Namespaces
Variants
Actions

Difference between revisions of "EHP spectral sequence"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
An inductive procedure to calculate the homotopy groups of spheres (cf. [[Spheres, homotopy groups of the|Spheres, homotopy groups of the]]). It has the attractive feature that the input for the calculation is always a result of an earlier calculation once one inputs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e1100201.png" />. It has been used to calculate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e1100202.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e1100203.png" />. The last of these calculations are contained in [[#References|[a3]]] and complete references can be found there. It is constructed by splicing together fibrations discovered by I.M. James [[#References|[a2]]] at the prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e1100204.png" /> and at odd prime numbers by H. Toda [[#References|[a4]]].
+
<!--
 +
e1100201.png
 +
$#A+1 = 52 n = 2
 +
$#C+1 = 52 : ~/encyclopedia/old_files/data/E110/E.1100020 EHP spectral sequence
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
In this paragraph the case of the prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e1100205.png" /> is discussed, and all spaces are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e1100206.png" />-local and all groups should be considered as completed at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e1100207.png" />. James showed that there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e1100208.png" />-local [[Fibration|fibration]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e1100209.png" />. This gives rise to long exact sequences (cf. [[Exact sequence|Exact sequence]])
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002010.png" /></td> </tr></table>
+
An inductive procedure to calculate the homotopy groups of spheres (cf. [[Spheres, homotopy groups of the|Spheres, homotopy groups of the]]). It has the attractive feature that the input for the calculation is always a result of an earlier calculation once one inputs  $  \pi _ {1} ( S  ^ {1} ) $.
 +
It has been used to calculate  $  \pi _ {n + i }  ( S  ^ {n} ) $
 +
for  $  i \leq  31 $.
 +
The last of these calculations are contained in [[#References|[a3]]] and complete references can be found there. It is constructed by splicing together fibrations discovered by I.M. James [[#References|[a2]]] at the prime number  $  2 $
 +
and at odd prime numbers by H. Toda [[#References|[a4]]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002011.png" /></td> </tr></table>
+
In this paragraph the case of the prime number  $  2 $
 +
is discussed, and all spaces are  $  2 $-
 +
local and all groups should be considered as completed at  $  2 $.  
 +
James showed that there is a  $  2 $-
 +
local [[Fibration|fibration]]  $  S  ^ {n} \rightarrow \Omega S ^ {n + 1 } \rightarrow \Omega S ^ {2n + 1 } $.  
 +
This gives rise to long exact sequences (cf. [[Exact sequence|Exact sequence]])
  
The first mapping is usually called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002012.png" />, the second <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002013.png" /> and the connecting [[Homomorphism|homomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002014.png" />. The EHP spectral sequence is obtained by defining the filtration of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002015.png" /> by
+
$$
 +
\dots \rightarrow \pi _ {s + t }  ( S  ^ {t} ) \rightarrow \pi _ {s + t + 1 }  ( S ^ {t + 1 } ) \rightarrow
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002016.png" /></td> </tr></table>
+
$$
 +
\rightarrow
 +
\pi _ {s + t + 1 }  ( S ^ {2t + 1 } ) \rightarrow \dots .
 +
$$
  
This gives rise to a [[Spectral sequence|spectral sequence]] with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002017.png" /> and converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002018.png" />. There are several important features of this spectral sequence. First, note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002019.png" /> is itself a result of another calculation using this spectral sequence for smaller values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002020.png" />. The value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002021.png" /> is called the stem and the filtration parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002022.png" /> gives the sphere of origin of a class. The name a homotopy class receives from this spectral sequence is called the Hopf invariant of the class. Letting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002023.png" /> go to infinity gives a spectral sequence which calculates the stable homotopy groups (cf. [[Stable homotopy group|Stable homotopy group]]). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002024.png" /> term does not have a special name, but there is the following result. For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002026.png" />, and odd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002028.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002029.png" /> [[Vector space|vector space]] [[#References|[a1]]]. The differential is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002030.png" />.
+
The first mapping is usually called $  E $,
 +
the second  $  H $
 +
and the connecting [[Homomorphism|homomorphism]]  $  P $.  
 +
The EHP spectral sequence is obtained by defining the filtration of $  \pi _ {s + n }  ( S  ^ {n} ) $
 +
by
  
