All Questions
9,056 questions
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
5
votes
2
answers
944
views
What is $TC(\Sigma^\infty \Omega X)$?
I know that for $X$ a connected space, $THH(\Sigma^\infty \Omega X) = \Sigma^\infty \Lambda X$, the suspension spectrum of the free loop space of $X$. The computation can be carried out in spaces and ...
- 25.2k
10
votes
1
answer
943
views
Cyclic spaces and S^1-equivariant homotopy theory
I'm trying to understand the relationship between cyclic spaces and S1-equivariant homotopy theory. More precisely, I only care about S1-spaces up to equivalence of fixed point spaces for the finite ...
- 25.2k
5
votes
3
answers
1k
views
Computation of Joins of Simplicial Sets
It turns out that joins of simplicial sets are fairly easy to define, but hard to manage. In lots of cases, we'd like to compute what a join is, does it look like a horn?, a boundary?, etc? and ...
1
vote
1
answer
256
views
N_3 and N_4 periodic and pseudo Anosov auto-homeomorphisms
It is well know that the genus three non orientable surface, N3, has only periodic and reducible auto-homeomorphisms, meanwhile the surface N4 is the first non orientable surface with pseudo Anosov ...
- 345
37
votes
3
answers
5k
views
Topological Langlands?
In a workshop about the geometry of $\mathbb{F}_1$ I attended recently, it came up a question related to a mysterious but "not-so-secret-anymore" seminar about... an hypothetical Topological Langlands ...
- 2,992
7
votes
3
answers
1k
views
Joins of simplicial sets
Why doesn't the join operation on the category of simplicial sets commute up to unique isomorphism? I mean, aren't products and coproducts commutative up to isomorphism? That leads me to conclude at ...
3
votes
3
answers
447
views
Representations of finite commutative band semigroups
I think it's clear that commutative semigroups S that are also bands, i.e. $e^2 = e$ for all e, correspond to finite posets (consider the elements of the semigroups as sets, where the intersection of ...
- 2,216
53
votes
6
answers
8k
views
Why is the standard definition of cocycle the one that _always_ comes up??
This question might not have a good answer. It was something that occurred to me yesterday when I found myself in a pub, needing to do an explicit calculation with 2-cocycles but with no references ...
- 41.4k
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
24
votes
1
answer
3k
views
Characteristic classes of sphere bundles over spheres in terms of clutching functions
I'm trying to understand Milnor's proof of the existence of exotic 7-spheres.
Milnor finds his examples among $S^{3}$ bundles over $S^{4}$ (with structure group $SO(4)$ ). Such a bundle can be ...
- 6,546
6
votes
4
answers
1k
views
What is known about the intersection pairing on H^{mid}?
When we restrict to the torsion-free part of the cohomology of a manifold, the intersection pairing is nondegenerate. In dimension 2n, this gives a bilinear form on the free part of Hn (symmetric if ...
- 6,069
11
votes
2
answers
2k
views
The De Rham Cohomology of $\mathbb{R}^n - \mathbb{S}^k$
I'm reading Madsen and Tornehave's "From Calculus to Cohomology" and tried to solve this interesting problem regarding knots.
Let $\Sigma\subset \mathbb{R}^n$ be homeomorphic to $\mathbb{S}^k$, show ...
- 357
0
votes
1
answer
314
views
Homology of symmetric groups
Let $S_n$ denote the symmetric group on $n$ letters, and let $S_n(p)$ denote a Sylow $p$-subgroup. Why is the image of $H_i(S_n(p))$ in $H_i(S_n)$ the $p$-primary part of $H_i(S_n)$?
- 803
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
10
votes
3
answers
2k
views
How are multiplicative sequences related to formal power series and genera of manifolds?
Let $B$ be the graded ring $\bigoplus_i B^i$ (with $B^k B^l \subset B^{k+l}$), and $B_f$ the multiplicative group of all formal sums $1 + b_1 + b_2 + \cdots$ where $b_i \in B^i$ for all $i$.
The idea ...
- 5,530
3
votes
1
answer
361
views
Is the coproduct of fibrant spectra fibrant again?
