# EHP spectral sequence

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/EHP_spectral_sequence
> Markdown URL: https://mediated.wiki/source/EHP_spectral_sequence.md
> Source: https://en.wikipedia.org/wiki/EHP_spectral_sequence
> Source revision: 1139769679
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In [mathematics](/source/mathematics), the '''EHP spectral sequence''' is a [spectral sequence](/source/spectral_sequence) used for inductively calculating the [homotopy groups of spheres](/source/homotopy_groups_of_spheres) localized at some [prime](/source/prime_number) ''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 [George W. Whitehead](/source/George_W._Whitehead) named 3 of the maps of his sequence "E" (the first letter of the German word "Einhängung" meaning "suspension"), "H" (for [Heinz Hopf](/source/Heinz_Hopf), as this map is the second Hopf–James invariant), and "P" (related to [Whitehead product](/source/Whitehead_product)s).

For <math>p = 2</math> the spectral sequence uses some exact sequences associated to the fibration {{harv|James|1957}}
:<math>S^n(2)\rightarrow \Omega S^{n+1}(2)\rightarrow \Omega S^{2n+1}(2)</math>,
where <math>\Omega</math> stands for a loop space and the (2) is [localization of a topological space](/source/localization_of_a_topological_space) at the prime 2.  This gives a spectral sequence with <math>E_1^{k,n}</math> term equal to 
: <math>\pi_{k+n}(S^{2 n - 1}(2))</math> 
and converging to <math>\pi_*^S(2)</math> (stable homotopy groups 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}}: 
:<math>\widehat S^{2n}(p)\rightarrow \Omega S^{2n+1}(p)\rightarrow \Omega S^{2pn+1}(p)</math>
:<math> S^{2n-1}(p)\rightarrow \Omega \widehat S^{2n}(p)\rightarrow \Omega S^{2pn-1}(p)</math>
where <math>\widehat S^{2n}</math> is the <math>(2np-1)</math>-skeleton of the loop space <math>\Omega S^{2n+1}</math>. (For <math>p = 2</math>, the space <math>\widehat S^{2n}</math> is the same as <math> S^{2n}</math>, so Toda's fibrations at <math>p = 2</math> are the same as the James fibrations.)

==References==
{{sfn whitelist|CITEREFMahowald2001}}
*{{citation|first=Ioan M.|last= James|authorlink=Ioan James| title=On the suspension sequence|journal=[Annals of Mathematics](/source/Annals_of_Mathematics) |volume=65  |year=1957|pages= 74–107|doi=10.2307/1969666|issue=1|jstor=1969666|mr=0083124}}
*{{springer|id=E/e110020|title=EHP spectral sequence|first=Mark|last=Mahowald|authorlink=Mark Mahowald}}
*{{citation|first= Nobuyuki|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|doi= 10.3792/pjaa.53.202|issue= 6|doi-access= free}}
* {{citation|first = Douglas C. |last = Ravenel|authorlink=Douglas Ravenel| title = Complex cobordism and stable homotopy groups of spheres|edition= 2nd| url = http://www.math.rochester.edu/people/faculty/doug/mu.html|publisher = AMS Chelsea| year = 2003|isbn = 0-8218-2967-X}}
* {{citation|last = Toda| first = Hiroshi|authorlink=Hiroshi Toda| title = Composition methods in homotopy groups of spheres| publisher = [Princeton University Press](/source/Princeton_University_Press) | year = 1962| isbn = 0-691-09586-8}}
*{{citation
|last=Whitehead|first= George W.|authorlink=George W. Whitehead
|title=On the Freudenthal theorems
|journal=[Annals of Mathematics](/source/Annals_of_Mathematics) | series = Second Series |volume=57|year=1953|pages= 209–228
|doi=10.2307/1969855|issue=2|jstor=1969855|mr=0055683
}}

Category:Spectral sequences

{{topology-stub}}

---
Adapted from the Wikipedia article [EHP spectral sequence](https://en.wikipedia.org/wiki/EHP_spectral_sequence) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/EHP_spectral_sequence?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.