For odd prime numbers, the situation is slightly more complicated. In this paragraph all spaces are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002031.png" />-local for a fixed odd prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002032.png" /> and all groups are completed at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002033.png" />. First there is the result of James, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002034.png" />. This reduces the calculation for even-dimensional spheres to that of odd-dimensional spheres. In order to get Toda's version of the EHP sequence, one introduces a modified even sphere. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002035.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002036.png" />-skeleton of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002037.png" />. Then Toda showed that the following are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002038.png" />-local fibrations:
+
$$
 +
\pi _ {s + 1 }  ( S  ^ {1} ) \rightarrow \dots \rightarrow \pi _ {s + t }  ( S  ^ {t} ) \rightarrow \dots \rightarrow \pi _ {s + n }  S  ^ {n} .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002039.png" /></td> </tr></table>
+
This gives rise to a [[Spectral sequence|spectral sequence]] with  $  E _ {s,t }  ^ {1} = \pi _ {s + t }  ( S ^ {2t - 1 } ) $
 +
and converging to  $  \pi _ {s + n }  ( S  ^ {n} ) $.
 +
There are several important features of this spectral sequence. First, note that  $  E _ {s,t }  ^ {1} $
 +
is itself a result of another calculation using this spectral sequence for smaller values of  $  s $.
 +
The value of  $  s $
 +
is called the stem and the filtration parameter  $  t $
 +
gives the sphere of origin of a class. The name a homotopy class receives from this spectral sequence is called the Hopf invariant of the class. Letting  $  n $
 +
go to infinity gives a spectral sequence which calculates the stable homotopy groups (cf. [[Stable homotopy group|Stable homotopy group]]). The  $  E  ^ {2} $
 +
term does not have a special name, but there is the following result. For all  $  ( s,t ) $,
 +
$  s > 0 $,
 +
and odd  $  t $,
 +
$  E _ {s,t }  ^ {2} $
 +
is an  $  F _ {2} $[[
 +
Vector space|vector space]] [[#References|[a1]]]. The differential is  $  {d _ {r} } : {E  ^ {r} _ {s,t }  } \rightarrow {E  ^ {r} _ {s - 1,t - r }  } $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002040.png" /></td> </tr></table>
+
For odd prime numbers, the situation is slightly more complicated. In this paragraph all spaces are  $  p $-
 +
local for a fixed odd prime number  $  p $
 +
and all groups are completed at  $  p $.
 +
First there is the result of James,  $  \pi _ {s + 2n }  ( S ^ {2n } ) \simeq \pi _ {s + 2n - 1 }  S ^ {2n - 1 } \oplus \pi _ {s + 2n }  ( S ^ {4n - 1 } ) $.
 +
This reduces the calculation for even-dimensional spheres to that of odd-dimensional spheres. In order to get Toda's version of the EHP sequence, one introduces a modified even sphere. Let  $  {S \widetilde{ {}}  } ^ {2n } = ( \Omega S ^ {2n + 1 } ) ^ {\langle  {( p - 1 ) ( 2n ) } \rangle } $,
 +
the  $  ( 2n ) ( p - 1 ) $-
 +
skeleton of  $  \Omega S ^ {2n + 1 } $.  
 +
Then Toda showed that the following are  $  p $-
 +
local fibrations:
  
As in the prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002041.png" /> case, one can fit the long exact sequences in homotopy together to get a spectral sequence. It is associated to the filtration
+
$$
 +
S ^ {2n - 1 } \rightarrow \Omega {S \widetilde{ {}}  } ^ {2n } \rightarrow \Omega S ^ {2n ( p ) - 1 } .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002042.png" /></td> </tr></table>
+
$$
 +
{S \widetilde{ {}}  } ^ {2n } \rightarrow \Omega S ^ {2n + 1 } \rightarrow \Omega S ^ {2n ( p ) + 1 } .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002043.png" /></td> </tr></table>
+
As in the prime  $  2 $
 +
case, one can fit the long exact sequences in homotopy together to get a spectral sequence. It is associated to the filtration
  
As before, one can use Toda's sequences to identify the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002044.png" />-term: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002046.png" />. This spectral sequence converges to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002047.png" />-local homotopy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002048.png" />. As before, the input is the result of an earlier calculation. There is also the classical result that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002049.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002050.png" /> vector space. The parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002051.png" /> refers to the stem and the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002052.png" /> refers to the sphere of origin. Here, this refers to the odd sphere or the modified even sphere of origin. The name a class gets in this spectral sequence is also called the Hopf invariant.
+
$$
 +
\pi _ {s + 1 }  S  ^ {1} \rightarrow \dots \rightarrow \pi _ {s + 2t }  {S \widetilde{ {}}  } ^ {2t } \rightarrow
 +
$$
 +
 
