In this paper we study the subgroup of the Picard group of Voevodsky’s category of geometric motives $\operatorname{DM}_{\text{gm}}(k;\mathbb{Z}/2)$ generated by the reduced motives of affine quadrics. Our main tools here are the functors of Bachmann [On the invertibility of motives of affine quadrics, Doc. Math. 22 (2017), 363–395], but we also provide an alternative method. We show that the group in question can be described in terms of indecomposable direct summands in the motives of projective quadrics over $k$ . In particular, we describe all the relations among the reduced motives of affine quadrics. We also extend the criterion of motivic equivalence of projective quadrics.

In this article we construct symmetric operations for all primes (previously known only for $p=2$ ). These unstable operations are more subtle than the Landweber–Novikov operations, and encode all $p$ -primary divisibilities of characteristic numbers. Thus, taken together (for all primes) they plug the gap left by the Hurewitz map $\mathbb{L}{\hookrightarrow}\mathbb{Z}[b_{1},b_{2},\ldots ]$ , providing an important structure on algebraic cobordism. Applications include questions of rationality of Chow group elements, and the structure of the algebraic cobordism. We also construct Steenrod operations of tom Dieck style in algebraic cobordism. These unstable multiplicative operations are more canonical and subtle than Quillen-style operations, and complement the latter.