--- date: 2022-04-05 23:42 modification date: Tuesday 5th April 2022 23:42:25 title: stable homotopy groups of spheres aliases: - stable homotopy groups of spheres created: 2022-04-05T23:42 updated: 2024-01-01T23:13 --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags: - #homotopy/stable-homotopy #homotopy/of-spheres - Refs: - Overall summaries - Doug Ravenel's book on the homotopy groups of spheres - Kochman's book. - Mark Mahowald for results using the Adams spectral sequence - Doug Ravenel for the Adams-Novikov spectral sequence. - [https://www.youtube.com/watch?v=jie-ww7RBWY](https://www.youtube.com/watch?v=jie-ww7RBWY) - [http://mathematics.stanford.edu/wp-content/uploads/2013/08/Victor-Honors-Thesis-2013.pdf](http://mathematics.stanford.edu/wp-content/uploads/2013/08/Victor-Honors-Thesis-2013.pdf) - [http://archive.ymsc.tsinghua.edu.cn/pacm_download/293/8755-61sphere_final.pdf](http://archive.ymsc.tsinghua.edu.cn/pacm_download/293/8755-61sphere_final.pdf) - [https://web.stanford.edu/~amwright/HomotopyGroupsOfSoheres.pdf](https://web.stanford.edu/~amwright/HomotopyGroupsOfSoheres.pdf) - Links: - #todo/create-links --- # stable homotopy groups of spheres [Suspension](https://people.math.wisc.edu/~maxim/spseq.pdf#page=15): ![](attachments/Pasted%20image%2020220422205505.png) ## References Links: [J-homomorphism](J-homomorphism.md) # Motivating Problems - The [Kervaire invariant](Kervaire%20invariant%201.md)problem - Classifying manifolds up to [framed cobordism](framed%20cobordism) - Distinct [smooth structures](smooth%20structures.md) on spheres - Let $M$ be a closed $n$-manifold. Suppose $M$ is homotopy equivalent to $S^n$. Is $M$ homeomorphic to $S^n$? - For which $n$ does there exist a unique smooth structure on $S^n$? - Let $M \in \Mfd^{\smooth}_n$ be homeomorphic to $S^n$. Is $M$ diffeomorphic to $S^n$? # Motivation: Stable Homotopy Groups of Spheres ## Cobordism If one understood even the stable homotopy groups of spheres very well, one would therefore have a near complete understanding of the group of [smooth structures](smooth%20structures.md) on the $n\dash$-sphere for $n\neq 4$. ## Kervaire Invariant One of the most recent spectacular advances in algebraic topology was the solution of (most of) the [framed](framed.md) manifolds and stable homotopy groups of spheres. Things used to solve this classical problem: [orthogonal spectra](orthogonal%20spectra.md) ## Classification **Question**: Let $M$ be a closed $n$-manifold. Suppose $M$ is homotopy equivalent to $S^n$. Is $M$ homeomorphic to $S^n$? **Answer**: Yes in all dimensions. **Question**: For which $n$ does there exist a unique smooth structure on $S^n$? **Answer**: - For $n= 3$, yes, by Moise every closed [smooth structure](smooth%20structure.md). In particular, the 3-sphere has a unique smooth structure. - **For n= 4, this question is wildly open.** - For $n\geq 5$, Milnor constructed an [smooth structure](smooth%20structure.md) smooth structure on $S^7$. Kervaire and Milnor [27] showed that the answer is "no" in general for $n\geq 5$. **Question**: For which $n$ does there exist a unique smooth structure on $S^n$? - Kervaire and Milnor reduced Question 1.5 to a computation of the stable homotopy groups of spheres. In fact, Kervaire and Milnor constructed the $\Theta_n \in \Grp$ of [homotopy spheres](homotopy%20spheres). This classifies [smooth structures](smooth%20structures.md) on $S^n$ for $n\geq 5$. ## The Unknown The homotopy group $\pi_{n+k}(S^k)$ is a finite group except 1. For $n=0$ in which case $\pi_k(S^k)=\ZZ$; 2. For $k=2m$ and $n=2m−1$ in which case $\pi_{4m−1}(S^{2m})≃Z\oplus F_m$ for $F_m$ a finite group. # Results - The [K3 surfaces](K3%20surfaces.md) plays an important role in the third stable homotopy group of spheres. - It can be viewed as the source of the [24](24) in the group $\pi_3 \SS = \ZZ/{24}$. ## Computations - Table of $\pi_{n+k}S^n$: [http://www.math.nus.edu.sg/~matwujie/homotopy_groups_sphere.html](http://www.math.nus.edu.sg/~matwujie/homotopy_groups_sphere.html) - It is well-known that the third stable homotopy group of spheres is cyclic of order [24](24). - It is also well-known that the quaternionic [bundle](bundle), suspends to a generator of $/pi_8(S^5)=\pi^{st}_3$. - It is well-known that the complex [Hopf map](Hopf%20map) $\eta: S^3 \to S^2$ suspends to a generator of $\pi_4(S^3] = \pi_1 \SS = \ZZ_2$. - For this, there is a reasonably elementary argument, see