 +
$$
 +
\rightarrow
 +
\pi _ {s + 2t + 1 }  S ^ {2t + 1 } \rightarrow \dots \rightarrow \pi _ {s + 2n + 1 }  S ^ {2n + 1 } .
 +
$$
 +
 
 +
As before, one can use Toda's sequences to identify the $  E  ^ {1} $-
 +
term: $  E _ {s,2t }  ^ {1} = \pi _ {s+2t }  ( S ^ {2tp - 1 } ) $
 +
and $  E _ {s,2t + 1 }  ^ {1} = \pi _ {s + 2t + 1 }  ( S ^ {2tp + 1 } ) $.  
 +
This spectral sequence converges to the $  p $-
 +
local homotopy of $  S ^ {2n + 1 } $.  
 +
As before, the input is the result of an earlier calculation. There is also the classical result that $  E _ {s,t }  ^ {2} $
 +
is an $  F _ {p} $
 +
vector space. The parameter $  s $
 +
refers to the stem and the parameter $  t $
 +
refers to the sphere of origin. Here, this refers to the odd sphere or the modified even sphere of origin. The name a class gets in this spectral sequence is also called the Hopf invariant.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Barratt,  F. R. Cohen,  B. Gray,  M. Mahowald,  W. Richter,  "Two results on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002053.png" />-local EHP spectral sequence"  ''Proc. Amer. Math. Soc.'' , '''123'''  (1995)  pp. 1257–1261</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.M. James,  "On the suspension sequence"  ''Ann. of Math.'' , '''65'''  (1957)  pp. 74–107</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Oda,  "On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002054.png" />-components of the unstable homotopy groups of spheres, I–II"  ''Proc. Japan Acad. Ser. A Math. Sci.'' , '''53'''  (1977)  pp. 202–218</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Toda,  "Composition methods in homotopy groups of spheres" , ''Ann. Math. Studies'' , '''49''' , Princeton Univ. Press  (1962)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Barratt,  F. R. Cohen,  B. Gray,  M. Mahowald,  W. Richter,  "Two results on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002053.png" />-local EHP spectral sequence"  ''Proc. Amer. Math. Soc.'' , '''123'''  (1995)  pp. 1257–1261</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.M. James,  "On the suspension sequence"  ''Ann. of Math.'' , '''65'''  (1957)  pp. 74–107</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Oda,  "On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110020/e11002054.png" />-components of the unstable homotopy groups of spheres, I–II"  ''Proc. Japan Acad. Ser. A Math. Sci.'' , '''53'''  (1977)  pp. 202–218</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Toda,  "Composition methods in homotopy groups of spheres" , ''Ann. Math. Studies'' , '''49''' , Princeton Univ. Press  (1962)</TD></TR></table>

Latest revision as of 19:36, 5 June 2020


An inductive procedure to calculate the homotopy groups of spheres (cf. Spheres, homotopy groups of the). It has the attractive feature that the input for the calculation is always a result of an earlier calculation once one inputs $ \pi _ {1} ( S ^ {1} ) $. It has been used to calculate $ \pi _ {n + i } ( S ^ {n} ) $ for $ i \leq 31 $. The last of these calculations are contained in [a3] and complete references can be found there. It is constructed by splicing together fibrations discovered by I.M. James [a2] at the prime number $ 2 $ and at odd prime numbers by H. Toda [a4].

In this paragraph the case of the prime number $ 2 $ is discussed, and all spaces are $ 2 $- local and all groups should be considered as completed at $ 2 $. James showed that there is a $ 2 $- local fibration $ S ^ {n} \rightarrow \Omega S ^ {n + 1 } \rightarrow \Omega S ^ {2n + 1 } $. This gives rise to long exact sequences (cf. Exact sequence)

$$ \dots \rightarrow \pi _ {s + t } ( S ^ {t} ) \rightarrow \pi _ {s + t + 1 } ( S ^ {t + 1 } ) \rightarrow $$

$$ \rightarrow \pi _ {s + t + 1 } ( S ^ {2t + 1 } ) \rightarrow \dots . $$

The first mapping is usually called $ E $, the second $ H $ and the connecting homomorphism $ P $. The EHP spectral sequence is obtained by defining the filtration of $ \pi _ {s + n } ( S ^ {n} ) $ by

$$ \pi _ {s + 1 } ( S ^ {1} ) \rightarrow \dots \rightarrow \pi _ {s + t } ( S ^ {t} ) \rightarrow \dots \rightarrow \pi _ {s + n } S ^ {n} . $$

