Questions tagged [homotopy-groups-of-sphere]
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32 questions
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Induced map of degree $k$ self map of a sphere in the higher homotopy groups
Let $f:S^n\rightarrow S^n$ be a degree $k$ map. Then $f$ induces maps $\pi_l(S^n)\rightarrow \pi_l(S^n)$.
I believe that in the stable range ($l\leq 2n-2$) this map is multiplication by $k$. Unstably ...
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3
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1
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Freudenthal suspension homomorphism
I asked this question in MathStackExchange a couple of months ago, with little feedback, hence I try here.
The Hopf invariant $h(f)$ of a mapping $f:\mathbb S^{2m-1}\to \mathbb S^m$ is a homotopy ...
- 203
14
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1
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What is $\pi_{23}(S^2)$?
The first $22$ homotopy groups of the $2$-sphere were worked out by Toda in 1962, but I cannot find any results extending that to any higher homotopy groups of $S^2$.
Are any more of these groups ...
- 827
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Contractibility of infinite dimensional spheres and some other infinite dimensional manifolds
It is known that spheres in Banach spaces are contractible according to
Yoav Benyamini, Yaki Sternfeld, "Spheres in infinite-dimensional normed spaces are Lipschitz contractible", ...
- 593
8
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0
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360
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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
10
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1
answer
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Do elements of every order occur in homotopy groups of spheres?
It is known from Serre's classical result that every p-torsion occurs in the homotopy groups of every sphere. Is it known: do elements of every order occur in homotopy groups of spheres?
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Linear $S^{2k}$-bundles over $S^{4k}$
By the classification of Dold and Whitney, linear $S^2$-bundles over $S^4$ are classified by their first Pontryagin class $p_1$, which takes the value $4\lambda$ for the bundle corresponding to $\...
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1
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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
16
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1
answer
1k
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Easiest proof of computability of homotopy groups of spheres
Has it gotten easier to prove all homotopy groups of spheres are computable? I don’t care if the computation is inefficient, what’s the easiest proof? Are we still stuck doing Postnikov towers?
- 827
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Bigraded endomorphisms of the motivic sphere over a field
In An introduction to $\mathbb A^1$-homotopy theory ([1]) and On the motivic $\pi_0$ of the sphere spectrum ([2]) Morel describes a computation of $\bigoplus_{n\in \mathbb Z} [S^0, \mathbb G_m^{\wedge ...
- 15.9k
8
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Hopf invariants of elements from spherical fibrations
Let $G_n$ be the space of homotopy-equivalences of $S^{n-1}$. Evaluation produces a map $G_{n} \to S^{n-1}$. For $n = 2m+1$, I would like to understand the induced map on $\pi_{4m-1}$. More precisely, ...
- 11.9k
1
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1
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463
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Can a restriction of a null-homotopic spherical map be null-homopotic?
Let $n,q$ be positive integers. We are interested to the cases where $n>q$.
Let $F:\mathbb B^n\to\mathbb S^{q-1}$ be a continuous (differentiable, if needed) map, such that $F(1,0^{n-1})=(1,0^{q-1})...
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2
answers
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What clues originally hinted at stability phenomena in algebraic topology?
If you didn't know anything about stabilization phenomena in algebraic topology and were trying to discover/prove theorems about the homotopy theory of spaces, what clues would point you toward ...
- 443
8
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0
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869
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What is known about homotopy groups of spheres?
I'm looking for a list/table/survey of what is known (and what is not known) about homotopy groups of spheres, for example: which are known, which are known stably, which are known primally, non-$0$ ...
1
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1
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Give a null-homotopy of $2\eta :S^4\to S^3$ in coordinates
where $\eta$ is the suspension of the hopf fibration.
When I say "in coordinates" I mean that $2\eta$ comes from choosing an explicit representation of $\eta :S^3\to S^2$, suspending it, composing ...
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6
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What is the intuition for higher homotopy groups not vanishing?
The homotopy groups of the spheres $S^n$ (see Wikipedia) vanish for the circle $S^1$ as, naively speaking, there are not higher order holes to be grasped by higher order homotopy groups. This ...
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28
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1
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Modern survey of unstable homotopy groups?
Toda no doubt made some big strides when computing unstable homotopy groups $\pi_{n+k}(S^n)$ for $k < 20$ which his collaborators later improved upon.
The methods he used are documented in his ...
