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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):
More concretely, (James 57) constructed maps
(for Hopf as in Hopf invariant) and showed that 2-locally these fit with $E$ into homotopy fiber sequences
(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
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
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 $p$.
For James’ fiber sequence, the essential property required of $H$ is to realize the isomorphism
The remaining corollaries then follow using the fact cohomology pullback is a ring homomorphism, and the $mod 2$ Leray-Serre spectral sequence.
Using the James model of $\Omega\Sigma X$ as a quotient space of $colim_n X^n$, a candidate $H$ is constructed by recursion:
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.
Starting with the three-legged cospan $X \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} *$, construct the cube of all pushouts
Construct pullbacks in some pair of parallel squares, and compare them by naturality
where $\star$ is reduced join. On the other hand, the natural transformations $\Sigma\Omega \to 1$ give natural maps, e.g.
The composite
is a candidate $H$.
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 (Behrems 10)
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
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
Review includes
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
Wikipedia, EHP spectral sequence
Last revised on April 29, 2019 at 13:29:45. See the history of this page for a list of all contributions to it.