answers
5k
views
Construction of the Stiefel-Whitney and Chern Classes
I've seen two constructions of these characteristic classes. The first comes from Milnor and Stasheff's book and involves the Thom isomorphism and (at least for me) the rather mysterious Steenrod ...
- 3,968
36
votes
3
answers
2k
views
Defining $SU(n)$ in HoTT
From a recent answer by Mike Shulman, I read:
"HoTT is (among other things) a foundational theory, on roughly the same ontological level as ZFC, whose basic objects can be regarded as $\infty$-...
- 43.2k
36
votes
2
answers
5k
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Is the fundamental group functor a left-adjoint?
Theorem 1B.9 in Hatcher's Algebraic Topology says that for a (pointed) connected CW complex $X$ and group $G$, there is a bijection $\text{Hom}(\pi_1(X), G) \cong [X,K(G,1)]$, where $\pi_1(X)$ is the ...
- 1,446
36
votes
4
answers
5k
views
Compact open topology on $\mathrm{Homeo}(X)$
Let $X$ and $Y$ be topological spaces. Define the compact open topology on the set $\mathrm{M}(X,Y)$ of continuous maps from $X$ to $Y$ via the subbase $[K,O]$ of all maps $f:X\rightarrow Y$ s.t. $f(K)...
- 2,751
36
votes
2
answers
4k
views
Timeline of cohomology (1935 to 1938)
There was a recent question on intuitions about sheaf cohomology, and I answered in part by suggesting the "genetic" approach (how did cohomology in general arise?). For historical material specific ...
- 12.6k
36
votes
2
answers
1k
views
Is there an analog of Sperner's lemma for the Hopf invariant?
Recall that Sperner's lemma is essentially a combinatorial version of the topological statement "A map from $S^n$ to $S^n$ with degree one cannot be nullhomotopic."
My question is, does there exist ...
- 8,493
36
votes
0
answers
1k
views
Functor that maps to both $KO^n$ and $KO^{-n}$
(my question is also meaningful for complex K-theory, but since Kn(X) is always isomorphic to K-n(X), it's less interesting)
I start by recalling the analytic definition of KO-theory:
The following ...
- 43.2k
35
votes
9
answers
5k
views
Covering maps in real life that can be demonstrated to students
Edit: I've really enjoyed everyone's examples (especially the pictures!), but I was mostly looking for a general theorem. For instance, a similar statement to mine is, Can the mapping cylinder of ...
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
35
votes
5
answers
9k
views
Intuition behind Alexander duality
I was wondering if anyone could offer some intuition for why Alexander duality holds. Of course, the proof is easy enough to check, and it is also easy to work out many examples by hand. However, I ...
- 361
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
35
votes
4
answers
3k
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References for sign conventions in homological algebra
There is no shortage of sign conventions in homological algebra. And once these conventions are set out, there is no shortage of diagrams where an obvious commutative diagram on the underlying ...
35
votes
5
answers
3k
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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
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
35
votes
4
answers
4k
views
An intelligent ant living on a torus or sphere – Does it have a universal way to find out?
I wanted to ask a question about topological invariants and whether they are connected in a fundamental or universal way. I am not an expert in topology, so please let me ask this question by way of a ...
- 6,937
35
votes
1
answer
4k
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Why is persistent cohomology so much faster than persistent homology
I refer to this paper: de Silva, Vin; Morozov, Dmitriy; Vejdemo-Johansson, Mikael. Dualities in persistent (co)homology. Inverse Problems 27 (2011), no. 12, 124003, 17 pp. (Journal link, arXiv link).
...
- 1,229
35
votes
2
answers
3k
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What do loop groups and von Neumann algebras have to do with elliptic cohomology?
Recall that complex $K$-theory is a cohomology theory on topological spaces, which can be described in several equivalent ways:
Given a finite complex $X$, $K^0(X)$ is the Grothendieck group of ...
- 25.6k
35
votes
3
answers
1k
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Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$
We fix $G=\mathrm{SL}_3(\mathbf{R})$.
Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b_2(\Gamma)=0$?
Here the second Betti number $b_2(\Gamma)$ is both the dimension of the ...
- 63.9k
35
votes
1
answer
2k
views
Are there topological versions of the idea of divisor?
I am trying to extract a particular, more lightweight and more focussed at the same time, case of my recent question Which of the physics dualities are closest in essence to the Spanier-Whitehead ...
- 18.4k
35
votes
1
answer
4k
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Applications of arithmetic topology to number theory
There is a well-known analogy between 3-manifolds and number fields, with knots corresponding to prime ideals. Are there any results in number theory that have been proven using topology through this ...
- 1,046
35
votes
2
answers
3k
views
Equivalent descriptions of Hodge conjecture?
I would like to know equivalent descriptions of the Hodge conjecture (with references).
Dan Freed's Version:
Consider a topological cycle (boundary less chains that are free to deform) on a ...
35
votes
1
answer
1k
views
Finding the octonionic analog of the K3 surface, via (almost) hyperkahler geometry?
The K3 manifold is an amazing object in mathematics which plays an important role in several fields ranging from the study of smooth 4-manifolds to algebraic geometry to differential geometry and ...
- 27.5k
35
votes
1
answer
3k
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Manifolds admitting CW-structure with single n-cell
Let $M$ be a topological $n$-manifold, closed and connected (not necessarily oriented):
When does $M$ not admit (up to homotopy-type) a CW-structure with a single $n$-cell?
By classification of ...
- 17.5k