All Questions
1,239 questions
50
votes
2
answers
2k
views
Can I wrap a suitcase with hair ties
Is there a nontrivial link in a big solid torus that is trivial in the ambient Euclidean space such that each circle is unknot and has a sufficiently small length?
It is motivated by a question that ...
- 45k
49
votes
2
answers
6k
views
Why should I care about topological modular forms?
There seems to be a lot of recent activity concerning topological modular forms (TMF), which I gather is an extraordinary cohomology theory constructed from the classical theory of modular forms on ...
- 491
49
votes
4
answers
7k
views
Elegant proof that any closed, oriented 3-manifold is the boundary of some oriented 4-manifold?
I'm looking for an elegant proof that any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$.
- 1,709
48
votes
0
answers
17k
views
What is the current understanding regarding complex structures on the 6-sphere?
In October 2016, Atiyah famously posted a preprint to the arXiv, "The Non-Existent Complex 6-Sphere" containing a very brief proof $S^6$ admits no complex structure, which I immediately read and ...
- 2,995
47
votes
6
answers
6k
views
Can we actually find any fixed points with Brouwer's theorem?
Background
At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is ...
- 15.5k
43
votes
5
answers
5k
views
How can you tell if a space is homotopy equivalent to a manifold?
Is there some criterion for whether a space has the homotopy type of a closed manifold (smooth or topological)? Poincare duality is an obvious necessary condition, but it's almost certainly not ...
- 31.2k
43
votes
9
answers
29k
views
Applications of knot theory
An answer of André Henriques' inspired the following closely related CW question. Parts of the following is extracted from his answer and my comments.
I regularly teach a knot theory class. ...
42
votes
11
answers
17k
views
Blackboard rendering of math fonts
I learned most of my math font rendering from watching others (for example, I draw ζ terribly). In most cases it is passable, but I'm often uncomfortable using fonts like Fraktur on the board. ...
- 52.7k
41
votes
7
answers
5k
views
Simplicial objects
How should one think about simplicial objects in a category versus actual objects in that category? For example, both for intuition and for practical purposes, what's the difference between a [...
- 21k
41
votes
1
answer
6k
views
Classification of surfaces and the TOP, DIFF and PL categories for manifolds
A surface is simply a 2-manifold. The classification theorem for compact connected surfaces (with boundary) is commonly regarded in the categories TOP, DIFF and PL. Well known proofs (e.g. via ...
- 2,136
39
votes
1
answer
5k
views
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 ...
- 21.4k
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
2
answers
13k
views
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
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
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
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
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
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
2
answers
2k
views
Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?
Let $k$ be a field and $V$ a $k$-vector space. Then there is a map $V \to V^{\ast \ast}$, where $V^{\ast}$ is the dual vector space. If we are in ZFC and $\dim V$ is infinite, then this map is not ...
- 156k
34
votes
4
answers
8k
views
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
3k
views
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
1
answer
2k
views
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
2k
views
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
33
votes
20
answers
5k
views
Do names given to math concepts have a role in common mistakes by students?
Perhaps this question overlaps with similar ones, ... but I want to focus on a particular possible cause of confusion. I notice that students are often confused by the concepts of "infinite" and "...
32
votes
7
answers
8k
views
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
2
answers
2k
views
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^...
- 64k
31
votes
3
answers
2k
views
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
31
votes
1
answer
4k
views
For which varieties is the natural map from the Chow ring to integral cohomology an isomorphism?
My apologies if this question is too naive.
Let $X$ be a smooth projective complex variety. There is a natural map $A^{\bullet}(X) \to H^{2\bullet}(X)$ of graded rings from the Chow ring of $X$ to ...
- 118k
29
votes
6
answers
4k
views
Concrete example of $\infty$-categories
I've seen many different notions of $\infty$-categories: actually I've seen the operadic-globular ones of Batanin and Leinster, and the opetopic, and eventually I'll see the simplicial ones too. ...
- 3,349
29
votes
4
answers
4k
views
Model structure on Simplicial Sets without using topological spaces
The category of simplicial sets has a standard model structure, where the weak equivalences are those maps whose geometric realization is a weak homotopy equivalence, the cofibrations are ...
- 1,292
28
votes
4
answers
4k
views
Classifying Space of a Group Extension
Consider a short exact sequence of Abelian groups -- I'm happy to assume they're finite as a toy example:
$$
0 \to H \to G \to G/H \to 0\ .
$$
I want to understand the classifying space of $G$. Since ...
- 4,217
28
votes
5
answers
4k
views
Modern source for spectra (including ring spectra)
I am looking for a modern introduction to spectra that improves on the treatment by Adams in his "Stable Homotopy and Generalized Homology" notes (by improves I mean taking into account what ...
27
votes
5
answers
7k
views
References for "modern" proof of Newlander-Nirenberg Theorem
Hi,
I'm starting to prepare a graduate topics course on Complex and Kahler manifolds for January 2011. I want to use this course as an excuse to teach the students some geometric analysis. In ...
