All Questions
8,725 questions
6
votes
0
answers
532
views
Can one calculate Ext's between microlocalized perverse sheaves/D-modules using topology?
So, I know one really good technique for calculating Ext's between perverse sheaves/D-modules using topology: the convolution algebra formalism, worked out in great detail in the book of Chriss and ...
- 44.7k
5
votes
2
answers
2k
views
Homotopy Pushouts via Model Structure in Top
As far as I know, one way to take a homotopy colimit in a model category is to replace (up to acyclic fibration) all arrows in the diagram with cofibrations, and take the strict colimit of the ...
- 1,033
15
votes
3
answers
3k
views
H-space structure on infinite projective spaces
Any Eilenberg-MacLane space $K(A,n)$ for abelian $A$ can be given the structure of an $H$-space by lifting the addition on $A$ to a continuous map $K(A\times A,n)=K(A,n)\times K(A,n)\to K(A,n)$.
Does ...
- 258
20
votes
2
answers
1k
views
Are non-empty finite sets a Grothendieck test category?
A "test category" is a certain kind of small category $A$ which turns out to have the following property: the category $\widehat{A}$ of presheaves of sets on $A$ admits a model category structure, ...
- 27.2k
11
votes
1
answer
1k
views
Why does the internal singular simplicial space realize to the same thing as the discrete singular simplicial set?
There are two version of the singular simplicial space of a topological space $X$, one discrete and one internal. At least if X is nice, both of them have homotopy equivalent geometric realizations (...
- 27.5k
14
votes
1
answer
961
views
Founding of homological without quite involving derived categories
I am looking at the foundations of homological algebra, e.g. the introduction
of Ext and Tor, and am unsatisfied. The references I look at start with
"this is called a projective module, this is ...
- 27.9k
23
votes
3
answers
6k
views
Does homology detect chain homotopy equivalence?
Is the following true: If two chain complexes of free abelian groups have isomorphic homology modules then they are chain homotopy equivalent.
- 425
39
votes
4
answers
7k
views
Two kinds of orientability/orientation for a differentiable manifold
Let $M$ be a differentiable manifold of dimension $n$. First I give two definitions of Orientability.
The first definition should coincide with what is given in most differential topology text books, ...
- 7,442
4
votes
1
answer
972
views
relationship between borromean rings and hanging-a-picture-from-three-nails puzzle?
I recently heard the following puzzle: There are three nails in the wall, and you want to hang a picture by wrapping a wire attached to the picture around the nails so that if any one nail is removed ...
- 6,069
21
votes
3
answers
1k
views
What's the analogue of the Hilbert class field in the following analogy?
There's a wonderful analogy I've been trying to understand which asserts that field extensions are analogous to covering spaces, Galois groups are analogous to deck transformation groups, and ...
- 118k
7
votes
5
answers
980
views
Killing the torsion in homotopy
Origin
This question was asked by John Baez in This Week's Finds in Mathematical Physics (Week 286). Therefore, please don't upvote this question (unless you really want to), but do upvote the ...
- 15.1k
8
votes
4
answers
586
views
Examples of the varying strengths of topological invariants
In my first algebraic topology class, I remember being told that the simplest reason for homology was to distinguish spaces. For example, if is X=circle and a Y= wedge of a circle and a 2-sphere then ...
5
votes
1
answer
383
views
Killing Chern classes
Let $G$ be a compact connected Lie group and let $E\to B$ be a principal $G$-bundle. Suppose $a$ is a rational cohomology class of $E$ such that its pullback $b$ under an orbit inclusion map $G\to E$ ...
- 23.5k
15
votes
1
answer
1k
views
Stable ∞-categories as spectral categories
Let C be a stable ∞-category in the sense of Lurie's DAG I. (In particular I do not assume that C has all colimits.) Then C does have all finite colimits, the suspension functor on C is an ...
- 25.2k
7
votes
2
answers
2k
views
categorical homotopy colimits
let $hTop_*$ denote the homotopy category of pointed spaces. I believe that it has no pushouts, in general. the reason is that you can't expect the involved homotopies to be compatible. can anyone ...
- 63.1k
8
votes
2
answers
431
views
Formulas for vector fields on Grassmannians?
