All Questions
9,056 questions
7
votes
5
answers
979
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
14
votes
1
answer
2k
views
When are epimorphisms of algebraic objects surjective?
Let $C$ be the category of $\tau$-algebras for some type $\tau$. Consider the statements:
Every monomorphism is regular.
Every epimorphism in $C$ is surjective.
It is easy to see that 1. implies 2. ...
- 63.1k
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
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
38
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 ...
- 576
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
3
votes
3
answers
769
views
Reducible 3d torus bundles
Here reducible means that the mapping class for the fiber is a reducible auto-homeomorph in the sense of Nielsen-Thruston. So,
could anyone give me a hint to classify them?
In contrast, do you agree ...
- 345
16
votes
2
answers
817
views
Spin structures on 7-dimensional spherical space forms
Background
Let $M$ be a spin manifold and let $\Gamma$ be a finite group acting freely and isometrically on $M$ in such a way that $M/\Gamma$ is a smooth riemannian manifold. The quotient will be ...
- 33.3k
2
votes
3
answers
746
views
Two solid N_3 glued by its boundary
Let $N_3$ be the genus three non orientable surface. Do we have an analogous 3d manifold as the solid torus and the solid Klein bottle for $N_3$? I don't see how to extend the ideas related to the 3d ...
- 345
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
14
votes
1
answer
933
views
Smooth structures on PL 4-manifolds
Is it known whether $O(4) \to PL(4)$, the map from the orthogonal group to the group of piecewise linear homeomorphisms of $\mathbb{R}^4$, is a homotopy equivalence? By smoothing theory for PL ...
- 967
5
votes
2
answers
420
views
Connectivity after Geometric Realization?
Suppose that I have a map of simplicial spaces,
$ f: X_* \to Y_*$,
and that I know that the map on zero spaces $f_0: X_0 \to Y_0$ is n-connected. Can I conclude anything about the connectivity of ...
- 27.5k
2
votes
2
answers
1k
views
Periodic mapping classes of the genus two orientable surface
Please, any information on the periodic mapping classes of the genus two orientable surface, $O_2$, will be greatly thanked. We had been studying the topological structure of 3d surface bundles and ...
- 345
23
votes
9
answers
4k
views
What methods exist to prove that a finitely presented group is finite?
Suppose I have a finitely presented group (or a family of finitely presented groups with some integer parameters), and I'd like to know if the group is finite. What methods exist to find this out? I ...
- 1,861
36
votes
21
answers
6k
views
Generalizations of Planar Graphs
This is a follow up to Harrison's question: why planar graphs are so exceptional. I would like to ask about (and collect answers to) various notions, in graph theory and beyond graph theory (topology; ...
- 24.7k
21
votes
2
answers
2k
views
Topologically contractible algebraic varieties
From a post to The Jouanolou trick:
Are all topologically trivial (contractible) complex algebraic varieties necessarily affine? Are there examples of those not birationally equivalent to an affine ...
- 15.1k
7
votes
2
answers
639
views
Naive Z/2-spectrum structure on E smash E?
Let $E$ be a spectrum. Then $E \wedge E$ is a $\mathbb{Z}/2$-spectrum in the naivest possible sense, i.e., an object with $\mathbb{Z}/2$-action in the (∞,1)-category of spectra. Can I make it ...
- 25.2k
38
votes
2
answers
13k
views
Explanation for the Thom-Pontryagin construction (and its generalisations)
In 1950, Pontryagin showed that the n-th framed cobordism group of smooth manifolds was equal to n-th stable homotopy group of spheres:
$$ \lim_{k \to \infty} \pi_{n+k}(S^k) \cong \Omega_n^{\text{...
- 5,530