This gives rise to a spectral sequence with $ E _ {s,t } ^ {1} = \pi _ {s + t } ( S ^ {2t - 1 } ) $ and converging to $ \pi _ {s + n } ( S ^ {n} ) $. There are several important features of this spectral sequence. First, note that $ E _ {s,t } ^ {1} $ is itself a result of another calculation using this spectral sequence for smaller values of $ s $. The value of $ s $ is called the stem and the filtration parameter $ t $ gives the sphere of origin of a class. The name a homotopy class receives from this spectral sequence is called the Hopf invariant of the class. Letting $ n $ go to infinity gives a spectral sequence which calculates the stable homotopy groups (cf. Stable homotopy group). The $ E ^ {2} $ term does not have a special name, but there is the following result. For all $ ( s,t ) $, $ s > 0 $, and odd $ t $, $ E _ {s,t } ^ {2} $ is an $ F _ {2} $[[ Vector space|vector space]] [a1]. The differential is $ {d _ {r} } : {E ^ {r} _ {s,t } } \rightarrow {E ^ {r} _ {s - 1,t - r } } $.

For odd prime numbers, the situation is slightly more complicated. In this paragraph all spaces are $ p $- local for a fixed odd prime number $ p $ and all groups are completed at $ p $. First there is the result of James, $ \pi _ {s + 2n } ( S ^ {2n } ) \simeq \pi _ {s + 2n - 1 } S ^ {2n - 1 } \oplus \pi _ {s + 2n } ( S ^ {4n - 1 } ) $. This reduces the calculation for even-dimensional spheres to that of odd-dimensional spheres. In order to get Toda's version of the EHP sequence, one introduces a modified even sphere. Let $ {S \widetilde{ {}} } ^ {2n } = ( \Omega S ^ {2n + 1 } ) ^ {\langle {( p - 1 ) ( 2n ) } \rangle } $, the $ ( 2n ) ( p - 1 ) $- skeleton of $ \Omega S ^ {2n + 1 } $. Then Toda showed that the following are $ p $- local fibrations:

$$ S ^ {2n - 1 } \rightarrow \Omega {S \widetilde{ {}} } ^ {2n } \rightarrow \Omega S ^ {2n ( p ) - 1 } . $$

$$ {S \widetilde{ {}} } ^ {2n } \rightarrow \Omega S ^ {2n + 1 } \rightarrow \Omega S ^ {2n ( p ) + 1 } . $$

As in the prime $ 2 $ case, one can fit the long exact sequences in homotopy together to get a spectral sequence. It is associated to the filtration

$$ \pi _ {s + 1 } S ^ {1} \rightarrow \dots \rightarrow \pi _ {s + 2t } {S \widetilde{ {}} } ^ {2t } \rightarrow $$

$$ \rightarrow \pi _ {s + 2t + 1 } S ^ {2t + 1 } \rightarrow \dots \rightarrow \pi _ {s + 2n + 1 } S ^ {2n + 1 } . $$

As before, one can use Toda's sequences to identify the $ E ^ {1} $- term: $ E _ {s,2t } ^ {1} = \pi _ {s+2t } ( S ^ {2tp - 1 } ) $ and $ E _ {s,2t + 1 } ^ {1} = \pi _ {s + 2t + 1 } ( S ^ {2tp + 1 } ) $. This spectral sequence converges to the $ p $- local homotopy of $ S ^ {2n + 1 } $. As before, the input is the result of an earlier calculation. There is also the classical result that $ E _ {s,t } ^ {2} $ is an $ F _ {p} $ vector space. The parameter $ s $ refers to the stem and the parameter $ t $ refers to the sphere of origin. Here, this refers to the odd sphere or the modified even sphere of origin. The name a class gets in this spectral sequence is also called the Hopf invariant.

References

[a1] M. Barratt, F. R. Cohen, B. Gray, M. Mahowald, W. Richter, "Two results on the -local EHP spectral sequence" Proc. Amer. Math. Soc. , 123 (1995) pp. 1257–1261
[a2] I.M. James, "On the suspension sequence" Ann. of Math. , 65 (1957) pp. 74–107
[a3] N. Oda, "On the -components of the unstable homotopy groups of spheres, I–II" Proc. Japan Acad. Ser. A Math. Sci. , 53 (1977) pp. 202–218
[a4] H. Toda, "Composition methods in homotopy groups of spheres" , Ann. Math. Studies , 49 , Princeton Univ. Press (1962)
How to Cite This Entry:
EHP spectral sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=EHP_spectral_sequence&oldid=18009
This article was adapted from an original article by M. Mahowald (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article