Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
8,255
questions
3
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pushforward in non-multiplicative generalized cohomology theory
Let's fix a $(B,f)$ structure with Thom spectra $MB$. I'd like to know the condition for a not-necessarily-multiplicative generalized cohomology theory $E^*$ such that for a fibration $X\to Y$ with $(...
- 6,063
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1
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140
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formula of b_2 for minimal resolution of surface cyclic quotient singularity
We know the cyclic quotient singularity of type $\frac{1}{dn^2}(1,dna-1)$, where $n,a$ are coprime, has $Q-$Gorenstein smoothing. The second Betti number $b_2 = d-1$ on its smoothing. I'm wondering if ...
CommunityBot
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3
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Example of an unstable map between finite complexes which is the identity on homotopy but not homotopic to the identity?
Stably, phantom maps (nonzero maps which are zero on homotopy) exist, but it's not known if they exist between finite complexes (Freyd's Generating Hypothesis). Unstably, it's easy to find maps which ...
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1
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Crafting Suspension Spectra
There is a theorem by Hopkins and Smith which states that for every $n > 0$ there is an ideal $I_n = (v_0^{k_0}, \dots, v_n^{k_n})$ such that there exist a spectrum $X_n$ with the following ...
- 5,550
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2
answers
797
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Allowing $G$-CW complexes to have more general cells
Let $G$ a finite group. I've seen three options discussed for making $G$-cell complexes: in increasing generality, one might allow $X_n$ to be constructed from $X_{n-1}$ by attaching cells of the form ...
- 29.8k
23
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0
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571
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What is the symmetric monoidal functor from Clifford algebras to invertible K-module spectra?
There ought to be a symmetric monoidal functor from the symmetric monoidal $2$-groupoid whose objects are Morita-invertible real superalgebras (precisely the Clifford algebras), morphisms are ...
- 7,549
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2
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Classifying space as the geometric realization of the nerve of $G$ viewed as a small category
Let $C$ be a small category: we define its nerve $(N(C)_k)_k$ as the following simplicial set: $N(C)_0=Ob(C)$ (the set of objects), $N(C)_1=Mor(C)$ (the set of all morphisms) and $N(C)_k$ to be a set ...
- 33.9k
3
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1
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515
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Lefschetz Hyperplane theorem via Kodaira Vanishing
I'm trying to read the proof of the Lefschetz hyperplane theorem from Griffiths-Harris. They prove the theorem (on pages 156-157) using the Kodaira vanishing theorem. I have a basic question regarding ...
- 44k
6
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0
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Existence of $G$-map between finite $G$-simplicial complex
Let $X, Y$ be finite free $G$- simplicial complex. What kind of properties are necessary for existence a $G$-map,i.e, a continuous map which preserves $G$-action, from $X$ to $Y$? Does existence of ...
- 233
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(Torsion in) homology of free nilpotent groups
It is known that for free $k$-step nilpotent group on $r$ generators $N(r, k)$ its integral homology is torsion-free in degrees $\leq 3$ (obvious for 1 and 2, Igusa&Orr computations for 3). ...
- 4,436
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1
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Essential maps of spectra which are null when localized at any prime
There are maps of spaces which are not null-homotopic, but when localized at any prime become null. I don't know explicit constructions of any, but an example is given in Section 6 of Chapter 25 of ...
- 2,157
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6
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Concrete example of $\infty$-categories
I've seen many different notions of $\infty$-categories: actually I've seen the operadic-globular ones of Batanin and Leinster, and the opetopic, and eventually I'll see the simplicial ones too. ...
CommunityBot
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1
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Cup product on flat fiber bundles vs cup product on the corresponding Serre spectral sequence
Let $F \rightarrow E \rightarrow B$ be a flat fiber bundle, $E, F, B$ closed manifolds. Consider $H^*(E, \mathbb{Q})$ and the corresponding Serre spectral sequence with isomorphism $$(*) \ \ \ H^n(E;\...
- 51.5k
1
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1
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Can a planar tangle have an infinite number of input disks?
Can a planar tangle have an infinite number of input disks?
Some publications talk about cases with a finite number of input disks, while others do not say if it is finite or infinite.
So, is it ...
- 27.8k
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0
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Can the 2-complex associated to a finitely presented group be triangulated?
Let G be a finitely presented group. K is the 2-complex associated to G which is constructed as taught in Algebraic Topology. That is , 1-cells corresponding to generators and 2-cells corresponding to ...
- 325
5
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1
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Continuously varying the singularities of a vector field
An arc field on a topological space $X$ is a continuous function $\Psi: X \rightarrow X^{[0,1]}$ such that for every $x \in X$, the path $\Psi(x): [0,1] \rightarrow X$ (1) starts at $x$, (2) is ...
