All Questions
9,056 questions
34
votes
6
answers
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views
Why study finite topological spaces?
In rereading Thurston's essay On Proof and Progress in Mathematics I ran across this passage:
… this means that some concepts that I use freely and naturally in
my personal thinking are foreign to ...
- 737
34
votes
8
answers
6k
views
Applications of super-mathematics to non-super mathematics
Supergeometry and more broadly supermathematics has been around for few decades. Since its introduction by physicists, there has been an some mathematical interest in them.
Although interesting in its ...
34
votes
5
answers
3k
views
Do the signs in Puppe sequences matter?
A basic construction in homotopy is Puppe sequences. Given a map $A \stackrel{f}{\to} X$, its homotopy cofiber is the map $X\to X/A=X \cup_f CA$ from $X$ to the mapping cone of $f$. If we then take ...
- 31.2k
34
votes
4
answers
8k
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Cohomology of Flag Varieties
For $K$ a compact Lie-group with maximal torus $T$, I'd like to know the cohomology $\text{H}^{\ast}(K/T)$ of the flag variety $K/T$.
If I'm not mistaken, this should be isomorphic to the algebra of ...
- 2,756
34
votes
2
answers
2k
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vector bundle trivial over every compact subset, then it is globally trivial
Let $X$ be a non-compact metric space (though if the answer to the question is positive, then it probably also holds for more general spaces like, e.g., paracompact Hausdorff) and $E \to X$ a vector ...
- 2,998
34
votes
2
answers
5k
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Example Wanted: When Does Čech Cohomology Fail to be the same as Derived Functor Cohomology?
I want to know exactly how derived functor cohomology and Cech cohomology can fail to be the same.
I started worrying about this from Dinakar Muthiah's answer to an MO question, and Brian Conrad's ...
- 27.5k
34
votes
1
answer
2k
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Early stabilization in the homotopy groups of spheres
Thanks to Freudenthal we know that $\pi_{n+k}(S^n)$ is independent of $n$ as soon as $n \ge k+2$. However, I was looking at the table on Wikipedia of some of the homotopy groups of spheres and noticed ...
- 13.5k
34
votes
2
answers
3k
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What is the relation between the sphere spectrum and supersymmetry?
In this this google+ post of Urs Schreiber, he says: "Grading over the sphere spectrum is supersymmetry" and then he redirect us to the abstract idea of superalgebra (in nLab).
Are there some ...
- 1,720
34
votes
3
answers
8k
views
Different way to view action of fundamental group on higher homotopy groups
There are a couple of ways to define an action of $\pi_1(X)$ on $\pi_n(X)$. When $n = 1$, there is the natural action via conjugation of loops. However, the picture seems to blur a bit when looking at ...
34
votes
1
answer
4k
views
Construction of Morava E-theory
I'm wrapping up a summer project that involved a computation in Morava $E$-theory. As background knowledge I had to look into how the Johnson-Wilson theories $E(n)$ and Morava $K$-theories were ...
- 6,308
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
34
votes
2
answers
933
views
If $A$ is the ring of continuous functions on a genus $g$ surface, can the genus of $X$ be seen by simple algebra in $A$?
I was describing to a friend the result that a compact Hausdorff space is determined up to homeomorphism up to by its ring of continuous functions, and he asked how one could see the genus of a ...
- 1,462
34
votes
1
answer
2k
views
Is every ''group-completion'' map an acyclic map?
I start with a longer discussion which will result in a precise version of the question. I am puzzled about an issue with the
Quillen plus construction. I have seen outstanding experts being confused ...
- 20.9k
33
votes
4
answers
6k
views
What (if anything) happened to Intersection Homology?
In the early 1990's, Gil Kalai introduced me to a very interesting generalization of homology theory called intersection homology, which existed for like 10 years back then I believe. Defined ...
- 6,734
33
votes
5
answers
8k
views
Coefficients in cohomology
(Sorry if this is too elementary for this site)
I’m having some trouble understanding sheaf cohomology. It’s supposed to provide a theory of cohomology “with local coefficient”, and allow easy ...
- 473
33
votes
5
answers
4k
views
(Short) Exact sequences with no commutative diagram between them
This question was asked by a student (in a slightly different form), and I was unable to answer it properly. I think it's quite interesting.
The problem is to produce an example of the following ...
- 2,287
33
votes
6
answers
4k
views
What is the intuition for higher homotopy groups not vanishing?
