stable homotopy groups of spheres

Suspension: attachments/Pasted%20image%2020220422205505.png

References

Links: J-homomorphism

Motivating Problems

The Kervaire invariantproblem
Classifying manifolds up to framed cobordism
Distinct smooth structures 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 $M \in {\mathsf{Mfd}}^{{\mathsf{sm}}}_n$ 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 on the $n{\hbox{-}}$-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 manifolds and stable homotopy groups of spheres.

Things used to solve this classical problem: orthogonal spectra

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. 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 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 $\Theta_n \in {\mathsf{Grp}}$ of homotopy spheres. This classifies smooth structures on $S^n$ for $n\geq 5$.

The Unknown

The homotopy group $\pi_{n+k}(S^k)$ is a finite group except

For $n=0$ in which case $\pi_k(S^k)={\mathbf{Z}}$;
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 Unsorted/K3 surfaces plays an important role in the third stable homotopy group of spheres.
- It can be viewed as the source of the 24 in the group $\pi_3 {\mathbb{S}}= {\mathbf{Z}}/{24}$.

Computations

Table of $\pi_{n+k}S^n$: 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.
It is also well-known that the quaternionic bundle, suspends to a generator of $/pi_8(S^5)=\pi^{st}_3$.
It is well-known that the complex Hopf map $\eta: S^3 \to S^2$ suspends to a generator of $\pi_4(S^3] = \pi_1 {\mathbb{S}}= {\mathbf{Z}}_2$.
- For this, there is a reasonably elementary argument, see e.g. > Bredon, Topology and Geometry, page 465.
Complete or nearly complete calculations for $\pi {\mathbb{S}}$ localized at a Morava K-theory have been made by Toda, Goerss-Henn-Mahowald-Rezk, and Mark Behrens.
Computer calculations of $\operatorname{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 The stable homotopy groups (in degree $n$) of spheres are the same as stably framed manifolds (of dimension $n$).
The Pontryagin-Thom construction 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.

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: \begin{align*} \cdot \to \cdot \to \cdot \leadsto {\mathbf{Z}}/{2}^{\oplus 3} \quad \\ \cdot \to \cdot \leadsto {\mathbf{Z}}/2^{\oplus 2} .\end{align*}
The Adams Spectral Sequence instead.
There are several open problems related to differentials and invariants the arise from this spectral sequence
- E.g. what are the permanent cycles?
- The Adams-Novikov spectral sequence 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 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 $H_* {\Omega}^\infty \Sigma^\infty S^0$, 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 $n+1$-manifolds. Assume $n>4$, so h-cobordism classes are diffeomorphism classes.

Every stable framing (missing something). Hence (by Pontrayagin-Thom $S$ 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 $J: \pi_n(SO(k)) \to \pi_{n+k}(S_k](J-? /pi_{n+k}(S_k)$. So we get an injective homomorphism $\theta_n/ \operatorname{bP}_{n+1}\to \operatorname{coker}J$ which is onto e.g. for $n$ odd.

We don’t know $\operatorname{coker}(J)$ in high dimensions, so we don’t know the order of $\theta_n/ \operatorname{bP}_{n+1}$. But Serre’s finiteness theorem for the stable stems tells us that $Θ_n/bP_{n+1}$ is finite!

The subgroup $\operatorname{bP}_{n+1}$ is analyzed via surgery and the h-cobordism theorem. There’s a nice summary in Lück’s Basic introduction to surgery theory. #resources/summaries

We have $\operatorname{bP}_{odd} = 0$. There’s a formula for $\operatorname{bP}_{4p}$ involving Bernoulli numbers number numerators; this comes from a known (thanks to Adams) part of the stable stems, namely ???

Finally, $\operatorname{bP}_{4p+2}$ is at most $Z_2$. Here $S$ bounds a Kervaire invariant 1.

Browder showed that the Kervaire invariant can be one only when $4p+2=2l−2$ for some $l$, and Hill-Hopkins-Ravenel have shown that $l\leq 7$.

Conclusion: $\operatorname{bP}_{4p+2}$ is $Z_2$ except in dimensions $6, 14, 30, 62,$ and possibly $126$, where it’s zero.

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