EHP spectral sequence

In mathematics, the EHP spectral sequence is a spectral sequence used for inductively calculating the homotopy groups of spheres localized at some prime p. In the area of Mathematics known as Homological algebra, especially in Algebraic topology and Group cohomology, a spectral sequence is a In the mathematical field of Algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 It is described in more detail in Ravenel (2003, chapter 1. 5) and Mahowald (2001). It is related to the EHP long exact sequence of 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 (James 1957)

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

(where Ω stands for a loop space and the (2) is localization of a topological space at the prime 2). In mathematics well behaved topological spaces can be localized at primes in a similar way to the Localization of a ring at a prime This gives a spectral sequence with E₁^k,n term π_k+n(S²ⁿ⁻¹(2)) and converging to π_*^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 Oda (1977) to calculate the first 31 stable homotopy groups of spheres.

For arbitrary primes one uses some fibrations found by Toda (1962):

$\widehat S^{2n}(p)\rightarrow \Omega S^{2n+1}(p)\rightarrow \Omega S^{2pn+1}(p)$

$S^{2n-1}(p)\rightarrow \Omega \widehat S^{2n}(p)\rightarrow \Omega S^{2pn-1}(p)$

where $\widehat S^{2n}$ is the 2np − 1 skeleton of the loop space $Ω S 2 n + 1$ . (For p = 2, $\widehat S^{2n}$ is the same as $S 2 n$ , so Toda's fibrations at p = 2 are same same as the James fibrations. )

References

James, I. M. (1957), “On the suspension sequence”, Ann. of Math. 65: 74–107, <http://links.jstor.org/sici?sici=0003-486X%28195701%292%3A65%3A1%3C74%3AOTSS%3E2.0.CO%3B2-T>
Mahowald, M. (2001), “EHP spectral sequence”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
Oda, N. The Encyclopaedia of Mathematics is a large reference work in Mathematics. (1977), “On the 2-components of the unstable homotopy groups of spheres, I–II”, Proc. Japan Acad. Ser. A Math. Sci. 53: 202–218
Ravenel, Douglas C. (2003), Complex cobordism and stable homotopy groups of spheres (2nd edition ed. ), AMS Chelsea, ISBN 0-8218-2967-X, <http://www.math.rochester.edu/people/faculty/doug/mu.html>
Toda, Hirosi (1962), Composition methods in homotopy groups of spheres, Princeton University Press, ISBN 0-691-09586-8
Whitehead, George W. (1953), “On the Freudenthal theorems.”, Ann. of Math. (2) 57: 209-228, MR 0055683, <http://links.jstor.org/sici?sici=0003-486X%28195303%292%3A57%3A2%3C209%3AOTFT%3E2.0.CO%3B2-F>

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