The homotopy groups of the spheres $S^n$ (see Wikipedia) vanish for the circle $S^1$ as, naively speaking, there are not higher order holes to be grasped by higher order homotopy groups. This ...
- 679
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
33
votes
3
answers
6k
views
(co)homology of symmetric groups
Let $S_n=\{\text{bijections }[n]\to[n]\}$ be the n-th symmetric group. Its (co)homology will be understood with trivial action. What are the $\mathbb{Z}$-modules $H_k(S_n;\mathbb{Z})$? Using GAP, we ...
- 1,589
33
votes
3
answers
3k
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No matter how many algebraic invariants we attach to topological spaces, there will always be nonhomeomorphic spaces agreeing on all their invariants
A while ago a professor of mine said something along the lines of
No matter how many algebraic invariants we attach to topological spaces, there will always be nonhomeomorphic spaces agreeing on ...
- 2,788
33
votes
3
answers
2k
views
The probabilistic method outside of discrete mathematics
The probabilitic method is a genius idea in combinatorics, graph theory etc, where instead of constructing something by hand, you construct the thing randomly and show that there is a positive ...
33
votes
2
answers
2k
views
What happened to the last work Gaunce Lewis was doing when he died?
In 2006, Gaunce Lewis died at the age of 56. He'd done important work setting up equivariant stable homotopy theory, and I think it's fair to say his work was far ahead of its time. In recent years, ...
- 30.3k
33
votes
1
answer
2k
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For which maps $S^1\to S^1$ is the winding number defined?
There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number:
• Continuous maps:
Using the unique path lifting property of the universal covering map $\mathbb R\to S^...
- 43.2k
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
33
votes
1
answer
740
views
Equivalence of topological Hochschild homology and Mac Lane homology via an equivalence $QA\simeq HA \wedge_{\mathbb{S}} H\mathbb{Z}$
Mac Lane homology is a homology theory for (not necessarily commutative) rings. Given a ring $A$, Eilenberg and Mac Lane define its cubical construction $QA$ to be a certain connective chain complex, ...
- 498
33
votes
0
answers
2k
views
Is there a (discrete) monoid M injecting into its group completion G for which BM is not homotopy equivalent to BG?
For a (discrete) monoid $M$, the classifying space $BM$ is the
geometric realization of the nerve of the one object category whose
hom-set is $M$. (This definition gives the usual classfiying space
...
- 5,309
33
votes
0
answers
2k
views
Is there software to compute the cohomology of an affine variety?
I have some affine varieties whose cohomology (topological, with $\mathbb{C}$ coefficients) I would like to know. They are very nice, they are all of the form $\mathbb{A}^n \setminus \{ f=0 \}$ for ...
- 156k
32
votes
7
answers
8k
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Are there two non-homotopy equivalent spaces with equal homotopy groups?
Could someone show an example of two spaces $X$ and $Y$ which are not of the same homotopy type, but nevertheless $\pi_q(X)=\pi_q(Y)$ for every $q$? Is there an example in the CW complex or smooth ...
- 2,479
32
votes
3
answers
2k
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Is the Hurewicz theorem ever used to compute abelianizations?
The Hurewicz theorem tells us that if $X$ is a path-connected space then $H_1(X, \, \mathbb{Z})$ is isomorphic to the abelianisation of $\pi_1(X)$. This gives a potential method for computing the ...
user81684
32
votes
8
answers
2k
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Noncommutative rational homotopy type
Ok, this question is much less ambitious than it might sound, but still:
Two commutative differential graded algebras (cdga's) are quasi-isomorphic if they can be connected by a chain of cdga quasi-...
- 23.5k
32
votes
2
answers
2k
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When is there a submersion from a sphere into a sphere?
(First posted on math.SE, with no answers.)
That is:
For which positive integers $n, k \ge 1$ does there exist a submersion $S^{n+k} \to S^k$?
The discussion at this math.SE question has ...
- 118k
32
votes
5
answers
4k
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Some intuition behind the five lemma?
Slightly simplified, the five lemma states that if we have a commutative diagram (in, say, an abelian category)
$$\require{AMScd}
\begin{CD}
A_1 @>>> A_2 @>>> A_3 @>>> A_4 @...
- 1,412
32
votes
3
answers
1k
views
Complex projective manifolds are homeomorphic if homotopy equivalent
If two complex projective manifolds are homotopy equivalent are they homeomorphic?
user164740
32
votes
4
answers
7k
views
Computational software in Algebraic Topology?
I was wondering if there is any good software out there that allows you to do specific computations in algebraic topology. For example:
Create a simplicial complex/set and ask questions about its ...
