EHP spectral sequence



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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 Ω Σ S 0=Ω 𝕊\Omega^\infty \Sigma^\infty S^0 = \Omega^\infty \mathbb{S} of the sphere spectrum by suspensions (German: Einhängung):

Ω nS nEΩ n+1S n+1. \Omega^n S^n \stackrel{E}{\longrightarrow} \Omega^{n+1} S^{n+1} \,.

More concretely, (James 57) constructed maps

ΩS nHΩS 2n1 \Omega S^n \stackrel{H}{\longrightarrow} \Omega S^{2n-1}

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

Ω n+2S 2n+1PΩ nS nEΩ n+1S n+1HΩ n+1S 2n+1. \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 PP is by definition the homotopy fiber of EE, 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

s,tπ s+t(Ω s+1S s+1) i π s+t(Ω s+1S s+1) s,tπ t+s(Ω s+1S 2s+1). \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 1E_1-page given by

E 1 k,n=π k+n(S 2n1)π 𝕊. 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 pp.

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

Constructions of HH

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

(themapH) cohomologypullback:H 2n(Ω𝕊 2n+1)H 2n(Ω𝕊 n+1). (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 mod2mod 2 Leray-Serre spectral sequence.

Via the James model

Using the James model of ΩΣX\Omega\Sigma X as a quotient space of colim nX ncolim_n X^n , a candidate HH is constructed by recursion:

H([x 0,,x n+1])=H([x 0,,x n])#([x 0x n+1,,x nx n+1]) 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 ix j x_i\wedge x_j w.r.t. x kx lx_k\wedge x_l depends only on the relative orders of i,j,k,li,j,k,l, so that HH is well-defined on the quotient space ΩΣXΩΣ(XX)\Omega\Sigma X \to \Omega \Sigma(X\wedge X).

In particular, the restriction to J 2XJ_2 X factors through XXΩΣXX X\wedge X \to \Omega\Sigma X\wedge X as the cofiber of the inclusion XJ 2XX \to J_2 X. In the case X𝕊 nX\simeq \mathbb{S}^n, the desired cohomology isomorphism is immediate.

Via pushout/pullback comparisons

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

X * * ΣX ΣX ΣXΣX * ΣX \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

ΩΣX * Ω((ΩΣX)(ΩΣX)) ΣX ΣX ΣXΣX * ΣX \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 ΣΩ1 \Sigma\Omega \to 1 give natural maps, e.g.

Ω(evτev):Ω((ΩΣX)(ΩΣX))Ω(XX).\Omega( ev \circ \tau \circ ev ) : \Omega((\Omega\Sigma X) \star (\Omega\Sigma X)) \to \Omega( X \star X) .

The composite

ΩΣXΩ((ΩΣX)(ΩΣX))Ω(XX) \Omega \Sigma X \to \Omega((\Omega\Sigma X)\star(\Omega\Sigma X)) \to \Omega(X\star X)

is a candidate HH.


Original articles include

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


See also:

Last revised on November 29, 2020 at 15:23:01. See the history of this page for a list of all contributions to it.