The Wikipedia article on (real) Grassmannians gives a simple argument that the Euler characteristic satisfies a recurrence relation $$\chi G_{n,r} = \chi G_{n-1,r-1} + (-1)^r \chi G_{n-1,r}$$. This ...
- 44.4k
15
votes
2
answers
973
views
Infinity de Rham quasi-isomorphism
This question is similar to Do chains and cochains know the same thing about the manifold? in the sence that both deal with a natural "comparison" quasi-isomorphism that does not preserve the ring ...
- 23.5k
45
votes
10
answers
4k
views
effective teaching
Eric Mazur has a wonderful video describing how physics is taught at many universities and his description applies word for word to the way I learned mathematics and the way it is still being taught, ...
3
votes
1
answer
858
views
Any reason why K_23(Z) has order 65520?
I'm rereading my notes and they mention that $K_{23}(\mathbb Z) = \mathbb Z/(65520)$
This looks like a good point to stop and ask whether there is any explanation for this $K$-group of integers (23 ...
- 15.1k
20
votes
5
answers
2k
views
Equivalence of ordered and unordered cech cohomology.
Given a topological space X and a finite cover X = $\cup X_i$, one can define Cech cohomology of a sheaf of abelian groups F with respect to the cover $\{X_i\}$ in two different ways:
(Ordered): ...
- 10.5k
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
7
votes
4
answers
686
views
Realizing complexes with bases as cellular complexes
This is a question a friend of mine asked me some time ago. I suspect the answer is "no" but can't prove it.
Every free complex of abelian groups is isomorphic to the reduced cellular complex of some ...
- 23.5k
17
votes
4
answers
2k
views
Characteristic classes in generalized cohomology theories?
Hello,
'ordinary' Stiefel-Whitney classes are elements of the singular cohomology ring and are constructed using the Thom isomorphism and Steenrod squares. So I think they should exist for any (...
- 173
3
votes
1
answer
299
views
disagreement between two definitions of the singular boundary map
Hi everyone, I have a little problem with the definition of singular boundary map in singular homology theory. It appears to be some disagreement between two authors. The first one is Hatcher in his '...
- 105
11
votes
4
answers
1k
views
Equivariant singular cohomology
One can define the $G$-equivariant cohomology of a space $X$ as being the ordinary singular cohomology of $X \times_G EG$ --- I think this is due to Borel? (See e.g. section 2 of these notes)
...
- 21k
12
votes
3
answers
2k
views
How does one find vanishing algebraic cycles?
I have a question, related to what I asked before.
Let's consider a smooth hyperplane section $X$ of a smooth projective variety $Y$ over $\mathbb C$.
According to Weak Lefschetz theorem, cohomology ...
- 1,783
19
votes
4
answers
3k
views
What are the fibrant objects in the injective model structure?
If C is a small category, we can consider the category of simplicial presheaves on C. This is a model category in two natural ways which are compatible with the usual model structure on simplicial ...
- 27.5k
18
votes
1
answer
943
views
Do chains and cochains know the same thing about the manifold?
This question was inspired by Poincaré quasi-isomorphism
Let $M$ be a closed oriented $n$-manifold. The cap product with the fundamental class of $M$ induces an isomorphism $H^i(M,\mathbf{Z})\to ...
- 23.5k
8
votes
2
answers
1k
views
Poincaré quasi-isomorphism
Suppose we have a simplicial combinatorial manifold (just a triangulated manifold) and its Poincaré dual cell complex.
Corresponding homology simplicial and homology cell complexes are quasi-...
- 1,482
39
votes
3
answers
6k
views
Why do finite homotopy groups imply finite homology groups?
Why does a space with finite homotopy groups [for every n] have finite homology groups? How can I proof this [not only for connected spaces with trivial fundamental group]? The converse is false. $\...
- 2,782
23
votes
13
answers
7k
views
Pedagogical question about linear algebra
Last semester I taught a linear algebra class that is intended to introduce young students (at a sophmore-junior level) to "abstract mathematics". It seems that a major conceptual hurdle for many of ...
23
votes
4
answers
4k
views
Curriculum reform success stories at an "average" research university
Greetings all,
There's a never-ending story that many of us have sunk our teeth into. How do we go about teaching subjects like calculus and analysis "well?" Most universities that I'm familiar ...
