COMPACTLY SUPPORTED A 1 $\mathbb {A}^1$ double struck upper A Superscript 1-EULER CHARACTERISTICS OF SYMMETRIC POWERS OF CELLULAR VARIETIES

Abstract The compactly supported A 1 $\mathbb {A}^1$ double struck upper A Superscript 1 -Euler characteristic, introduced by Hoyois and later refined by Levine and others, is an analogue in motivic homotopy theory of the classical Euler characteristic of complex topological manifolds. It is an invariant on the Grothendieck ring of varieties K 0 ( Var k ) $\mathrm {K}_0(\mathrm {Var}_k)$ normal upper K 0 left parenthesis upper V a r Subscript k Baseline right parenthesis taking values in the Grothendieck-Witt ring GW ( k ) $\mathrm {GW}(k)$ upper G upper W left parenthesis k right parenthesis of the base field k . The former ring has a natural power structure induced by symmetric powers of varieties. In a recent preprint, the first author and Pál construct a power structure on GW ( k ) $\mathrm {GW}(k)$ upper G upper W left parenthesis k right parenthesis and show that the compactly supported A 1 $\mathbb {A}^1$ double struck upper A Superscript 1 -Euler characteristic respects these two power structures for 0 $0$ 0 -dimensional varieties, or equivalently étale k -algebras. In this paper, we define the class Sym k $\mathrm {Sym}_k$ upper S y m Subscript k