All Questions
9,056 questions
27
votes
1
answer
4k
views
connectivity of the group of orientation-preserving homeomorphisms of the sphere
In the paper "Local Contractions and a Theorem of Poincare" Sternberg has mentioned the following question which was open when the paper was written:
Is the group of orientation-preserving ...
- 6,224
27
votes
2
answers
2k
views
Cobordism of orbifolds?
Is it possible to setup classical cobordism theory in the context of orbifolds? For example, let's consider the free abelian group generated by oriented smooth orbifolds and quotient by those which ...
- 18.7k
27
votes
2
answers
3k
views
Teaching the fundamental group via everyday examples
This question is a "prequel" to a similar question about homology. Both questions were inspired by seeing a talk, by Tadashi Tokieda, about the interesting physics that appears in toys.
What ...
27
votes
2
answers
3k
views
Euler Characteristic of a manifold with non-vanishing vector field,
A friend of mine recently asked me if I knew any simple, conceptual argument (even one that is perhaps only heuristic) to show that if a triangulated manifold has a non-vanishing vector field, then ...
- 15.3k
27
votes
3
answers
2k
views
What do whitehead towers have to do with physics?
First let me say something that I don't completely understand, since I do not know enough physics. If I say anything wrong, someone please tell me:
For the spinning particle, there is a sigma-model, ...
- 15.5k
27
votes
2
answers
796
views
Is there a flat manifold with trivial first homology?
Is there a closed flat manifold whose fundamental group has trivial abelianization?
The famous Hantzsche–Wendt flat manifold has fundamental group with finite abelianization.
- 29.1k
27
votes
1
answer
3k
views
Mixed Hodge structure on the rational homotopy type
A mixed Hodge structure (mHs) on a commutative differential graded algebra (cgda) over $\mathbf{Q}$ is a mixed Hodge structure on the underlying vector space such that the product and the differential ...
- 23.5k
27
votes
2
answers
1k
views
Is being simply connected very rare?
Essentially, my question is how strong a restriction it is to be simply connected.
Here is a way of making this precise: Let's say we want to count simplicial complexes (of dimension 2, though that ...
- 752
27
votes
2
answers
2k
views
Need for support and guidance for my near future as a PhD student (or: has stable homotopy theory become an overly algebraic theory?)
The question in brackets in the title is my main mathematical question, but does not reflect my initial motivation for writing this post. It is in fact above all for personal reasons that I'm ...
- 395
27
votes
2
answers
2k
views
Combinatorics of K(Z,2)?
Anybody knows a semi-simplicial model for $K(Z,2)$ having finite number of simplexes in any dimension? With some regular description? I have heard about big activity on triangulating $CP^n$ but this ...
- 1,482
27
votes
1
answer
1k
views
n-categorical description of Chern classes
The Chern classes of a rank $n$ vector bundle on $X$ are obtained from composing the associated classifying map in $[X, BU(n)]$ with the maps $BU(n) \to B^{2i} \mathbb{Z}$ corresponding to the ...
- 805
27
votes
0
answers
1k
views
Spectral sequences as deformation theory
I believe that running the spectral sequence of a filtered complex / spectrum $ \cdots \to F_n \to F_{n+1} \to \cdots$ can be viewed as doing deformation theory in some very primitive "derived ...
- 63.9k
27
votes
0
answers
1k
views
Computational complexity of topological K-theory
I am a novice with K-theory trying to understand what is and what is not possible.
Given a finite simplicial complex $X$, there of course elementary ways to quickly compute the cohomology of $X$ with ...
- 666
26
votes
3
answers
3k
views
Third differential in Atiyah Hirzebruch spectral sequence
Does any one know why $d_3: H^* (X, K^0(point))\rightarrow H^{*+3}(X,K^0(point))$ is actually extended $Sq^3$ to $\mathbb{Z} $ coefficient.
- 1,003
26
votes
2
answers
2k
views
Does a "Chern character" exist for any generalized cohomology theory?
The Chern character is a ring homomorphism from complex K-theory to the usual cohomology.
1) I wonder if there are "Chern character"-like ring homomorphisms from other generalized cohomology theories ...
- 1,525
26
votes
1
answer
1k
views
Spheres with the same homotopy groups
What is known about the existence of other pairs of spheres (such as $S^2$ and $S^3$) whose homotopy groups coincide starting from some index.
A sufficient condition for this is the existence of a ...
- 6,962
26
votes
1
answer
3k
views
Are there "principal" bundles $S^1 \to S^3 \to S^2$ other then Hopf's? (They would be necessarily not locally trivial)
It is well known that the only principal locally trivial fiber bundle $S^1 \to S^3 \to S^2$ is Hopf map $h$ (see, for example, [1]).
What if we drop the local triviality but mantain a "principality" ...
- 1,123
26
votes
6
answers
3k
views
How to get convinced that there are a lot of 3-manifolds?
