All Questions
9,056 questions
3
votes
0
answers
129
views
Which spectra have a homotopy-universal connective quotient?
Prefatory remark: This is a repost of a previous question, to which Tyler Lawson supplied a lovely $\infty$-categorical answer. The example that motivated the question was specifically about the ...
- 54.6k
1
vote
0
answers
137
views
Is $\pi_m(M) = 0$ if $\pi_m(M-X) = 0$ for a low-dimensional subset $X$?
I am doing a problem where I am stuck at this point.
Let $M$ be a connected smooth manifold of dimension $n$ and let $X$ be any subset of $M$. Assume that there is a positive integer $m$ such that $n&...
9
votes
1
answer
756
views
Does there exist a GRR-like generalization of the AS Index Theorem?
The Hirzebruch Riemann-Roch Theorem (HRR) expresses an analytic/algebraic invariant, namely the Euler-Poincaré characteristic of a vector bundle $V$ over a compact complex/algebraic manifold $X$, as ...
- 1,020
15
votes
1
answer
2k
views
How to motivate constructible sheaves
I'm writing some notes for some students which just finished a first course in scheme theory. There I would like to talk about constructible sheaves, but I found it hard to give a compelling ...
- 771
4
votes
0
answers
196
views
Valuations and (semi)norms on ring spectra
Valuations and seminorms on rings play a big role in number theory and analytic geometry, with seminorms being heavily used in Berkovich geometry and valuations featuring heavily in adic geometry.
Let'...
- 11.8k
14
votes
1
answer
919
views
What is known about exotic spheres up to stable diffeomorphism?
In even dimensions $n=2k$ we can define two smooth manifolds $M$ and $N$ to be stably diffeomorphic if they become diffeomorphic after the connect sum with $r$ many copies of $S^k \times S^k$ for some ...
- 27.5k
3
votes
0
answers
99
views
What does homotopy invariance mean for twisted K-theory?
In ordinary K-theory, homotopy invariance means that if $f,g \colon X \to Y$ are homotopic maps then their induced maps on K-theory are equal: $f^* = g^* \colon K(Y) \to K(X)$.
My question is how to ...
- 293
6
votes
1
answer
417
views
Do $h$-cobordism groups arise from a 'Thom-like' spectrum?
Not thinking about $h$-cobordism, one usually defines a cobordism between manifolds, realizes it is an equivalence relation, chooses an appropriate class of structured manifolds (framed, unoriented, ...
- 609
13
votes
1
answer
624
views
Ultracategories with one object
Historically, the theory of ultracategories was invented by Makkai to prove a strong conceptual completeness theorem for first-order logic, roughly: if $T$ and $S$ are two first-order theories such ...
- 133
9
votes
1
answer
235
views
Links and non-orientable surfaces
Let $\Sigma \subset \mathbb{R}^3$ be a compact embedded surface with boundary $\partial \Sigma$ and $i:\Sigma\setminus \partial\Sigma \to \mathbb{R}^3 \setminus \partial\Sigma$ the inclusion.
Is the ...
- 443
3
votes
1
answer
197
views
Homology of iterated loop spaces on odd--dimensional spheres
For prime $p$ let $E_p[\dots]$ and $P_p[\dots]$ be the external and polynomial $\mathbb{Z}_p$--algebras.
It is known that for $n\geqslant 1$ and odd $p$ where is an isomorphism
of primitively ...
- 191
3
votes
1
answer
114
views
What is this cochain complex about, whose $H^1 = \mathbb{R}$?
$\DeclareMathOperator\QEnd{QEnd}$Let $C^n$ be the set of functions $\mathbb{Z}^n \to \mathbb{Z}$, and $B^n$ the set of bounded such functions. For $a_1,...,a_{n+1} \in \mathbb{Z}$, the differential of ...
- 1,346
2
votes
1
answer
1k
views
Most efficient way of getting a brief overview of the current active research areas in Algebraic Topology
I'd be applying for a Ph.D. at various grad schools in the U.S. in the coming months and while I know I'd like to pursue research in the field of Algebraic Topology, I am not knowledgeable enough yet ...
4
votes
0
answers
202
views
Fibrations and Euler characteristics with bad fundamental group
Consider a fibration $F\to E\to B$ where $H^i(F;\mathbb{Q})$ and $H^i(B;\mathbb{Q})$ are finite-dimensional, and they vanish for $i\gg 0$, and $B$ is connected. However, we do not assume that $B$ is ...
- 56.9k
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
0
votes
3
answers
400
views
Are two different definitions for Čech cohomology equivalent?
In Spanier's book Algebraic Topology (Chapter 6 section 7) he defines Čech cohomology in terms of the nerves of open coverings.
I wish to know if this is equivalent, for a topological space A closed ...
41
votes
5
answers
11k
views
Mathematically mature way to think about Mayer–Vietoris
This question is short but to the point: what is the "right" abstract framework where Mayer-Vietoris is just a trivial consequence?
