View source for EHP spectral sequence

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]]].

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]])

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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

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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" />.

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:

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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

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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.

====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>