Define an $S^{1}$-spectrum $E$ to be a sequence of pointed simplicial sets $E_{n},\\ n=0,1,2...$ with assembly morphisms $\sigma_{n}:S^{1}\wedge E_{n}\rightarrow E_{n+1}$.
An $S^{1}$-spectrum $E$ is ...
- 51
9
votes
5
answers
1k
views
References/literature for pushouts in category of commutative monoids? [ed. - amalgams]
This is more of a request for pointers to relevant literature than a question per se. I am, erm, looking at a paper which uses a kind of iterated pushout construction to obtain a commutative monoid ...
- 25.8k
9
votes
3
answers
1k
views
Integration in equivariant K-theory
Let F be a smooth classifying space for K-theory (ordinary or equivariant). If X is a smooth compact manifold and W is a real vector space of dimension n, there is an integration map from the ...
- 226
46
votes
6
answers
7k
views
Why does one think to Steenrod squares and powers?
I'm studying Steenrod operations from Hatcher's book. Like homology, one can use them only knowing the axioms, without caring for the actual construction. But while there are plenty of intuitive ...
- 14.7k
13
votes
2
answers
2k
views
Combinatorics of the Stasheff polytopes
First a little background for those unaware. The Stasheff polytopes (or associahedra) are certain convex polytopes that arise in the theory of $A_\infty$-algebras. There is one polytope for each $n\...
- 3,423
1
vote
1
answer
870
views
Simplicial set notation and vocabulary question.
Notation question:
What does $(\Delta^1)^{ \{1, \ldots,n-1 \}}$ denote? UPDATE: I (David Speyer) tried to fix the LaTeX. Please see if I got it right.
Vocabulary question:
Suppose $z:\Delta^{n+1} \...
50
votes
10
answers
14k
views
Definition of "simplicial complex"
When I think of a "simplicial complex", I think of the geometric realization of a simplicial set (a simplicial object in the category of sets). I'll refer to this as "the first definition".
However, ...
- 21k
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 ...
5
votes
1
answer
638
views
Spanier-Whitehead dual and Hopf fibration
Consider a map of spheres $f:S^n\to S^m$ covered by a map of trivial $\mathbb R^k$-bundles.
In other words, we take the trivial rank $k$ vector bundle over $S^m$ and pull it to $S^n$ via $f$. Consider ...
- 29.1k
53
votes
4
answers
14k
views
Explanation for the Chern character
The Chern character is often seen as just being a convenient way to get a ring homomorphism from K-theory to (ordinary) cohomology.
The most usual definition in that case seems to just be to define ...
- 5,530
13
votes
4
answers
3k
views
Circle bundles over $RP^2$
Does anybody know if orientable, closed $3$-manifolds that are circle bundles over $RP^2$ have been classified?
One can determine the isomorphism classes of bundles using obstruction theory, but I am ...
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
8
votes
1
answer
637
views
Cohomology map induced by the group actions on homogeneous vector bundles
Here is a topological question which seems quite elementary. The answer to this question may be useful e.g. in estimating the orders of the automorphism groups of some algebraic varieties and in ...
- 23.5k
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
4
votes
3
answers
2k
views
Homotopy groups of smooth manifolds?
For a fixed $d$, is there a relationship between the homotopy groups of smooth $d$-manifolds?
The $d=1$ case is trivial, but I already don't know how to approach $d=3$ (I should have said that the ...
- 15.1k
7
votes
2
answers
268
views
What are the fibres of a representable simplicial sheaf (in the Nisnevich topology)
Let $k$ be a field and $X$ a smooth $k$-scheme. We consider now the pointed (constant) simplical Nisnevich-sheaf $X_{+}$ that is represented by $X$. Let now $\nu\in U$ be a point of a smooth scheme, ...
- 71
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
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
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
8
votes
5
answers
1k
views
Examples of left reversible semigroups
I am looking for concrete examples of cancellative, left reversible semigroups. Left reversible semigroups are also called "Ore semigroups". See this wikipedia page for the definition of a left ...
- 550
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