ADAMS SPECTRAL SEQUENCE

In mathematics, the 'Adams spectral sequence' is a spectral sequence introduced by Frank Adams, to provide a computational tool in stable homotopy theory. It was a reformulation using homological algebra, and an extension, of a technique called 'killing homotopy groups' applied by the French school of Henri Cartan and Jean-Pierre Serre.
The Adams spectral sequence concerns cohomology groups, of well-behaved topological spaces, such as CW complexes. It involves the structure of
:''H''
^★ (''X''),
for such a space ''X'', the graded abelian group of singular cohomology, as a module over the Steenrod algebra. For this formulation, therefore, it is necessary to take ''H''
^★ as cohomology with coefficients in 'Z'/''p'''Z', where ''p'' is a fixed prime number. Then the spectral sequence is built up from the Ext groups
:''Ext''_''A''^''r''(''H''
^★ (''Y''), ''H''
^★ (''X'')).
These acquire a second grading from the grading on ''H''
^★ (''Y''). The point of the construction is that the sequence converges to:
:[''X'',''Y'']
which is the set of homotopy classes of mappings from ''X'' to ''Y''. In fact the abutment is expressed as ''stable'' classes, from ''X'' to suspensions of ''Y''; and with respect to the fixed prime ''p'', this sequence will only pick up the ''p''-power torsion elements. If we know that ''X'' is the suspension of a space, then [''X'',''Y''] inherits a group structure, if ''X'' is the double suspension of a space then [''X'',''Y''] is in fact an abelian group.
The sequence itself is not an algorithmic device, but lends itself to problem solving in particular cases. It does explain a general way to 'bridge' from the point of view of cohomology, which is generally accessible, to that of homotopy theory which is harder to handle.

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External link

External link

★ Book chapter (PDF)