The Dyer--Lashof algebra and the~Steenrod algebra for generalized homology and cohomology

V. A. Smirnov

Sbornik. Mathematics, Tome 190 (1999) no. 12, pp. 1807-1842

Résumé

An analogue $\mathbb R$ of the Dyer–Lashof algebra $R$ and an analogue $\mathbb A$ of the Steenrod algebra $A$ are defined for generalized homology and cohomology theories. It is shown that if there is an $E_\infty$-multiplicative structure on a spectrum $\mathbb H$, then on the corresponding generalized cohomology $\mathbb H^*(X)$ of a topological space $X$ there is an action $\mathbb A\otimes \mathbb H^*(X)\to \mathbb H^*(X)$ of the Steenrod algebra, while if the space $X$ is an $E_\infty$-space, then on the generalized homology $\mathbb H^*(X)$ there is an action $\mathbb R\otimes \mathbb H_*(X)\to \mathbb H_*(X)$ of the Dyer–Lashof algebra. These actions are computed for cobordism of topological spaces. A connection is established between the Steenrod operations and the Landweber–Novikov operations.