7
votes
2
answers
541
views
(Co-) Homology associated to Waldhausen K-Theory
Waldhausen K-Theory takes as input a Waldhausen category C and produces a spectrum K(C). I would like to know what is known about generalized (co-) homology theories that can be realized by this ...
- 1,273
11
votes
4
answers
958
views
Geometry of the multilagrangian Grassmannian
Let's introduce the following variety $MG(3,6)$, which is a "multisymplectic" analog of a Lagrangian Grassmannian $LG(3,6)$.
Consider a 3-form $\omega = dx1 \wedge dx2 \wedge dx^3 - dx4 \wedge dx5 \...
- 1,783
3
votes
2
answers
465
views
Branched coverings over orbifolds with reflector lines
It is well known that if $F\to B$ is a $n$-finite branched covering over an orbifold with cone-points then the orbifold Euler's characteristics are related via $\chi(F)=n(\chi(B)-\sum_i^r\frac{a_i-1}{...
- 345
32
votes
8
answers
2k
views
Noncommutative rational homotopy type
Ok, this question is much less ambitious than it might sound, but still:
Two commutative differential graded algebras (cdga's) are quasi-isomorphic if they can be connected by a chain of cdga quasi-...
- 23.5k
5
votes
2
answers
357
views
Truncated exact sequence of homotopy groups
This is a question about a name of a very useful lemma,
that permits one in particular to show that smooth birational complex projective
varieties have isomorphic fundamental groups.
If this lemma ...
- 28.9k
40
votes
4
answers
3k
views
Chain homotopy: Why du+ud and not du+vd?
When one wants to prove that a morphism $f_*$ between two chain complexes $\left(C_*\right)$ and $\left(D_*\right)$ is zero in homology, one of the standard approaches is to look for a chain homotopy, ...
- 33.8k
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
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
39
votes
4
answers
4k
views
What manifolds are bounded by RP^odd?
Real projective spaces $\mathbb{R}P^n$ have $\mathbb{Z}/2$ cohomology rings $\mathbb{Z}/2[x]/(x^{n+1})$ and total Stiefel-Whitney class $(1+x)^{n+1}$ which is $1$ when $n$ is odd, so it follows that ...
- 586
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
21
votes
1
answer
767
views
The density hex
Gale famously showed that the determinacy of n-player, n-dimensional Hex is equivalent to the Brouwer fixed point theorem in n dimensions.
We can (and Gale does) view this as saying that if you d-...
- 12.6k
2
votes
1
answer
510
views
Are the C(S^n, S^n)'s homeomorphic ?
Let m, n > 1. Is it true that C(S^m, S^m), and C(S^n, S^n) are homeomorphic ?
[both endowed with the sup metric (or equivalently the compact-open topology)]
Generally, C(S^n, S^n), with n >= 1, is a ...
- 4,060
14
votes
2
answers
790
views
Finding cocycles that square to zero
Suppose $x$ is a chosen class in the singular cohomology (integer coefficients) of a space $X$. I'm thinking primarily of classes of odd degree on a simply connected space. What are necessary ...
- 13.2k
15
votes
1
answer
2k
views
Graded commutativity of cup in Hochschild cohomology
I am trying to get used to Hochschild cohomology of algebras by proving its properties. I am currently trying to show that the cup product is graded-commutative (because I heard this somewhere); ...
- 33.8k
11
votes
5
answers
7k
views
(infinity,1)-categories directly from model categories
Edit & Note: I'm declaring a convention here because I don't feel like trying to fix this in a bunch of spots: If I said model category and it doesn't make sense, I meant a model-category "model" ...
19
votes
6
answers
3k
views
Simplicial homotopy book suggestion for HTT computations
I'm struggling through Lurie's Higher Topos Theory, since it appears that someone reading through the book is expected to be somewhat comfortable with simplicial homotopy theory. The main trouble I'...
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
5
votes
2
answers
666
views
HNN extensions which are free products
which HNN-extensions are free products? this question is related with another still unsolved about Nielsen-Thruston-reducibility and connected-sum-irreducibility of 3d-torus- bundles...
- 345