All Questions
1,239 questions
21
votes
1
answer
7k
views
Kuenneth-formula for group cohomology with nontrivial action on the coefficient
For a trivial action on the coefficient, we have the following Kuenneth formula
for group cohomology:
$$
H^n(G_1 \times G_2; M) \cong
[\oplus_{i= 0}^n H^i(G_1;M) \otimes_M H^{n-i}(G_2;M)]
\oplus [\...
- 4,826
19
votes
3
answers
3k
views
When does the tangent bundle of a manifold admit a flat connection?
Let $M$ be a smooth manifold, and let $TM$ denote its tangent bundle. Under what conditions does $TM$ admit a flat connection $\omega$?
Edit: Formerly, I asked about a flat connection on the frame ...
- 8,512
18
votes
1
answer
2k
views
Points on a sphere
Wonder whether any of you know where it was that the following pearl of topology first appeared:
Prove that at any instant of time you can find three isothermal points on the surface of the Earth ...
- 8,842
15
votes
3
answers
1k
views
Linking topological spheres
Is there a simple proof of the fact that:
If $A\subset S^3$ is homeomorphic to $S^1$, then there is a circle $B$
embedded into $S^3\setminus A$ that such that the circles $A$ and $B$
are ...
- 28k
14
votes
2
answers
3k
views
How to prove that a projective variety is a finite CW complex?
Let $X$ be a (singular) projective variety, in other words something given by a collection of polynomial equations in $\mathbb CP^n$ or $\mathbb RP^n$. How can one prove it is a finite $CW$ complex?
...
- 28.9k
14
votes
1
answer
2k
views
Finite dimensional real division algebras
A celebrated theorem of Milnor and Kervaire asserts that any finite dimensional (not necessarily associative, unital) division algebra over the real numbers has dimension 1,2,4 or 8. This result is ...
- 2,550
8
votes
1
answer
715
views
What is the homotopy category of the sphere spectrum?
Is there a known explicit description of the abelian $2$-group $\mathsf{Ho}(\mathbb{S})\overset{\mathrm{def}}{=}\mathsf{Ho}(QS^0)\cong\Pi_{\leq1}(QS^0)$?
- 11.8k
7
votes
1
answer
506
views
$G$ cocycle split to a coboundary in $J$, via a group extension
Consider a generic nontrivial $d$-cocycle $\omega_d^G \in H^d(G,U(1))$ in the cohomology group of a group $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. In otherwords, here the $d$-cocycle $\...
- 10.5k
7
votes
2
answers
1k
views
Is there a "spectral exterior algebra" construction in higher algebra?
Given a ring spectrum $R$ and an $R$-module $E$, we have the spectral symmetric algebra $\mathrm{Sym}_R(E)$ of $E$ over $R$, defined by
$$
\begin{align*}
\mathrm{Sym}_R(E) &\overset{\mathrm{def}}{=...
- 11.8k
231
votes
4
answers
16k
views
Is $\mathbb R^3$ the square of some topological space?
The other day, I was idly considering when a topological space has a square root. That is, what spaces are homeomorphic to $X \times X$ for some space $X$. $\mathbb{R}$ is not such a space: If $X \...
- 5,275
154
votes
7
answers
85k
views
Where to buy premium white chalk in the U.S., like they have at RIMS? [closed]
While not a research-level math question, I'm sure this is a question of interest to many research-level mathematicians, whose expertise I seek.
At RIMS (in Kyoto) in 2005, they had the best white ...
150
votes
31
answers
70k
views
What are the most misleading alternate definitions in taught mathematics?
I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...
144
votes
24
answers
19k
views
Occurrences of (co)homology in other disciplines and/or nature
I am curious if the setup for (co)homology theory appears outside the realm of pure mathematics. The idea of a family of groups linked by a series of arrows such that the composition of consecutive ...
140
votes
7
answers
34k
views
Is the boundary $\partial S$ analogous to a derivative?
Without prethought, I mentioned in class once that the reason the symbol $\partial$
is used to represent the boundary operator in topology is
that its behavior is akin to a derivative.
But after ...
- 151k
122
votes
7
answers
15k
views
Topology and the 2016 Nobel Prize in Physics
I was very happy to learn that the work which led to the award of the 2016 Nobel Prize in Physics (shared between David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz) uses Topology. In ...