- 289
14
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1
answer
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Whitehead products in homotopy groups of spheres
Here is what I know about Whitehead products in homotopy groups of spheres:
$[\mathrm{id}_{S^{2n}},\mathrm{id}_{S^{2n}}]$ has Hopf invariant (EDIT: $\pm$) two.
No element that survives into the ...
- 934
3
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1
answer
429
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A projective (or free) $\mathbb{Z}\pi_1$-module
Suppose that $Z$ is a finite wedge of spheres containing circles and there exist maps $f:Y\to Z$ and $g:Z\to Y$ so that $g\circ f\simeq 1_Y$. Assume that there exists a map $h:X\to Y$ which induces ...
- 139
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702
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Homotopy groups of spheres and differential forms
The only infinite homotopy groups of spheres are $\pi_n(\mathbb{S}^n)$ and $\pi_{4n-1}(\mathbb{S}^{2n})$. This is a well known result of Serre. In both cases the nontriviality of these groups can be ...
- 28k
4
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1
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A space homotopy dominated by a wedge of spheres
Recall that the space $A$ is homotopy dominated by $X$ if there are maps $f:A\longrightarrow X$ and $g:X\longrightarrow A$ such that $gf\simeq id_A$.
Suppose that $X$ is a wedge of some spheres and $...
- 1,182
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1
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Spherical Harmonics on $S^3$ [closed]
My understanding is that harmonic analysis on the circle ($S^1$) leads to Fourier Series/Integrals whereas harmonic analysis on the sphere ($S^2$) leads to Spherical Harmonics. If we take the next ...
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0
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The order of $im(\nu'_*)\subseteq \pi_*S^3$
The 3-sphere $S^3$ has homotopy 2-exponent 4. That is, any 2-torsion element $\alpha\in\pi_*S^3$ has order at most 4. This bound is sharp, for example the Blakers-Massey element $\nu'\in\pi_6S^3$ has ...
- 5,596
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1
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752
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Algebraic structure on homotopy groups of spheres
It is about a "conjecture" I heard (when I was student). There would exist an algebraic structure on the homotopy groups of spheres such that this algebraic structure would be the free algebraic ...
- 3,901
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Samelson Products in $SO(n)$
Given a topological group $G$ one forms the commutator $c\colon G\times G\rightarrow G$, $(x,y)\mapsto xyx^{-1}y^{-1}$. This map then factors through the smash $G\wedge G$. This map is the most ...
- 5,596
66
votes
1
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Is there an octonionic analog of the K3 surface, with implications for stable homotopy groups of spheres?
The infamous K3 surface has many constructions in many fields ranging from algebraic geometry to algebraic topology. Its many properties are well known. For this question I am really interested in the ...
- 27.5k
8
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1
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cohomology of iterated loop space on spheres
In the book The homology of iterated loop spaces, the homology Hopf algebra
(1)
$$
H_*(\Omega^n \Sigma^n X;\mathbb{Z}_p)
$$
for primes $p\geq 2$ is obtained on p. 226, Thm. 3.2. In particular, the ...
- 2,223
7
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1
answer
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Homotopy groups of an infinite wedge of 2-spheres
I know Hilton's result about a finite wedge of spheres, and I know that certain homotopy groups (such as the third homotopy group) can be directly calculated for an infinite wedge too.
My question is ...
- 459
11
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0
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513
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The spheres operad
I have a rather naive question. Consider the space of all maps
$$ S^{j_1} \times S^{j_2} \times \cdots \times S^{j_k} \to S^n $$
for all possible natural numbers $n, k, j_1, \cdots , j_k$.
This ...
- 44.4k
7
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1
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Detecting homotopy nontriviality of an element in a torsion homotopy group
I have a map, constructed geometrically, $S^4 \to S^3$. I suspect that it is a representative for the generator $\eta_3\in \pi_4(S^3) \simeq \mathbb{Z}_2$, but I am not 100% sure ($\eta_3$ is defined ...
- 35.5k
14
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0
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How to see the quaternionic hopf map generates the stable 3-stem?
I am looking for a direct proof that the quaternionic hopf map generates (after suspension) the 3rd stable homotopy group of spheres. There are some related MO questions, for example:
third-stable-...
- 27.5k
14
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1
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"Small" maps from sphere to sphere
Start with a continuous map $f:S^{n+k} \rightarrow S^n$ (round unit spheres). The graph of $f$ lives in $S^{n+k}\times S^n$ and suppose it has a surface area (as a subspace of co-dimension $n$). Now ...
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