My question is rather philosophical : without using advanced tools as Perlman-Thurston's geometrisation, how can we get convinced that the class of closed oriented $3$-manifolds is large and that ...
- 2,696
26
votes
4
answers
1k
views
Conjuring phantoms by hand?
A map $f:X\to Y$ of CW-complexes is called a phantom if $f$ restricted to the $n$-skeleton of $X$ is contractible for all $n$. The first non-trivial example of such a map, with $X=\Sigma\mathbb{P}^\...
- 23.5k
26
votes
3
answers
1k
views
Proving that a function's image contains (1/n,...,1/n)
This question is a follow-up to a previous question answered by Neil Strickland:
Map from simplex to itself that preserves sub-simplices
Let $B$ denote the closed unit ball in $\mathbb{R}^2$ and let ...
- 645
26
votes
2
answers
1k
views
A characterisation of $\mathbb{P}^n$
Let $X$ be a projective variety (so, with some (edit: fixed nondegenerate closed) embedding) with the following curious property: for every hyperplane section $H$, we have that $X-H \cong \mathbb{A}^n$...
- 437
26
votes
2
answers
2k
views
Structure of Hopf algebras - trouble understanding an old paper
UPDATE: I am grateful to Peter May for the accepted answer, which makes most of the details below irrelevant. However, I will leave them in place for the record.
I am trying to understand the proof ...
- 56.9k
26
votes
3
answers
2k
views
Reverse mathematics of (co)homology?
Background
Exercise 2.1.16b in Hartshorne (homework!) asks you to prove that if $0 \rightarrow F \rightarrow G \rightarrow H \rightarrow 0$ is an exact sequence of sheaves, and F is flasque, then $0 \...
- 12.1k
26
votes
2
answers
2k
views
Loop Spaces as Generalized Smooth spaces or as Infinite dimensional Manifolds?
There are two ways to define smooth mapping spaces and I want to know how they compare.
Let's take the concrete special case of free loops spaces. I think this is the most studied example so will ...
- 27.5k
26
votes
1
answer
2k
views
A refinement of Serre's finiteness theorem on unstable homotopy groups of spheres
Serre's finiteness theorem says if $n$ is an odd integer, then $\pi_{2n+1}(S^{n + 1})$ is the direct sum of $\mathbb{Z}$ and a finite group. By looking at the table of homotopy groups, say on ...
- 513
26
votes
5
answers
2k
views
Surprising properties of closed planar curves
In https://arxiv.org/abs/2002.05422 I proved with elementary topological methods that a smooth planar curve with total turning number a non-zero integer multiple of $2\pi$ (the tangent fully turns a ...
- 405
26
votes
4
answers
789
views
Ring of closed manifolds modulo fiber bundles
Let $R$ be the ring which is generated by homeomorphism classes $[M]$ of compact closed manifolds (of arbitrary dimension) subject to the relations that
$$[F]\cdot [B] = [E]$$
if there exists a fibre ...
- 25.5k
26
votes
1
answer
831
views
Are complex-oriented ring spectra determined by their formal group law?
To every complex-oriented ring spectrum $E$ there is associated a formal group law, which is a power series $F_E(x,y)\in E_*[[x,y]]$.
Suppose $E$ and $F$ are two complex-oriented ring spectra and ...
- 2,052
26
votes
1
answer
942
views
Closed manifold with non-vanishing homotopy groups and vanishing homology groups
Is there a closed connected $n$-dimensional topological manifold $M$ ($n\geq 2$) such that $\pi_i(M)\neq 0$ for all $i>0$ and $H_i(M, \mathbb{Z})=0$ for $i\neq 0$, $n$? The manifold $S^1\times S^2$ ...
user137767
26
votes
1
answer
1k
views
Is the $\infty$-topos $Sh(X)$ hypercomplete whenever $X$ is a CW complex?
It can be shown (see Is every paracompact, Hausdorff, locally contractible space homotopy equivalent to a CW complex?) that if $X$ is a locally contractible paracompact Hausdorff space such that the $\...
- 9,481
26
votes
1
answer
1k
views
From the perspective of bordism categories, where does the ring structure on Thom spectra come from?
To fix ideas, let's consider the Thom spectrum of framed bordism $M$, the spectrum whose homotopy groups are the framed bordism groups. $M$ has a ring spectrum structure inducing the product of ...
- 118k
26
votes
2
answers
5k
views
Cohomology of Lie groups and Lie algebras
The length of this question has got a little bit out of hand. I apologize.
Basically, this is a question about the relationship between the cohomology of Lie groups and Lie algebras, and maybe ...
- 23.5k
26
votes
1
answer
940
views
Are (semi)simple Lie groups some sort of "homotopy quotient groups" of their maximal tori?
Warning: non-specialist writing, some rubbish possible.
The formula $h^*(BG)\cong h^*(BT)^W$ valid for complex oriented cohomology of the classifying space of a compact Lie group $G$ with maximal ...
