All Questions
9,056 questions
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
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
152
votes
13
answers
22k
views
Why is the fundamental group of a compact Riemann surface not free ?
Consider a compact Riemann surface $X$ of genus $g$.
It is well-known that its fundamental group $\pi_1(X)$ is the free group on the generators $a_1,b_1,...,a_g,b_g$ divided out by the normal ...
- 47.5k
149
votes
7
answers
23k
views
Homotopy groups of Lie groups
Several times I've heard the claim that any Lie group $G$ has trivial second fundamental group $\pi_2(G)$, but I have never actually come across a proof of this fact. Is there a nice argument, ...
- 4,014
147
votes
21
answers
23k
views
Are there examples of non-orientable manifolds in nature?
Whilst browsing through Marcel Berger's book "A Panoramic View of Riemannian Geometry" and thinking about the Klein bottle, I came across the sentence:
"The unorientable surfaces are never discussed ...
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 ...
141
votes
0
answers
13k
views
Grothendieck-Teichmüller conjecture
(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmüller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here $...
- 7,527
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
128
votes
12
answers
12k
views
Spectral sequences: opening the black box slowly with an example
My friend and I are attempting to learn about spectral sequences at the moment, and we've noticed a common theme in books about spectral sequences: no one seems to like talking about differentials.
...
- 13.5k
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 ...
119
votes
6
answers
10k
views
What properties make $[0,1]$ a good candidate for defining fundamental groups?
The title essentially says it all. Consider the category $\mathfrak{Top}_2$ of triples $(J,e_0,e_1)$ where $J$ is a topological space, and $e_i \in J$. There is an obvious generalization of the ...
- 5,759
112
votes
6
answers
10k
views
Counterexamples in algebraic topology?
In this thread
Books you would like to read (if somebody would just write them...),
I expressed my desire for a book with the title "(Counter)examples in Algebraic Topology".
My reason for doing so ...
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
104
votes
10
answers
24k
views
Motivation for algebraic K-theory?
I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like ...
98
votes
10
answers
14k
views
equivalence of Grothendieck-style versus Cech-style sheaf cohomology
Given a topological space $X$, we can define the sheaf cohomology of $X$ in
I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$)
or
II. the Čech ...
- 1,871
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
93
votes
9
answers
37k
views
Is Mac Lane still the best place to learn category theory?
For a student embarking on a study of algebraic topology, requiring a knowledge of basic category theory, with a long-term view toward higher/stable/derived category theory, ...
Is Mac Lane still ...
93
votes
3
answers
11k
views
What is homology anyway?
Disclaimer: I don't feel qualified to ask this question and yet it's been troubling me for some time now and I lost my patience and decided to ask to get some kind of answer. If there are any stupid ...
- 7,789
90
votes
5
answers
7k
views
Algorithm or theory of diagram chasing
One of the standard parts of homological algebra is "diagram chasing", or equivalent arguments with universal properties in abelian categories. Is there a rigorous theory of diagram chasing, and ...
- 56.6k
89
votes
5
answers
16k
views
Why higher category theory?
This is a soft question.
I am an undergrad and is currently seriously considering the field of math I am going into in grad school. (perhaps a little bit late, but it's better late then never.) I ...
87
votes
11
answers
14k
views
What is Quantization ?
I would like to know what quantization is, I mean I would like to have some elementary examples, some soft nontechnical definition, some explanation about what do mathematicians quantize?, can we ...
86
votes
16
answers
9k
views
Teaching homology via everyday examples
What stories, puzzles, games, paradoxes, toys, etc from everyday life are better understood after learning homology theory?
To be more precise, I am teaching a short course on homology, from ...
86
votes
4
answers
15k
views
Etale cohomology -- Why study it?
I know (at least I think I know) that some of the main motivating problems in the development of etale cohomology were the Weil conjectures. I'd like to know what other problems one can solve using ...
- 2,799
85
votes
23
answers
11k
views
Solving algebraic problems with topology
Often, topologists reduce a problem which is - in some sense - of geometric nature, into an algebraic question that is then (partiallly) solved to give back some understanding of the original problem.
...
83
votes
7
answers
7k
views
Computational complexity of computing homotopy groups of spheres
At various times I've heard the statement that computing the group structure of $\pi_k S^n$ is algorithmic. But I've never come across a reference claiming this.
Is there a precise algorithm ...
- 44.4k
83
votes
0
answers
3k
views
Which finite abelian groups aren't homotopy groups of spheres?
Someone asked me if all finite abelian groups arise as homotopy groups of spheres. I strongly doubted it, and I bet ten bucks that $\mathbb{Z}_5$ is not $\pi_k(S^n)$ for any $n,k$. But I don't know ...
- 22.3k
82
votes
12
answers
15k
views
Compelling evidence that two basepoints are better than one
This question is inspired by an answer of Tim Porter.
Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his ...
- 22.1k
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
80
votes
15
answers
15k
views
Why torsion is important in (co)homology ?
I've once been told that "torsion in homology and cohomology is regarded by topologists as a very deep and important phenomenon". I presume an analogous statement could be said in the context of ...
80
votes
7
answers
12k
views
Cubical vs. simplicial singular homology
Singular homology is usually defined via singular simplices, but Serre in his thesis uses singular cubes, which he claims are better adapted to the study of fibre spaces. This young man (25 years old ...
- 47.5k
80
votes
2
answers
7k
views
Vladimir Voevodsky's works
Vladimir Voevodsky has made several contributions in abstract algebraic geometry, focused on the homotopy theory of schemes, algebraic K-theory, and interrelations between algebraic geometry, and ...
80
votes
1
answer
3k
views
Topological cobordisms between smooth manifolds
Wall has calculated enough about the cobordism ring of oriented smooth manifolds that we know that two oriented smooth manifolds are oriented cobordant if and only if they have the same Stiefel--...
- 18.2k
78
votes
12
answers
12k
views
Why aren't representations of monoids studied so much?
It seems to me like every book on representation theory leaps into groups right away, even though the underlying ideas, such as representations, convolution algebras, etc. don't really make explicit ...
- 2,392
78
votes
1
answer
5k
views
The topology of Arithmetic Progressions of primes
The primary motivation for this question is the following: I would like to extract some topological statistics which capture how arithmetic progressions of prime numbers "fit together" in a manner ...
- 15.5k
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
72
votes
3
answers
8k
views
Where do all these projection formulas come from?
I have been intrigued for a long time by the formal similarity of results from different areas of mathematics. Here are some examples.
Set theory Given a map $f:X\to Y$ and subsets $X' \subset X, Y'\...
- 47.5k
71
votes
10
answers
25k
views
Nice proof of the Jordan curve theorem?
As a student, I was taught that the Jordan curve theorem is a great example of an intuitively clear statement which has no simple proof.
What is the simplest known proof today?
Is there an intuitive ...
- 1,843
70
votes
28
answers
7k
views
Examples where it's useful to know that a mathematical object belongs to some family of objects
For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon:
(1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, ...
70
votes
6
answers
8k
views
third stable homotopy group of spheres via geometry?
It is ''well-known'' that the third stable homotopy group of spheres is cyclic of order $24$. It is also ''well-known'' that the quaternionic Hopf map $\nu:S^7 \to S^4$, an $S^3$-bundle, suspends to a ...
- 20.9k
68
votes
12
answers
29k
views
Algebraic topology beyond the basics: any texts bridging the gap?
Peter May said famously that algebraic topology is a subject poorly served by its textbooks. Sadly, I have to agree. Although we have a freightcar full of excellent first-year algebraic topology texts ...
68
votes
9
answers
10k
views
List of Classifying Spaces and Covers
I am looking for a list of classifying spaces $BG$ of groups $G$ (discrete and/or topological) along with associated covers $EG$; there does not seem to be such cataloging on the web. Or if not a ...
68
votes
3
answers
21k
views
Properly Discontinuous Action
When looking definition, and theorems related to Properly discontinuous action of a group $G$ on a topological space $X$, it is different in different books (Topology and Geometry-Bredon, Complex ...
- 1,236
66
votes
8
answers
10k
views
What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$
I would like to know if there is a known necessary and sufficient
property on an open subset of $\mathbb{R}^n$ to be diffeomorphic to $\mathbb{R}^n$ :
For example :
Are all open star-shaped subsets ...
- 677
66
votes
5
answers
8k
views
Does homology have a coproduct?
Standard algebraic topology defines the cup product which defines a ring structure on the cohomology of a topological space. This ring structure arises because cohomology is a contravariant functor ...
- 661
66
votes
4
answers
6k
views
Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?
Similarly is the complement of any countable set in $\mathbb R^3$ simply connected?
Reading around I found plenty of articles discussing the path connectedness $\mathbb R^2 \setminus \mathbb Q^2$ and ...
- 1,187
66
votes
1
answer
2k
views
Is there an octonionic analog of the K3 surface, with implications for stable homotopy groups of spheres?
The infamous K3 surface has many constructions in many fields ranging from algebraic geometry to algebraic topology. Its many properties are well known. For this question I am really interested in the ...
- 27.5k
64
votes
1
answer
4k
views
A dictionary of Characteristic classes and obstructions
I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians.
In an effort to ...
- 7,789
63
votes
5
answers
18k
views
What is modern algebraic topology(homotopy theory) about?
At a basic level, algebraic topology is the study of topological spaces by means of algebraic invariants. The key word here is "topological spaces". (Basic) algebraic topology is very useful in other ...