e.g. > Bredon, Topology and Geometry, page 465. - Complete or nearly complete calculations for $\pi \SS$ localized at a [Morava K-theory](Morava%20K-theory) have been made by Toda, Goerss-Henn-Mahowald-Rezk, and Mark Behrens. - Computer calculations of $\Ext$: Robert Bruner or Christian Nassau. - The unstable and stable homotopy groups $\pi_i(S^3)$ for $i\leq 64$ are apparently computed in: > Curtis, Edward B.,Mahowald, Mark, The unstable Adams spectral sequence for $S^3$, Algebraic topology (Evanston, IL, 1988), 125-162, Contemp. Math., 96, Amer. Math. Soc., Providence, RI, 1989. ## Cobordism - [J-homomorphism](J-homomorphism.md) The stable homotopy groups (in degree $n$) of spheres are the same as [stably framed](framed.md) manifolds (of dimension $n$). - The [Pontryagin-Thom construction](Pontryagin-Thom%20construction) shows that the stable homotopy groups of spheres in degree $n$ are the same as the groups of stably framed manifolds of dimension $n$ up to cobordism. - In dimension 3 the generator is given by $\nu = (S^3,Lie)$, the 3-sphere with its [quaternions](quaternions). # Unsorted ## How to read the stem diagrams - Each one is for a fixed $p$, for example at $p=2$ each dot depicts a factor of 2 and vertical lines denote additive extensions. For example, for vertical dots: $$ \cdot \to \cdot \to \cdot \leadsto \ZZ/{2}^{\oplus 3} \quad \\ \cdot \to \cdot \leadsto \ZZ/2^{\oplus 2} .$$ - The [Adams Spectral Sequence](Adams%20Spectral%20Sequence) instead. - There are several open problems related to differentials and invariants the arise from this [spectral sequence](spectral%20sequence.md) - E.g. what are the permanent cycles? - The [Adams-Novikov spectral sequence](Adams-Novikov%20spectral%20sequence) ends up being cleaner, fewer differentials! > Hatcher: Connections between homotopy groups of spheres and low-dimensional geometry and topology have traditionally been somewhat limited, with the [Hopf bundle](Hopf bundle) being the thing that comes most immediately to mind. A fairly recent connection is Soren Galatius' theorem that the homology groups of $Aut(F_n)$ (the automorphism group of a free group) are isomorphic in a stable range of dimensions to $H_* \Loop^\infty \Sigma^\infty S^0$, the space whose homotopy groups are the stable homotopy groups of spheres. ## Relation to Classification of Manifolds > Hatcher: Kervaire-Milnor theory ("Groups of Homotopy Spheres") and Pontryagin-Thom show that our knowledge/ignorance about the stable homotopy groups of spheres is reflected in knowledge/ignorance about classification of manifolds. In each dimension $n$, one has a group $\theta_n$ of smooth $n$-manifolds that are homotopy $n$-spheres, up to [framed](framed.md) $n+1$-manifolds. Assume $n>4$, so [h-cobordism](h-cobordism.md) classes are diffeomorphism classes. Every [stable framing](stable%20framing) (missing something). Hence (by [Pontrayagin-Thom](Pontrayagin-Thom.md) $S$ is a regular fiber of a map $S_{n+k}\to S_k$ for $k\gg 0$ whose class in $\pi_{n+k}(S_k)$ is the obstruction to $S$ (with chosen stable framing]] being a framed boundary. Changing the stable framing amounts to adding something in the [J-homomorphism](J-homomorphism.md) $J: \pi_n(SO(k)) \to \pi_{n+k}(S_k](J-? /pi_{n+k}(S_k)$. So we get an injective homomorphism $\theta_n/ \bP_{n+1}\to \coker J$ which is onto e.g. for $n$ odd. We don't know $\coker(J)$ in high dimensions, so we don't know the order of $\theta_n/ \bP_{n+1}$. But [Serre's finiteness theorem](Serre's%20finiteness%20theorem) for the stable stems tells us that $Θ_n/bP_{n+1}$ is finite! The subgroup $\bP_{n+1}$ is analyzed via [surgery](surgery.md) and the [h-cobordism theorem](h-cobordism%20theorem). There's a nice summary in Lück's Basic introduction to surgery theory. #resources/summaries We have $\bP_{odd} = 0$. There's a formula for $\bP_{4p}$ involving [Bernoulli numbers](Bernoulli%20numbers) number numerators; this comes from a known (thanks to Adams) part of the stable stems, namely ??? Finally, $\bP_{4p+2}$ is at most $Z_2$. Here $S$ bounds a [Kervaire invariant 1](Kervaire%20invariant%201.md). Browder showed that the Kervaire invariant can be one only when $4p+2=2l−2$ for some $l$, and [Hill-Hopkins-Ravenel](Hill-Hopkins-Ravenel.md) have shown that $l\leq 7$. **Conclusion**: $\bP_{4p+2}$ is $Z_2$ except in dimensions $6, 14, 30, 62,$ and possibly $126$, where it's zero. ![](attachments/Pasted%20image%2020220209191032.png)