27
votes
1
answer
4k
views
connectivity of the group of orientation-preserving homeomorphisms of the sphere
In the paper "Local Contractions and a Theorem of Poincare" Sternberg has mentioned the following question which was open when the paper was written:
Is the group of orientation-preserving ...
- 6,224
27
votes
3
answers
4k
views
"Dirty" proof that Eilenberg-MacLane spaces represent cohomology?
The standard approach to proving that $H^n(X; G)$ is represented by $K(G, n)$ seems to be to prove that $\text{Hom}(X, K(G, n))$ defines a cohomology theory and then use Eilenberg-Steenrod uniqueness. ...
- 2,168
27
votes
3
answers
6k
views
Why is BG infinite dimensional for G finite ?
If $G \neq \lbrace 1 \rbrace$ is a finite group with classifying space $BG$
then there are infinitely many i such that $H^i(BG,\mathbb{Z}) \neq 0$. This
can be found, for example, there:
Non-...
- 2,160
27
votes
3
answers
2k
views
Configuration space of little disks inside a big disk
The space of configurations of $k$ distinct points in the plane
$$F(\mathbb{R}^2,k)=\lbrace(z_1,\ldots , z_k)\mid z_i\in \mathbb{R}^2, i\neq j\implies z_i\neq z_j\rbrace$$
is a well-studied object ...
- 35.9k
26
votes
1
answer
1k
views
From the perspective of bordism categories, where does the ring structure on Thom spectra come from?
To fix ideas, let's consider the Thom spectrum of framed bordism $M$, the spectrum whose homotopy groups are the framed bordism groups. $M$ has a ring spectrum structure inducing the product of ...
- 118k
26
votes
3
answers
3k
views
Third differential in Atiyah Hirzebruch spectral sequence
Does any one know why $d_3: H^* (X, K^0(point))\rightarrow H^{*+3}(X,K^0(point))$ is actually extended $Sq^3$ to $\mathbb{Z} $ coefficient.
- 1,003
25
votes
2
answers
3k
views
generalisations of the Seifert-van Kampen Theorem?
I have been reading Jacob Lurie's book "Higher Algebra", version May 8, 2011. One is grateful to him for covering such a lot of ground and for making it all so readily available.
My attention was ...
- 12.3k
24
votes
2
answers
4k
views
complement of a totally disconnected closed set in the plane
While preparing a course in complex analysis, I stumbled over a remark in Dudziak's book on removable sets, namely that any totally disconnected $K \subset\subset {\mathbb C}$ must have a connected ...
24
votes
3
answers
4k
views
Plus construction considerations.
In order to realise the K-groups of a ring as the homotopy groups of some space associated to that ring, Quillen proposed the following (roughly-sketched) construction:
Recall that $K_1(R) = GL(R)/E(...
- 1,105
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
23
votes
2
answers
3k
views
Calculating Mayer-Vietoris efficiently
This is a question whose motivation and framing seem to involve a lot of topology, but which I suspect comes down to some simple and standard combinatorics that's probably recorded in a book somewhere....
- 156k
23
votes
6
answers
5k
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cohomology of BG, G compact Lie group
It has been stated in several papers that $H^{odd}(BG,\mathbb{R})=0$ for compact Lie group
$G$. However, I've still not found a proof of this. I believe that the proof is as follows:
--> $G$ compact ...
- 1,709
23
votes
1
answer
4k
views
The Dedekind eta function in physics
This interesting little fellow (a nice introduction is the video "Mock Modular Forms are Everywhere" by Cheng and Felder) popped up in some operator algebra (Witt / Virasoro Lie algebra) I ...
22
votes
6
answers
3k
views
Does every vector bundle allow a finite trivialization cover?
Suppose there is a vector bundle (smooth, with constant rank finite-dimensional fibres) over a (smooth, second-countable, Hausdorff, not necessarily connected) manifold $B$ of dimension $n$.
(a) Is ...
- 1,284
22
votes
1
answer
1k
views
Little disks operad and $Gal (\bar {Q}/Q)$
My question is simple:
How do the little disks operad and $Gal (\bar {Q}/Q)$ relate?
I realize that a huge amount of heavy-machinery can be brought into an answer to this, but I'm struggling with ...
- 2,734
22
votes
1
answer
1k
views
Word maps on compact Lie groups
Let $w=w(a,b)$ be a non-trivial word in the free group $F_2 = \langle a,b \rangle$ and $w_G \colon G \times G \to G$ be the induced word map for some compact Lie group $G$.
Murray Gerstenhaber and ...
- 25.5k
22
votes
1
answer
1k
views
Word problem for fundamental group of submanifolds of the 4-sphere
Given any finitely-presented group $G$, there are a few equivalent techniques for constructing smooth/PL 4-manifolds $M$ such that $\pi_1 M$ is isomorphic to $G$. For most constructions of these 4-...
- 44.4k