- 18.4k
26
votes
2
answers
2k
views
Are there geometrically formal manifolds, which are not rationally elliptic?
Formality of a space is meant in the sense of Sullivan, i.e. a space $X$ is called formal, if its commutative differential graded algebra of piecewise linear differential forms $(A_{PL}(X),d)$ is ...
- 2,974
26
votes
1
answer
2k
views
Infinity-categorical analogue of compact Hausdorff
Recently I became through this mathoverflow question aware of the article Codensity and the ultrafilter monad by Tom Leinster. There he shows that the ultrafilter monad on the category $\mathrm{Set}$ ...
- 12.1k
26
votes
1
answer
2k
views
What is to tmf as KR is to KO?
The $E_\infty$-ring spectrum $KU$ of complex K-theory carries a canonical involution induced from complex conjugation of complex vector bundles. The homotopy fixed points of this $\mathbb{Z}_2$-action ...
- 19.8k
26
votes
2
answers
2k
views
Euler characteristic and universal cover
Let $M$ be a compact manifold, let $\tilde{M}$ be its universal cover, and suppose that the Euler characteristic $\chi(\tilde{M})=0$.
My question is: does this imply that $\chi(M)=0$?
This is clear if ...
- 1,452
26
votes
1
answer
1k
views
Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle?
Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spaces" and I am trying ...
- 27.5k
26
votes
1
answer
615
views
What is the minimal dimension of a complex realising a group representation?
This question is inspired by this one, which was about representations that can be realised homologically by an action on a graph (i.e., a 1-dimensional complex).
Many interesting integral ...
- 10.9k
25
votes
4
answers
4k
views
A possible generalization of the homotopy groups.
The homotopy groups $\pi_{n}(X)$ arise from considering equivalence classes of based maps from the $n$-sphere $S^{n}$ to the space $X$. As is well known, these maps can be composed, giving arise to a ...
- 5,759
25
votes
4
answers
4k
views
How canonical is cofibrant replacement?
Quillen's original definition of a model category included noncanonical factorization axioms, one being that any map can be factored into a cofibration followed by an acyclic fibration. More recent ...
- 52.6k
25
votes
6
answers
5k
views
Is there a classification of open subsets of euclidean space up to homeomorphism?
I hope this question is reasonable enough to have a well known answer. i.e either there is a simple invariant (like the homotopy groups) that characterizes the homeomorphism type of such set among ...
- 2,027
25
votes
4
answers
5k
views
Integrals from a non-analytic point of view
I've mentioned before that I'm using this forum to expand my knowledge on things I know very little about. I've learnt integrals like everyone else: there is the Riemann integral, then the Lebesgue ...
- 6,348
25
votes
8
answers
2k
views
Avatars of the ring of symmetric polynomials
I'm collecting different apparently unrelated ways in which the ring (or rather Hopf algebra with $\langle,\rangle$) of symmetric functions $Z[e_1,e_2,\ldots]$ turns up (for a Lie groups course I will ...
- 20.7k
25
votes
3
answers
2k
views
Hermann Weyl's work on combinatorial topology and Kirchhoff's current law in Spanish
Hermann Weyl was one of the pioneers in the use of early algebraic/combinatorial topological methods in the problem of electrical currents on graphs and combinatorial complexes. The ...
- 2,630
25
votes
2
answers
2k
views
Steenrod operations in etale cohomology?
For $X$ a topological space, from the short exact sequence
$$ 0 \rightarrow \mathbb{Z}/2 \rightarrow \mathbb{Z}/4 \rightarrow \mathbb{Z}/2 \rightarrow 0 $$
we get a Bockstein homomorphism
$$H^i(X,...
- 2,809
25
votes
4
answers
6k
views
Singular Homology/Cohomology as a derived functor?
Hello,
Learning some Alg.geometry and Sheaf theory, I got used to the notion that cohomology arises naturally as a derived functor of some sort.
This has led me thinking, singular cohomology, from ...
- 1,893
25
votes
3
answers
2k
views
Are fundamental groups of aspherical manifolds Hopfian?
A group $G$ is Hopfian if every epimorphism $G\to G$ is an isomorphism. A smooth manifold is aspherical if its universal cover is contractible. Are all fundamental groups of aspherical closed smooth ...
- 32.4k
25
votes
3
answers
1k
views
What spaces $X$ do have $\text{End}(X) \cong \text{End}(\mathbb{R})$?
This is a follow-up on the following question. Let $\text{End}(X)$ denote the endomorphism monoid of a topological space $X$ (that is, the collection of all continuous maps $f:X\to X$ with composition)...
- 48.9k
25
votes
2
answers
2k
views
Who computed the third stable homotopy group?
I have spend some time with the geometric approach of framed cobordisms to compute homotopy classes, due to Pontryagin. He computed $\pi_{n+1}(S^n)$ and $\pi_{n+2}(S^n)$. After surveying the ...
- 7,583