All Questions
8,725 questions
17
votes
10
answers
109k
views
What are the qualities of a good (math) teacher? [closed]
In forming your answer you may treat the qualifier math or maths as optional, since part of the question is whether there is anything peculiar to the subject of mathematics that demands anything ...
22
votes
4
answers
2k
views
Functorial Whitehead Tower?
The Whitehead tower of a (pointed) space is a tower of spaces which successively kills the bottom homotopy groups. The first two spaces can be constructed functorially (at least for suitably nice ...
- 27.5k
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
4
votes
3
answers
2k
views
Mapping torus of a homotopy equivalence
The mapping torus $M_f$ of a homeomorphism $f$ of some topological space $X$ is a fiber bundle whose base is a circle and whose fiber is the original space $X$. If instead of a homeomorphism $f$ is ...
- 41
12
votes
2
answers
1k
views
Why are Delta-generated spaces locally presentable?
Does anybody understand why Delta-generated spaces are locally presentable? This is of course claimed by Jeff Smith, and there is a paper by Fajstrup and Rosicky
A convenient category for directed ...
- 3,685
15
votes
3
answers
2k
views
complex cobordism from formal group laws?
Reading Ravenel's "green book", I wonder about his question on p.15 "that the spectrum MU may be constructed somehow using formal group law theory without using complex manifolds or vector bundles. ...
- 10.8k
6
votes
2
answers
396
views
Reference for iterated homotopy fixed points?
What are (good) references for results about iterated homotopy fixed points? That is, suppose G is a topological group acting on a space (or spectrum) X, and H is a normal subgroup of G. Then one ...
- 3,103
12
votes
3
answers
4k
views
Notions of degree for maps $S^n \to S^n$?
In algebraic topology, we define a degree for a map $f: S^n \to S^n$ as where the induced map $f_*$ on the $n$-th homology group of $S^n$ sends $1$.
In differential topology, we have a different (...
- 502
8
votes
1
answer
575
views
Homotopy orbit spaces of representation spheres
Let $G$ be a finite group and $V$ be finite-dimensional real representation of $G$. Write $S^V$ for the one-point compactification of $V$, with induced $G$-action, viewed as a pointed space, and ...
- 25.2k
3
votes
1
answer
928
views
Simple applications of Atiyah-Bott localization
I am looking for some simple and concrete -- but still non-trivial and illustrative -- applications of Atiyah-Bott localization in the context of equivariant cohomology.
Do you know any good ones?
- 21k
8
votes
5
answers
1k
views
Braided Monoidal 2-categories with duals
Which categorifications give explicit braided monoidal 2-categories with duals?
This question is in response to Ben Webster's questions in recent history. The point is that given a braided monoidal 2-...
- 5,264
9
votes
3
answers
3k
views
Math History Question about the exponential function
While tutoring a student recently, I have come across the situation of explain logarithms by first introducing functions of the form $$f(x)= a^x$$ where $a \ge 0,x\in \mathbb{R}$. My student then ...
- 297
14
votes
2
answers
947
views
squares in stable homotopy
I noticed that the generator of the second stable stem b is the square of the generator of the first stable stem a, in the sense that if take two copies of a and smash product them together you get b ...
- 1,387
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
9
votes
1
answer
875
views
Analogue of Sperner's lemma for Lefschetz theorem?
Sorry if this is easy/well-known, I don't know much algebraic topology and I'm just curious about this question.
One of the easier proofs of the Brouwer fixed-point theorem (we'll say for n = 2 for ...
- 12.6k
24
votes
3
answers
4k
views
Subgroups of free abelian groups are free: a topological proof?
There is a well-known topological proof of the fact that subgroups of free groups are free. Many people, myself included, think it is easier and more natural than the purely algebraic proofs which ...
- 65.4k
14
votes
2
answers
1k
views
Exotic spheres and stable homotopy in all large dimensions?
Given that the kervaire invariant one problem has been solved in (almost) all dimensions....my question is whether there exists an exotic sphere in all sufficently lagre dimensions? Given the Kervaire-...
- 703
60
votes
6
answers
7k
views
Torsion in homology or fundamental group of subsets of Euclidean 3-space
Here's a problem I've found entertaining.
Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups is not ...
- 44.4k
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
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
18
votes
4
answers
2k
views
When can you desuspend a homotopy cogroup?
Any topological group $G$ has a classifying space, whose loopspace is a (homotopy) group which is homotopy equivalent to $G$ in a way that preserves the group structure. More generally, if $G$ is an $...
- 31.2k
16
votes
12
answers
11k
views
Are there any interesting connections between Game Theory and Algebraic Topology?
I've been learning game theory on my own and was just curious how it connected with previous things I've learned. So are there any interesting connections between Game Theory and Algebraic Topology? ...
- 809
2
votes
0
answers
526
views
How much of math could be taught without using mathematical notation? [closed]
Given that mathematics is not about number, and that it is not even about the cryptic notation used to describe mathematical problems, how much of mathematics could be taught without reference to ...
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
5
answers
1k
views
Explanation for E_8's torsion
To study the topology of Lie groups, you can decompose them into the simple compact ones, plus some additional steps, such as taking the cover if necessary. After that, the structure of $SO(n)$'s is ...
- 15.1k
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
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
6
votes
3
answers
2k
views
Mayer-Vietoris homotopy groups sequence of a pull-back of a fibration
This must be an elementary question: could somebody tell me a reference for the Mayer-Vietoris homotopy groups sequence of a pull-back of a fibration?
I'm working in the category of pointed ...
- 1,975
11
votes
3
answers
734
views
Relationship between universal coefficient theorem and $[K(\mathbb{Z},n), K(G,n)]$?