106
votes
4
answers
13k
views
What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?
In 1991, Kapranov and Voevodsky published a proof of a now famously false result, roughly saying that the homotopy category of spaces is equivalent to the homotopy category of strict infinity ...
- 42.4k
86
votes
44
answers
21k
views
Demystifying complex numbers
At the end of this month I start teaching complex analysis to
2nd year undergraduates, mostly from engineering but some from
science and maths. The main applications for them in future
studies are ...
80
votes
10
answers
11k
views
What are the uses of the homotopy groups of spheres?
Pete Clark threw down the challenge in his comment to my answer on Why the heck are the homotopy groups of the sphere so damn complicated?:
Have the homotopy groups of spheres ever been applied to ...
- 26.8k
76
votes
9
answers
15k
views
understanding Steenrod squares
There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called Steenrod squaring: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from ...
- 6,069
73
votes
1
answer
3k
views
Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure?
This question has been crossposted from Math.SE in the hopes that it reaches a larger audience here.
$\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an ...
- 9,580
72
votes
9
answers
9k
views
What is a continuous path?
I would like some help, because I am getting mad trying to answer the following
Question: Let $X$ be a topological space, what is a continuous path in $X$?
Well, maybe you're already getting ...
- 6,034
58
votes
10
answers
9k
views
de Rham cohomology and flat vector bundles
I was wondering whether there is some notion of "vector bundle de Rham cohomology".
To be more precise: the k-th de Rham cohomology group of a manifold $H_{dR}^{k}(M)$ is defined as the set of closed ...
- 1,939
51
votes
22
answers
19k
views
Why linear algebra is fun!(or ?)
Edit: the original poster is Menny, but the question is CW; the first-person pronoun refers to Menny, not to the most recent editor.
I'm doing an introductory talk on linear algebra with the ...
51
votes
5
answers
9k
views
Fundamental group as topological group
Background
Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the ...
- 63.1k
50
votes
4
answers
3k
views
To which extent can one recover a manifold from its group of homeomorphisms
Question. Suppose that $M$ is a closed connected topological manifold and $G$ is its group of homeomorphisms (with compact-open topology). Does $G$ (as a topological group) uniquely determine $M$?
One ...
- 31.2k
49
votes
8
answers
13k
views
How should one think about pushforward in cohomology?
Suppose f:X→Y. If I decorate that first sentence with appropriate adjectives, then I get a pushforward map in cohomology H*(X)→H*(Y).
For example, suppose that X and Y are oriented ...
- 8,866
49
votes
4
answers
4k
views
Why is there a duality between spaces and commutative algebras?
1) The category of affine varieties over $\mathbb{C}$ is equivalent to the opposite category of finitely generated reduced algebras over $\mathbb{C}$. The equivalence associates to an affine variety ...
- 9,481
48
votes
3
answers
13k
views
When is a Homology Class Represented by a Submanifold? [duplicate]
Possible Duplicate:
Cohomology and fundamental classes
Given an oriented manifold $M$ and an oriented submanifold $\phi:N\to M$ we can obtain a homology class $\phi_*[N]\in H_*(M)$ ...
- 2,283
48
votes
3
answers
9k
views
Connected sum of topological manifolds
A definition of the connected sum of two $n$-manifolds $M$ and $M'$ begins by considering two $n$-balls $B$ in $M$, $B'$ in $M'$, and glueing the varieties $M\setminus \mathring B$ and $M'\setminus \...
- 12.9k
46
votes
8
answers
11k
views
Non-examples of model structures, that fail for subtle/surprising reasons?
An often-cited principle of good mathematical exposition is that a definition should always come with a few examples and a few non-examples to help the learner get an intuition for where the concept's ...
- 21.3k
46
votes
5
answers
3k
views
‘Naturally occurring’ $K(\pi, n)$ spaces, for $n \geq 2$
[edited!] Given a group $\pi$ and an integer $n>1$, what are examples of Eilenberg–MacLane spaces $K(\pi, n)$ that can be constructed as "known" manifolds? (Or if not a manifold, say some ...
- 2,734
45
votes
8
answers
23k
views
An "advanced beginner's" book on algebraic topology?
