nLab EHP spectral sequence

Idea

The EHP spectral sequence (we follow Mahowald 85) is the spectral sequence for computation of homotopy groups of spheres induced from the filtration of the underlying homotopy type $\Omega^\infty \Sigma^\infty S^0 = \Omega^\infty \mathbb{S}$ of the sphere spectrum by suspensions (German: Einhängung):

$$\Omega^n S^n \stackrel{E}{\longrightarrow} \Omega^{n+1} S^{n+1} \,.$$

More concretely, (James 57) constructed maps

$$\Omega S^n \stackrel{H}{\longrightarrow} \Omega S^{2n-1}$$

(for Hopf as in Hopf invariant) and showed that 2-locally these fit with $E$ into homotopy fiber sequences

$$\Omega^{n+2} S^{2n+1} \stackrel{P}{\longrightarrow} \Omega^n S^n \stackrel{E}{\longrightarrow} \Omega^{n+1} S^{n+1} \stackrel{H}{\longrightarrow} \Omega^{n+1}S^{2n+1} \,.$$

(Here $P$ is by definition the homotopy fiber of $E$, the notation refers to Whitehead product.)

This “EHP-long homotopy fiber sequence” gives rise to the corresponding long exact sequence of homotopy groups and so to an exact couple of the form

$$\array{ \underset{s,t}{\oplus} \pi_{s+t}(\Omega^{s+1}S^{s+1}) && \stackrel{i}{\longrightarrow} && \pi_{s+t}(\Omega^{s+1}S^{s+1}) \\ & \nwarrow && \swarrow \\ && \underset{s,t}{\oplus} \pi_{t+s}(\Omega^{s+1}S^{2s+1}) } \,.$$

The corresponding spectral sequence is the EHP spectral sequence proper. It converges, 2-locally, to the stable homotopy groups of spheres, with $E_1$-page given by

$$E^{k,n}_1 = \pi_{k+n}(S^{2n-1}) \;\Rightarrow\; \pi_\bullet^{\mathbb{S}} \,.$$

For more general prime numbers than just 2, (Toda 62) found analogous fibrations, which hence give EHP spectral sequences for general $p$.

The EHP spectral sequence is often used used in the context of the Adams-Novikov spectral sequence for p-localization at some prime $p$.

Constructions of $H$

For James’ fiber sequence, the essential property required of $H$ is to realize the isomorphism

$$(the\:map\:H)^{cohomology\:pullback} : H^{2n}(\Omega\mathbb{S}^{2n+1})\overset{\sim}{\to} H^{2n}(\Omega \mathbb{S}^{n+1}) .$$

The remaining corollaries then follow using the fact cohomology pullback is a ring homomorphism, and the $mod 2$ Leray-Serre spectral sequence.

Via the James model

Using the James model of $\Omega\Sigma X$ as a quotient space of $colim_n X^n$, a candidate $H$ is constructed by recursion:

$$H([x_0 , \dots, x_{n+1}]) = H([x_0,\dots,x_n]) \# ([x_0 \wedge x_{n+1} , \dots , x_n\wedge x_{n+1}])$$

where $\#$ denotes concatenation and $\wedge$ smash product. One checks that the ordering of product terms $x_i\wedge x_j$ w.r.t. $x_k\wedge x_l$ depends only on the relative orders of $i,j,k,l$, so that $H$ is well-defined on the quotient space $\Omega\Sigma X \to \Omega \Sigma(X\wedge X)$.

In particular, the restriction to $J_2 X$ factors through $X\wedge X \to \Omega\Sigma X\wedge X$ as the cofiber of the inclusion $X \to J_2 X$. In the case $X\simeq \mathbb{S}^n$, the desired cohomology isomorphism is immediate.

Via pushout/pullback comparisons

Starting with the three-legged cospan $X \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} *$, construct the cube of all pushouts

$$\array{ X & & & \to & & & * \\ & \searrow & & & & \swarrow \\ & & * & \to & \Sigma X \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ & & \Sigma X & \to & \Sigma X \vee \Sigma X \\ & \nearrow & & & & \nwarrow \\ * & & & \to & & & \Sigma X }$$

Construct pullbacks in some pair of parallel squares, and compare them by naturality

$$\array{ \Omega\Sigma X & & & \to & & & * \\ & \searrow & & & & \swarrow \\ & & \Omega((\Omega\Sigma X)\star(\Omega\Sigma X)) & \to & \Sigma X \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ & & \Sigma X & \to & \Sigma X \vee \Sigma X \\ & \nearrow & & & & \nwarrow \\ * & & & \to & & & \Sigma X }$$

where $\star$ is reduced join. On the other hand, the natural transformations $\Sigma\Omega \to 1$ give natural maps, e.g.

$$\Omega( ev \circ \tau \circ ev ) : \Omega((\Omega\Sigma X) \star (\Omega\Sigma X)) \to \Omega( X \star X) .$$

The composite

$$\Omega \Sigma X \to \Omega((\Omega\Sigma X)\star(\Omega\Sigma X)) \to \Omega(X\star X)$$

is a candidate $H$.

Related concepts

• the Freudenthal suspension theorem may be obtained from the EHP spectral sequence;

• the Goodwillie spectral sequence of the identity functor at the point also computes homotopy groups of spheres, the interplay of the two is discussed in (Behrens 10)

References

Original articles include

• Ioan Mackenzie James, Reduced product spaces, Ann. of Math. (2) 62 (1955), 170-197.

• Ioan Mackenzie James, On the Suspension Sequence, Annals of Mathematics Second Series, Vol. 65, No. 1 (Jan., 1957), pp. 74-107 (jstor)

• Hiroshi Toda, Composition methods in homotopy groups of spheres, Princeton University Press (1962)

• Mark Mahowald, Lin’s theorem and the EHP sequence. Conference on algebraic topology in honor of Peter Hilton, Contemp. Math. 37 (1985), 115–119. Amer. Math. Soc., Providence, RI.

• Marcel Bökstedt, Anne Marie Svane, A generalization of the stable EHP spectral sequence (arXiv:1208.3938)

Relation to the Goodwillie spectral sequence is discussed in

• Mark Behrens, The Goodwillie tower and the EHP sequence (arXiv:1009.1125)

An algebraic version of the EHP spectral sequence for the Lambda-algebra and used for computation of the second page of the classical Adams spectral sequence (the Curtis algorithm), is discussed in

• Stanley Kochman, around prop. 5.2.6 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

Review:

• Michael Hopkins (notes by Akhil Mathew), Lectures 6,7 in: Spectra and stable homotopy theory, 2012 (pdf, pdf)

• Mark Mahowald, Doug Ravenel, section 7 of Towards a Global Understanding of the Homotopy Groups of Spheres (pdf)

• Doug Ravenel, chapter 1, section 5 of Complex cobordism and stable homotopy groups of spheres

• eom, EHP spectral sequence

See also:

• Wikipedia, EHP spectral sequence