In short, I'm wondering whether the universal coefficient theorem can be understood/reinterpreted by using maps of Eilenberg-MacLane spaces. This is a wishy-washy idea and I don't have evidence to ...
- 6,069
4
votes
1
answer
1k
views
properly interpreting Pi_0 in the homotopy exact sequence
Define the lens space L(m,n) as the quotient of S2m+1 by the action of the cyclic group ℤn⊂S1⊂ℂ*. We can create the infinite lens space L(∞,n) by a telescoping construction ...
- 6,069
16
votes
2
answers
2k
views
What is known about K-theory and K-homology groups of (free) loop spaces?
Calculating the homology of the loop space and the free loop space is reasonably doable. There exists the Serre spectral sequence linking the homology of the loop space and the homology of the free ...
- 8,167
13
votes
5
answers
1k
views
What kind of geometric operations "scale up" cohomology?
There's an obvious operation on the category of graded rings, given by "scaling up," multiplying the grading of every element by some fixed constant.
Does anyone know of an operation on the level of ...
- 44.7k
28
votes
4
answers
4k
views
(∞, 1)-categorical description of equivariant homotopy theory
I'm trying to learn a bit about equivariant homotopy theory. Let G be a compact Lie group. I guess there is a cofibrantly generated model category whose objects are (compactly generated weak ...
- 25.2k
24
votes
6
answers
2k
views
Simplicial model of Hopf map?
The Hopf fibration is a famous map $S^3\to S^2$ with fiber $S^1$, which is the generator in $\pi_3(S^2)$. We can model this map in terms simplicial sets by taking the singular simplicial sets of these ...
- 27.5k
18
votes
8
answers
3k
views
How to get product on cohomology using the K(G, n)?
This came up in the question about Eilenberg-MacLane spaces. Given the definition of K(G, n), it's easy to prove that there is a map ...
- 15.1k
19
votes
3
answers
4k
views
Cohomology and Eilenberg-MacLane spaces
This question is related to this question from Dinakar, which I found interesting but don't yet have the background to understand at that level.
Unless I'm mistaken, the rough statement is that $H^n(...
- 6,069
7
votes
2
answers
1k
views
One Point Compactification
Suppose X is a path-connected, locally compact, Hausdorff space and Y is its one-point compactification. Let G be the fundamental group of X and H be the fundamental group of Y. Is it true that the ...
- 205
2
votes
2
answers
369
views
Classifying space of a crossed complex
Brown defines the classifying space of a crossed complex in the following way.
Given a filtration X* of a space X, define the fundamental crossed complex by:
C_0 = X_0, C_1=\pi(X_1,X_0) (the ...
- 1,422
4
votes
5
answers
474
views
Classifying maps into homogeneous spaces up to homotopy
I'm still just a beginner in algebraic topology, but there's a specific problem I'd like to understand, which is how to classify maps from one space into another up to homotopy -- for instance, I've ...
- 13.6k
32
votes
5
answers
4k
views
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
21
votes
7
answers
4k
views
Whitehead for maps
I made the following claim over at the Secret Blogging Seminar, and now I'm not sure it's true:
Let $f: X \to Y$ and $g: X \to Y$ be two maps between finite CW complexes. If f and g induce the same ...
- 156k
4
votes
2
answers
2k
views
(how) are vector bundles and homotopy groups related?
Hello,
homotopic maps induce isomorphic pullback bundles, and so isomorphism classes of vector bundles over X correspond to homotopy classes of maps from the grassmannian to X. But I think, in the ...
- 51
58
votes
12
answers
29k
views
Homological Algebra texts
I would like to hear the communities' ideas on good Homological Algebra textbooks / references. The standard example is of course Weibel (which I'll leave for someone else to describe).
As usual, ...
3
votes
7
answers
4k
views
de Rham Cohomology of surfaces
Does anyone know a good book where I can find the computation of the de Rham Cohomology of surfaces in R^3 and other classical manifolds (higher dimensional spheres and projective spaces for example) ?...
- 1,058
5
votes
1
answer
190
views
Adapting families of diffeomorphisms to an open cover
Has anyone seen the following result in the literature? I've asked a few experts but so far I've come up with nothing.
Given a manifold M and an open cover {U_i} ...
- 12.8k
17
votes
5
answers
5k
views
How to determine the homotopy groups of the suspension of a space?
Let $SX$ be the suspension of CW complex. What are some results available to determine the homotopy groups of $SX$?
- 173
20
votes
5
answers
3k
views
Does the cohomology ring of a simply-connected space X determine the cohomology groups of ΩX?
One could try to apply the Eilenberg-Moore spectral sequence to the pullback diagram • → X ← •, obtaining a spectral sequence TorH•(X, R)(R, R) => H•(ΩX, R), but ...
- 25.2k
46
votes
11
answers
6k
views
What is the Cayley projective plane?
One can build a projective plane from $\Bbb R^n$, $\Bbb C^n$ and $\Bbb H^n$ and is then tempted to do the same for octonions. This leads to the construction of a projective plane known as $\Bbb OP^2$, ...
- 8,167
7
votes
2
answers
559
views
Can one calculate the (co)homology of the loopspace of a Lie group from its Lie algebra?
Compact connected simply-connected Lie groups have so much structure that you can calculate their cohomology from their Lie algebras using Lie algebra cohomology (certain Ext-groups) and similarly ...
- 8,167
15
votes
2
answers
1k
views
Are generalized cohomology theories a homotopy category of some category of invariants?
I was taught to think of generalized cohomology theories as the homotopy category of (symmetric) spectra. But is there also a category of 'invariants', that is, some category of contravariant functors ...
- 8,167