In
mathematics, the EHP spectral sequence is a
spectral sequence used for inductively calculating the
homotopy groups of spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure o ...
localized at some
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
''p''. It is described in more detail in and . It is related to the EHP long exact sequence of ; the name "EHP" comes from the fact that
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
Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry.
Early life and education
Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Elizabeth ( ...
, as this map is the second Hopf–James invariant), and "P" (related to
Whitehead products).
For
the spectral sequence uses some exact sequences associated to the fibration
:
,
where
stands for a loop space and the (2) is
localization of a topological space 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 construction was described by Dennis Sullivan in 1970 lecture notes that were finally published in .
The re ...
at the prime 2. This gives a spectral sequence with
term equal to
:
and converging to
(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 to calculate the first 31 stable homotopy groups of spheres.
For arbitrary primes one uses some fibrations found by :
:
:
where
is the
-skeleton of the loop space
. (For
, the space
is the same as
, so Toda's fibrations at
are the same as the James fibrations.)
References
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Spectral sequences
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