Steenrod algebra

The graded algebra $ A _ {p} $ over the field $ \mathbf Z _ {p} $ of all stable cohomology operations (cf. Cohomology operation) modulo $ p $. For any space (spectrum of spaces) $ X $, the group $ H ^ \star ( X; \mathbf Z _ {p} ) $ is a module over the Steenrod algebra $ A _ {p} $.

The Steenrod algebra is multiplicatively generated by the Steenrod operations (cf. Steenrod operation). Thus, the Steenrod algebra $ A _ {2} $ is a graded associative algebra, multiplicatively generated by the symbols $ Sq ^ {i} $ with $ \mathop{\rm deg} Sq ^ {i} = i $, which satisfy the Adem relation:

$$ Sq ^ {a} Sq ^ {b} = \ \sum _ { t } \left ( \begin{array}{c} b- t- 1 \\ a- 2t \end{array} \right ) Sq ^ {a+} b- t Sq ^ {t} , $$

$ a < 2b $, so that an additive basis (over $ \mathbf Z _ {2} $) of the Steenrod algebra $ A _ {2} $ consists of the operations $ Sq ^ {i _ {1} } \dots Sq ^ {i _ {r} } $, $ i _ {k} \geq 2i _ {k+} 1 $( the so-called Cartan–Serre basis). Similar results are true for $ A _ {p} $ with $ p > 2 $. Furthermore,

$$ ( A _ {p} ) ^ {i} \cong H ^ {i+} n ( K( \mathbf Z _ {p} , n); \ \mathbf Z _ {p} ) ,\ \ n \textrm{ large } , $$

where $ K( \mathbf Z _ {p} , n) $ is an Eilenberg–MacLane space. The multiplication

$$ K( \mathbf Z _ {p} , m) \wedge K( \mathbf Z _ {p} , n ) \rightarrow K( \mathbf Z _ {p} , m+ n) $$

induces the diagonal $ \Delta : A _ {p} \rightarrow A _ {p} \otimes A _ {p} $ in $ A _ {p} $, which is a homomorphism of algebras, and, consequently, turns $ A _ {p} $ 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 $ E $ is $ E _ \star ( E) $; cf. Generalized cohomology theories and Spectrum of spaces. The $ E _ {2} $- term of the Adams spectral sequence, cf. Spectral sequence, $ \mathop{\rm Ext} _ {E _ \star ( E) } ^ {s, t } ( E _ \star ( X), E _ \star ( Y) \Rightarrow [ X, Y] _ \star ) $ is a purely (homological) algebra construct obtained by regarding the homology groups as modules over the Hopf algebra $ E _ \star ( E) $.

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