- 35k
6
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1
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Connection between Stalling's end theorem and Seifert-van Kampen Theorem
Stalling`s end Theorem (a group has more than one end iff it splits over a finite subgroup) and the Seifert-van Kampen Theorem (the fundamental group of a 'decomposable' space is a free amalgamated ...
- 1,228
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2
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745
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Interpretation of the cohomology of compact lie groups and their classifying spaces in DAG?
I'll be using homological grading throughout this question.
Let $G$ be a compact connected lie group. The following isomorphisms are classical and can be proven using several methods:
$$H^{\bullet}(...
- 60.2k
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1
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Could an inverse of (weak) Morse inequality exists in some special case?
Could an inverse Morse inequality hold in some sense? More precisely I wish the following result to be true:
Problem
$M$ is a smooth simply connected compact manifold, $dim(M)=n$, $f$ is a morse ...
- 19.2k
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Plus construction considerations.
In order to realise the K-groups of a ring as the homotopy groups of some space associated to that ring, Quillen proposed the following (roughly-sketched) construction:
Recall that $K_1(R) = GL(R)/E(...
- 54.4k
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0
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Is this an $E_\infty$-algebra?
I have a particular kind of algebraic structure that's come up in my work. It's basically a chain complex equipped with a multiplication which is commutative and associative up to homotopy in a ...
- 39.3k
5
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0
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Is there any known lower bound on the dimension of sphere of origin?
For a given element $f\in\pi_n^s$ by Freudenthal theorem, it is known that $f$ pulls back to an element of $\pi_{2n+1}S^{n+1}$. At a prime $p$, whether or not if $f$ pulls back further depends on the ...
- 3,071
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Computing an explicit homotopy inverse for $B(*,H,*) \hookrightarrow B(*,G,G/H)$
Suppose that $G$ is a finite group, $M$ is a right $G$-set and $N$ is a left $G$-set. Then we have a simplicial set $B(M,G,N)$ whose $n$-simplicies are $M \times G^n \times N$.
Now suppose that $H \...
- 3,031
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0
answers
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Atiyah's paper "Non-existent complex 6-sphere"
I'm trying to understand the main idea of Atiyah's proof (https://arxiv.org/abs/1610.09366). Although there were discussions on MO year ago I couldn't find answers to my questions.
Consider the ...
- 171
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votes
1
answer
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Sullivan conjecture for compact Lie groups
Let $G$ be a topological group, and $M$ a connected compact smooth manifold. I'm studying
$$ \pi_0 (map (BG,M)). $$
For $G$ a finite group, we know that this is just a point by the Sullivan ...
- 10.6k
5
votes
1
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Simplicial complex construction from given Betti numbers?
Is it possible given a set of Betti numbers to construct a (possibly set of) simplicial complex with the given Betti-described topology? I understand there can be an infinity of simplicial complexes ...
- 15.5k
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What is the current understanding regarding complex structures on the 6-sphere?
In October 2016, Atiyah famously posted a preprint to the arXiv, "The Non-Existent Complex 6-Sphere" containing a very brief proof $S^6$ admits no complex structure, which I immediately read and ...
- 60.2k
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3
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Relationships between homology maps of cobordant manifolds
Let $W$ be a cobordism between $n$-manifolds $M$ and $N$, and $f\colon W \mapsto X$ be a map to some manifold $X$.
Does anybody know of any nice examples of general relationships between the images ...
- 9,802
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9
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Covering maps in real life that can be demonstrated to students
Edit: I've really enjoyed everyone's examples (especially the pictures!), but I was mostly looking for a general theorem. For instance, a similar statement to mine is, Can the mapping cylinder of ...
- 4,618
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240
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Plane Curve invariants via Contour Integrals
We learn in complex analysis class how to find the winding number of the contour $\Gamma$ around the origin.
\[ n = \frac{1}{2\pi i} \oint \frac{dz}{z} = \frac{1}{2\pi i} \oint d(\log z)
= \...
- 1,228
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1
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A groupoid which is homotopy equivalent to $BG$
Let $G$ be a finitely generated group, then its action groupoid $BG$ is a simplicial set. In fact $BG$ is the nerve of a groupoid where the set of objects is given by a point $*$ and the set of maps ...
- 29.8k
4
votes
1
answer
635
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Complexity of computing the Vietoris-Rips complex
For me it looks like computing the Vietoris-Rips complex from a data cloud is very similar to the clique problem in graph theory, which it NP-hard.
How do the two differ and what is the computational ...
- 13.5k
3
votes
1
answer
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Singularity of torus fixed points from combinatorial data
May I ask what are the relations between the geometry and combinatorics near a torus fixed point? Any references?
In particular, let $S$ be a scheme that is torus invariant with finitely many zero and ...
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5
votes
1
answer
194
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Homology of a limit of spectra + Cofiber
I have a countable sequence of finite suspension spectra $X_i$, whose $BP$-homology is a $BP_*(BP)$-comodule. Let's assume $BP_*(X_i) = \Sigma^{d_i} BP_* / (v_0^{k_0}, \dots v_i^{k_i}),$ for some $d_n$...