It has so happened that I have come this far knowing nothing on the subject of algebraic topology (as in homology theories of topological spaces and their applications). I've decided to finally read ...
44
votes
4
answers
5k
views
Integral cohomology (stable) operations
There have been a couple questions on MO, and elsewhere, that have made me curious about integral or rational cohomology operations. I feel pretty familiar with the classical Steenrod algebra and its ...
- 3,726
42
votes
3
answers
5k
views
The Origin(s) of Modular and Moduli
In mathematics and in physics, people use the terms "modular..." and "moduli space" very often. I was puzzled by the etymology, the origins and the similarity/equivalence/differences for these usages/...
- 10.5k
41
votes
3
answers
3k
views
Can the unsolvability of quintics be seen in the geometry of the icosahedron?
Q1. Is it possible to somehow "see" the unsolvability of quintic polynomials
in the $A_5$ symmetries of the icosahedron (or dodecahedron)?
Perhaps this is too vague a question.
Q2. Are there ...
- 151k
41
votes
1
answer
10k
views
Why not a Roadmap for Homotopy Theory and Spectra?
MO has seen plenty of roadmap questions but oddly enough I haven't seen one for homotopy theory. As an algebraic geometer who's fond of derived categories I would like some guidance on how to build up ...
- 1,162
41
votes
4
answers
4k
views
When is a submanifold of $\mathbf R^n$ given by global equations?
Let $M \subset \mathbf R^n$ be a (smooth) submanifold of dimension $d$. Under which conditions does there exist global equations defining $M$? By global equations I mean : does there exist a smooth ...
- 2,521
38
votes
4
answers
8k
views
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
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
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
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
31
votes
2
answers
2k
views
Why is the motivic category defined over the site of smooth schemes only?
Fix a base scheme $S$. Stable and unstable motivic categories over $S$ are defined as certain categories of higher stacks on the Nisnevich site $Sm_S$ of smooth schemes over $S$. Why smooth?
As a ...
- 64k
27
votes
6
answers
4k
views
Failure of smoothing theory for topological 4-manifolds
Smoothing theory fails for topological 4-manifolds, in that a smooth structure on a topological 4-manifold $M$ is not equivalent to a vector bundle structure on the tangent microbundle of $M$. Is ...
- 967
24
votes
2
answers
2k
views
How much of homotopy theory can be done using only finite topological spaces?
Let $X$ be a finite simplicial complex and let $B$ denote the set of barycenters of the simplices of $X$. McCord constructed a $T_0$ topology on $B$ with the property that the inclusion $B \to X$ is ...
- 29.2k
21
votes
3
answers
2k
views
Cohomology of fibrations over the circle: how to compute the ring structure?
This question is inspired by Cohomology of fibrations over the circle Moreover, it can be considered a subquestion of the above, but somehow it seems to me that some of the more interesting points ...
- 23.5k
21
votes
1
answer
1k
views
What is the group of pointed homotopy classes of maps from $S^3 \times S^3$ to $S^3$?
What is the group of pointed homotopy classes of maps from $S^3 \times S^3$ to $S^3$? The group structure induced by the group structure on the codomain. This question is a followup to Eric's answer ...
- 295
21
votes
8
answers
4k
views
Cogroup objects
Pretty much anyone who does algebra is familiar with group objects in categories, but what about cogroup objects? Most of what I've been able to find about them is that they "arise naturally in ...
- 16k
21
votes
2
answers
2k
views
Is there a high-concept explanation of the dual Steenrod algebra as the automorphism group scheme of the formal additive group?
Recall that for any space $X$, the cohomology $H^*X$ (always, in this post, with $\mathbb{Z}/2$-coefficients) has an action of the Steenrod algebra $\mathcal{A}$; that is, a natural morphism $\mathcal{...
- 25.6k
19
votes
2
answers
1k
views
Exotic smooth structures on Lie groups?
If a topological group $G$ is also a topological manifold, it is well-known (Hilbert's 5th Probelm) that there is a unique analytic structure making it a Lie group.
However, for a compact Lie group $...
- 783
19
votes
6
answers
3k
views
Diffeomorphism of 3-manifolds
Surgery theory aims to measure the difference between simple homotopy types and diffeomorphism types. In 3 dimensions, geometrization achieves something much more nuanced than that. Still, I wonder ...
- 13.2k