Steenrod algebra
The graded algebra over the field
of all stable cohomology operations (cf. Cohomology operation) modulo
. For any space (spectrum of spaces)
, the group
is a module over the Steenrod algebra
.
The Steenrod algebra is multiplicatively generated by the Steenrod operations (cf. Steenrod operation). Thus, the Steenrod algebra is a graded associative algebra, multiplicatively generated by the symbols
with
, which satisfy the Adem relation:
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, so that an additive basis (over
) of the Steenrod algebra
consists of the operations
,
(the so-called Cartan–Serre basis). Similar results are true for
with
. Furthermore,
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where is an Eilenberg–MacLane space. The multiplication
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induces the diagonal in
, which is a homomorphism of algebras, and, consequently, turns
into a Hopf algebra.
References
[1] | N.E. Steenrod, D.B.A. Epstein, "Cohomology operations" , Princeton Univ. Press (1962) |
[2] | J. Milnor, "The Steenrod algebra and its dual" Ann. of Math. , 67 (1958) pp. 150–171 |
[3] | M.K. Tangora, "Cohomology operations and applications in homotopy theory" , Harper & Row (1968) |
Comments
The analogue of the Steenrod algebra for a cohomology theory defined by a spectrum is
; cf. Generalized cohomology theories and Spectrum of spaces. The
-term of the Adams spectral sequence, cf. Spectral sequence,
is a purely (homological) algebra construct obtained by regarding the homology groups as modules over the Hopf algebra
.
References
[a1] | J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989) |
[a2] | R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) pp. Chapts. 18–19 |
[a3] | J.F. Adams, "Stable homotopy and generalized homology" , Univ. Chicago Press (1974) pp. Part III, Chapts. 12, 15 |
Steenrod algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steenrod_algebra&oldid=12348