- 435
32
votes
4
answers
5k
views
Visualizing how Cech cohomology detects holes
I think it's pretty intuitive how singular/simplicial cohomology detects "holes" in a space.
How can we directly visualize how and in what sense the Cech cohomology of a cover does this?
...
- 11.3k
32
votes
1
answer
2k
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Stable homotopy theory and physics
At various points in my life, I have held the following beliefs:
1) Stable homotopy theory is "easy" rationally, and "interesting" integrally.
2) The spectrum of topological modular forms (TMF) is ...
- 321
32
votes
2
answers
2k
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Converse to Stokes' Theorem
Does satisfying Stokes' Theorem imply that a form is linear?
Let $M$ be an $n$-manifold. A differential $k$-form $\omega \in \Omega^k M$ assigns to each point $x \in M$ a function $\omega_x : \Lambda^...
- 63.9k
32
votes
5
answers
9k
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How many binary operations are associative?
Let $X$ be a finite set of $n$ elements, and consider a binary operation $\odot: X \times X \rightarrow X$. There are $n^{n^2}$ such binary operations, as the $n \times n$ table entries can each
be ...
- 151k
32
votes
1
answer
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"Affine communication" for topological manifolds
There is a situation that comes up regularly in algebraic topology when giving proofs of facts about manifolds, like Poincare duality and the like. The typical sequence goes like this:
Prove ...
- 52.6k
32
votes
2
answers
2k
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Persistence barcodes and spectral sequences
Persistent homology is a well-developed tool which allows topological analysis of large data sets. From a topological perspective, the input is a filtered complex, and the output is a sequence of ...
- 35.9k
32
votes
2
answers
2k
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Unifying Geometry for Characteristic Classes
When working with characteristic classes (more concretely Chern classes), one finds at least four essentially distinct approaches:
Axiomatic Approach. See, for instance, Vector Bundles and K-Theory, ...
- 2,091
32
votes
3
answers
2k
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A Pachner complex for triangulated manifolds
A theorem of Pachner's states that if two triangulated PL-manifolds are PL-homeomorphic, the two triangulations are related via a finite sequence of moves, nowadays called "Pachner moves".
A ...
- 44.4k
32
votes
2
answers
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Open problems in algebraic topology and homotopy theory
Some time ago (I see it was initially written before 1999?) Mark Hovey assembled a list of open problems in algebraic topology. The list can be found here. Some of the problems I know about have been ...
32
votes
1
answer
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About a claim by Gromov on proper holomorphic maps
At p. 223 of his paper [G03], Mikhail Gromov makes the following claim:
Let $X$, $Y$ be two complex manifolds (not necessarily compact or Kähler) of the same dimension and having the same even Betti ...
- 66.3k
32
votes
0
answers
3k
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Microlocal geometry - A theorem of Verdier
(1) In "Geometrie Microlocale", Verdier states the following theorem.
Theorem: Let $E$ be a vector space and $F$ a constructible complex on $E$.
Then for $\ell$ a linear form on $E$, we have a ...
- 7,527
31
votes
9
answers
5k
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Why should I prefer bundles to (surjective) submersions?
I hope this question isn't too open-ended for MO --- it's not my favorite type of question, but I do think there could be a good answer. I will happily CW the question if commenters want, but I also ...
- 54.6k
31
votes
10
answers
8k
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What are some interesting problems in the intersection of Algebraic Number Theory and Algebraic Topology?
I'm a beginning graduate student and while my background is primarily in algebraic number theory, I've found myself a bit smitten with the subject of algebraic topology recently after only having read ...
31
votes
7
answers
3k
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Why are we interested in permutahedra, associahedra, cyclohedra, ...?
The following families of polytopes have received a lot of attention:
permutahedra,
associahedra,
cyclohedra,
...
My question is simple: Why?
As I understand, at least the latter two were ...
- 13.6k
31
votes
3
answers
1k
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Non embedding of $Y\times Y$ into $\mathbb{R}^3$
I know that this is a well known result, but where can I find a proof? I am also interested to see more general non-embedding results of this type.
Theorem. Let $Y$ be the union of two segments ...
- 28k
31
votes
3
answers
2k
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What are the higher homotopy groups of a K3 suface?
All K3 surfaces have the same homotopy type. What are their higher homotopy groups?
I know that $\pi_1$ is trivial, and $\pi_2$ is $\mathbb{Z}^{22}$.
Even if the answer isn't known in all degrees, I'...
- 15.4k