Questions tagged [homotopy-groups-of-sphere]

Filter by
Sorted by
Tagged with
1
vote
1answer
143 views

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})...
21
votes
2answers
774 views

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 ...
1
vote
0answers
262 views

What is known about homotopy groups of spheres?

I'm looking for a list/table 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-$0$ etc. ...
1
vote
1answer
143 views

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 ...
29
votes
6answers
3k views

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 ...
27
votes
1answer
612 views

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 ...
11
votes
1answer
521 views

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 ...
3
votes
1answer
395 views

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 ...
18
votes
0answers
563 views

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 ...
4
votes
1answer
251 views

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 $...
4
votes
1answer
615 views

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 ...
4
votes
0answers
75 views

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 ...
9
votes
1answer
567 views

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 ...
9
votes
0answers
228 views

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 ...
61
votes
1answer
2k views

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 ...
8
votes
1answer
561 views

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 ...
8
votes
1answer
1k views

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 ...
10
votes
0answers
420 views

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 ...
7
votes
1answer
295 views

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 ...
14
votes
0answers
647 views

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-...
14
votes
1answer
440 views

“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 ...