- 6,348
2
votes
0
answers
106
views
Lifting homology classes to the unit tangent bundle, a la Johnson
Let $M$ be a oriented smooth closed 2-manifold, and let $\gamma$ be an oriented smooth simple closed curve in $M$.
In Spin structures and quadratic forms on surfaces, Johnson definines a standard way ...
- 371
6
votes
1
answer
289
views
Relationship between the holonomy pseudogroup and holonomy homomorphism (foliation)
This question is surely a duplication of https://math.stackexchange.com/questions/4343635/relationship-between-the-holonomy-pseudogroup-and-holonomy-homomorphism-foliati , however, I got no replies. ...
- 295
3
votes
1
answer
160
views
Geometric vs cohomological dimension with families - on a proof of Lueck and Meintrup
Let $G$ be a discrete group, and let $\mathcal{F}$ be a family of subgroups of $G$ (closed under conjugation and taking subgroups). Then we may define the geometric and cohomological dimensions of $G$ ...
- 35.9k
1
vote
0
answers
96
views
Chain complexes indexed over measurable subsets of $\mathbb{R}$: Towards a measurable notion of Euler Characteristic
I have for a while tried to generalize the notion of a chain complex in a way to obtain a "continuous" or at least "measurable" notion of Euler Characteristic.
I have come up with ...
- 1,805
7
votes
2
answers
335
views
If $G$ is a topological group that contains a torsion element, then the classifying space $BG$ is infinite-dimensional?
We know that if $G$ is a topological group that contains a torsion element and $G$ satisfies additional conditions such as $G$ discrete or $G$ finite-dimensional, then the classifying space $BG$ is ...
- 10.5k
7
votes
1
answer
321
views
Is it possible to separate two linked (geometric) circles in $\Bbb R^3$ by a set homeomorphic to the 2-sphere (with arbitrarily “bad” homeomorphism)?
$A$ and $B$ are two linked (geometric) circles in $\Bbb{R}^3$. (Let, for definiteness, both have radius = 1, the first lies in the $z=0$ plane and its center is the origin of coordinates $(0,0,0)$, ...
10
votes
0
answers
213
views
Are topological theta series (taking values in tmf(N)) of lattices good for anything?
I'm going to start with Mike Hopkins' great survey article in the ICM on topological modular forms (https://arxiv.org/abs/math/0212397). In it, he outlines a construction, for even unimodular lattices,...
- 345
20
votes
2
answers
1k
views
The first unstable homotopy group of $Sp(n)$
Thanks to the fibrations
\begin{align*}
SO(n) \to SO(n+1) &\to S^n\\
SU(n) \to SU(n+1) &\to S^{2n+1}\\
Sp(n) \to Sp(n+1) &\to S^{4n+3}
\end{align*}
we know that
\begin{align*}
\pi_i(SO(...
- 19.3k
8
votes
1
answer
608
views
What is the right notion of a functor from an internal topological category to topological spaces?
Let $\mathcal C=(\mathcal O, \mathcal M)$ be a category internal to topological spaces. Thus $\mathcal O$ and $\mathcal M$ are topological spaces: the space of objects and the space of morphisms ...
- 10.9k
11
votes
1
answer
455
views
Asking whether there is a compact Lie group containing affine symplectic group
The affine symplectic group is interesting and important in physics. However, the Lie group is noncompact. In order to have some good properties (Basically, we need some good behavior of Haar measure) ...
- 149
2
votes
1
answer
163
views
Homological restrictions on certain $4$-manifolds
I am not very familiar with the non-compact $4$-manifold theory. So I apologize if the following question is very silly.
Let $X$ be a non-compact, orientable $4$ manifold that is homotopic to an ...
- 1,177
4
votes
1
answer
522
views
Singular cohomology of fields
Let's define the singular cohomologies of function fields of complex varieties, as the direct limit of the singular cohomologies of Zariski opens of the variety with analytic topology. So for a ...
- 5,901
14
votes
2
answers
1k
views
Very particular kind of 4-manifolds. Classification
Let $M$ be a smooth orientable compact connected (with boundary) manifold of dimension $4$. In addition $M$ is assumed to be aspherical and acyclic.
Question: is there a "classification" of ...
- 223
0
votes
0
answers
63
views
A construction that sort of merges two semigroups to build a new one
Suppose $H$ and $K$ are semigroups and assume without loss of generality that (the underlying sets of) $H$ and $K$ are disjoint. We can then extend the operations of both $H$ and $K$ to a binary ...
- 10.5k
25
votes
1
answer
2k
views
On a curious map from the complex projective plane into $S^5$
I have heavily edited the post (including the title), based on a comment by @GregoryArone that my map $f$ is not injective. In an earlier version of this post, I had thought to have constructed a ...
- 5,215
8
votes
0
answers
360
views
Worst-case complexity of calculating homotopy groups of spheres
Is the best known worst-case running time for calculating the homotopy groups of spheres $\pi_n(S^k)$ bounded by a finite tower of exponentials? How high is a tower? Does $O(2^{2^{2^{2^{n+k}}}})$ ...
