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EHP spectral sequence

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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