Adams spectral sequence

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(Redirected from Adams-Novikov 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/pZ, 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.

[edit] Adams-Novikov spectral sequence

The Adams-Novikov spectral sequence is a generalization of the Adams spectral sequence where ordinary cohomology is replaced by a generalized cohomology theory, often complex bordism or Brown-Peterson cohomology

[edit] See also

Adams filtration

[edit] References

Adams, J. Frank (1964), Stable homotopy theory., Lecture notes in mathematics, 3, Berlin-Göttingen-Heidelberg-New York: Springer-Verlag,, MR 0185597
Botvinnik, Boris (1992), Manifolds with Singularities and the Adams-Novikov Spectral Sequence, London Mathematical Society Lecture Note Series, ISBN 0521426081
McCleary, John (February 2001), A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics, 58 (2nd ed.), Cambridge University Press, doi:10.2277/0521567599, MR 1793722, ISBN 978-0-521-56759-6