- 827
0
votes
0
answers
58
views
Role of basins of attraction in the Morse decomposition
Let $M$ be a differentiable manifold and $F \in \mathcal{X}(M)$. We define a DS by
$$\dot{x}=F(x(t))$$
An ordered collection $\mathcal{M}=\left\{M_{1}, \ldots, M_{l}\right\}$ of compact subsets of ...
- 247
3
votes
1
answer
229
views
A pexiderization of the sine addition law on semigroups
Can we solve the follwing functional equation
$$f(xy)=g(x)h(y)+g(y)h(x)$$
on semigroups for unknown complex valued functions $f,g,h$ ?
21
votes
1
answer
837
views
What is the homotopy type of the poset of nontrivial decompositions of $\mathbf{R}^n$?
Consider the following partial order. The objects are unordered tuples $\{V_1,\ldots,V_m\}$, where each $V_i \subseteq \mathbf{R}^n$ is a nontrivial linear subspace and $V_1 \oplus \cdots \oplus V_m =...
- 1,025
3
votes
1
answer
167
views
Can the Picard-graded homotopy of a nonzero object be nilpotent?
Let $\mathcal C$ be a symmetric monoidal stable category such that the thick subcategory generated by the unit is all of $\mathcal C$ -- in particular, every object is dualizable (I'm particularly ...
- 64k
51
votes
5
answers
5k
views
What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism?
I know of two very general frameworks for describing generalizations of what a "cohomology theory" should be: Grothendieck's "six functors", and the theory of spectra.
In the former, one assigns to ...
- 40.2k
14
votes
1
answer
705
views
Is there a concrete description of the nontorsion elements in the homotopy groups of spheres?
By Serre's theorem, we know the only nontorsion parts of the homotopy groups of spheres occur as $\pi_n(S^n)$ and $\pi_{4n-1}(S^{2n})$. The first of these are trivial to describe, but the second have ...
- 1,949
37
votes
1
answer
1k
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Does there exist a continuous 2-to-1 function from the sphere to itself?
I am interested in the following question:
Does there exist a continuous function $f:S^2\to S^2$ such that, for any $p\in S^2$, $|f^{-1}(\{p\})|=2$?
I suspect the answer is no, but I don't know ...
- 513
10
votes
1
answer
418
views
Are all degree-1 cohomology operations Bocksteins?
I'm interested in cohomology operations (in ordinary cohomology)
$$H^i(-, G)\rightarrow H^{i+1}(-, H)\;,$$
that is, elements of
$$H^{i+1}(K(G, i), H)\;.$$
I know that $K(G, 1)=BG$, so for $i=1$, those ...
- 3,001
5
votes
1
answer
251
views
Monoid associated to $>2$-player Hackenbush
There is some literature on multiplayer combinatorial game theory, but as far as I can tell none of it follows the line of attack below. I'd love a pointer to a similar approach taken in the ...
- 20.5k
6
votes
1
answer
455
views
Homology and cohomology of free loop spaces
String topology, as well as Hochschild (co)homology, give a rich perspective on the homology and cohomology of a free loop space $LM$ of a manifold $M$.
Let $k$ be a field and let $M$ be $n$-...
- 512
19
votes
1
answer
989
views
Can the product of a 3-dimensional lens space with a circle be diffeomorphic to another such product when the lens spaces aren't diffeomorphic?
This is a question that I need to answer in order to resolve an issue for my dissertation and I am looking for a reference. Here is the precise statement of the question.
Suppose we have two three-...
- 293
44
votes
7
answers
22k
views
How do you show that $S^{\infty}$ is contractible?
Here I mean the version with all but finitely many components zero.
- 10.5k
4
votes
1
answer
165
views
$E^G_\ast(E)$ tensored with the rationals
Lemma 17.19 of Switzer's "Algebraic topology - Homology and Homotopy" states that $E_\ast(F)\otimes\mathbb{Q}$ is isomorphic to $\pi_\ast(E)\otimes\pi_\ast(F)\otimes\mathbb{Q}$. I wanted to ...
user493302
3
votes
0
answers
152
views
Equivariant classifying space and manifold models
The classifying space $BS^1$ for $S^1$-bundles can be taken to be the colimit of $\mathbb{CP}^n$ which are smooth manifolds and the inclusions $\mathbb{CP}^n \hookrightarrow \mathbb{CP}^{n+1}$ are ...
- 803
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
7
votes
2
answers
1k
views
How duality follows from a six functor formalism
For the sake of this question, we'll model a six functor formalism in the following way. Let $\mathsf{C}$ be a category of spaces (be it the category of schemes, or topological spaces) and consider a ...
- 771
7
votes
1
answer
410
views
Does a Gysin map depend on the choice of Thom class?
Let $f:X\rightarrow Y$ be a proper embedding between complex manifolds, then the normal bundle $N$ is complex which is in paticular $\textsf{spin}^c$. Hence we have a Thom class $\lambda_N$ and a Thom ...
user484289