- 51.5k
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3
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Given a link $L\subset S^3$ how to construct a link $L'$ whose complement have hyperbolic structure?
Thurston claimed that almost all closed 3-manifolds are hyperbolic. To support this, he said that every closed 3-manifold is obtained by Dehn surgery along some link whose complement is hyperbolic. ...
- 66.8k
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Has the structure of the 2-dimensional pin$^{\pm}$ bordism categories been written down?
If $H\to\mathrm O$ is a tangential structure (e.g. orientation, spin), let $\mathsf{Bord}_2^H$ denote the category whose objects are 1-dimensional manifolds with $H$-structure and whose morphisms $M_1\...
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8
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Why do we need model categories?
I cannot give a good answer to this question. And
2) Why this definition of model category is the right way to give a philosophy of homotopy theory? Why didn't we use any other definition?
3) Has ...
- 660
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0
answers
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Alexander modules and weight filtrations
$\def\Ext{\mathrm{Ext}}$I have the following situation: I have a complex algebraic variety $X$. I want to compute the weight filtration on various abelian covers $Y \to X$. As a concrete example, take ...
- 151k
2
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1
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Compute cohomology of flat fiber bundles - does this always work?
Edit: This has been answered in this thread Are there compact flat fiber bundles with "truly" infinite structure group?.
Setting
Let $p: E \rightarrow B$ be a flat fiber bundle with fiber ...
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2
answers
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Are these two notions of "dualizable" spectra equivalent?
A spectrum $X$ is dualizable if the natural map $$Map(X,\mathbb S) \wedge X \rightarrow Map(X,X)$$ is an equivalence of spectra. This is equivalent to having evaluation and coevaluation maps in the ...
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Equivalence of Flat Fiber Bundles vs Equivalence of Group Actions on the Fiber
Let's consider all flat fiber bundles with base space $B$ and fiber $F$, where $B$ and $F$ are compact and at least CW-complexes. (perhaps even topological/smooth manifolds if that helps)
All those ...
- 394
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0
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Two questions regarding flat fibre bundles and the corresponding group action on the fibre
Let $F$, $B$ be smooth, closed manifolds and $\phi:\pi_1(B) \rightarrow Aut(F)$ a smooth group action of the fundamental group of $B$ on $F$.
Consider the flat fibre bundle $E_\phi := \widetilde{B} \...
- 394
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votes
1
answer
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The integral cohomology of real projective space
I've run across a way of combining the integral cohomology of the real projective space $RP^\infty$ with its cohomology with twisted coefficients, that seems very simple and natural, but which I don't ...
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2
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239
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Topological Complexity $TC$ of two robots moving on number $8$
I have been working on my research as a student at Wilbur Wright College on Topological Complexity. We solved the problem of two robots moving on a circle and letter $T$ using Farber's theorem but ...
- 35k
5
votes
1
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Map between homology of spectra
Let $X$ be a suspension spectra whose $BP$-homology is infinitely generated
($BP_*(X) = \Sigma^d BP_*/I$, where $I$ has the form $I=(v_0^{i_0}, \dots , v_n^{i_n})$ such that the homology is a $BP_*(BP)...
- 51.5k
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votes
2
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References for computation of 2-primary stable 64-stem ${_2\pi_{64}^s}$?
I want to learn about the $2$-primary component of the stable homotopy groups of spheres in dimension $64$. Since the triviality of $61$-st stem has been proved just recently, I thought that either I ...
CommunityBot
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What is known about this cohomology operation?
Let $X$ be some space, $C^*(X,R)$ its cochain complex. Then there is a multiplication
$$
\mu : C^*(X,R) \otimes C^*(X,R) \rightarrow C^*(X,R)
$$
inducing the cup product, and a homotopy
$$H : C^*(...
- 51.5k
4
votes
1
answer
281
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What happens if we generalize the fundamental group to make knotted loops distinct?
The definition of an element of the fundamental group of a space $X$
based at point $p$
is
$$f:[0,1]\rightarrow X,\quad f(0)=p=f(1),$$
defined up to homotopy.
This homotopy allows self-intersection,
...
- 7,256
8
votes
1
answer
215
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Fanos with $\chi_{top} = 0$
Question 1:Do there exist smooth projective Fano varieties over $\mathbb{C}$ with topological Euler characteristic $0$?
Question 2:If so what is the lowest dimension in which such examples occur?
- 116
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Acyclic aspherical spaces with acyclic fundamental groups
A space $X$ (by which I mean a CW complex) is acyclic if its reduced singular homology $\tilde H_\ast(X;\Bbb Z)$ is trivial in all degrees.
A discrete group $\pi$ is said to be acyclic if its ...
- 20.6k