# EHP spectral sequence

EHP spectral sequence

In mathematics, the EHP spectral sequence is a spectral sequence used for inductively calculating the homotopy groups of sphereslocalized at some prime "p". It is described in more detail in harvtxt|Ravenel|2003|loc=chapter 1.5 and harvtxt|Mahowald|2001. It is related to the EHP long exact sequence of harvtxt|Whitehead|1953; the name "EHP" comes from the fact that Whitehead named 3 of the maps of his sequence "E", "H", and "P".

For "p" = 2 the spectral sequence uses some exact sequences associated to the fibration harv|James|1957:$S^n\left(2\right) ightarrow Omega S^\left\{n+1\right\}\left(2\right) ightarrow Omega S^\left\{2n+1\right\}\left(2\right)$(where &Omega; stands for a loop space and the (2) is localization of a topological space at the prime 2).This gives a spectral sequence with E1"k","n" term &pi;"k"+"n"("S"2"n"−1(2)) and converging to &pi;*"S"(2) (stable homotopy of spheres localized at 2). The spectral sequence has the advantage that the input is previously calculated homotopy groups. It was used by harvtxt|Oda|1977 to calculate the first 31 stable homotopy groups of spheres.

For arbitrary primes one uses some fibrations found by harvtxt|Toda|1962: :$widehat S^\left\{2n\right\}\left(p\right) ightarrow Omega S^\left\{2n+1\right\}\left(p\right) ightarrow Omega S^\left\{2pn+1\right\}\left(p\right)$:$S^\left\{2n-1\right\}\left(p\right) ightarrow Omega widehat S^\left\{2n\right\}\left(p\right) ightarrow Omega S^\left\{2pn-1\right\}\left(p\right)$where $widehat S^\left\{2n\right\}$ is the 2"np" − 1 skeleton of the loop space $Omega S^\left\{2n+1\right\}$. (For "p" = 2, $widehat S^\left\{2n\right\}$ is the same as $S^\left\{2n\right\}$, so Toda's fibrations at "p" = 2 are same same as the James fibrations.)

References

*citation|first=I.M.|last= James|title=On the suspension sequence|journal=Ann. of Math. |volume=65 |year=1957|pages= 74–107

*springer|id=E/e110020|title=EHP spectral sequence|first=M.|last=Mahowald
*citation|first= N.|last= Oda|title=On the 2-components of the unstable homotopy groups of spheres, I–II|journal= Proc. Japan Acad. Ser. A Math. Sci. |volume= 53 |year=1977|pages=202–218
*
*
*citation|id=MR|0055683
title=On the Freudenthal theorems.
journal=Ann. of Math. (2) |volume=57|year=1953|pages= 209-228

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