The Euler class group of a polynomial algebra II

Bott periodicity and calculus of Euler classes on spheres

We study maps from a smooth scheme to a motivic sphere in the Morel–Voevodsky \mathbb{A}^1 -homotopy category, i.e., motivic cohomotopy sets. Following Borsuk, we show that, in the presence of suitable hypotheses on the dimension of the source, motivic cohomotopy sets can be equipped with functorial abelian group structures. We then explore links between motivic cohomotopy groups, Euler class groups à la Nori–Bhatwadekar–Sridharan and Chow–Witt groups. We show that, again under suitable hypotheses on the base field {k} , if {X} is a smooth affine k -variety of dimension d , then the Euler class group of codimension d cycles coincides with the codimension d Chow–Witt group; the identification proceeds by comparing both groups with a suitable motivic cohomotopy group. As a byproduct, we describe the Chow group of zero cycles on a smooth affine k -scheme as the quotient of the free abelian group on zero cycles by the subgroup generated by reduced complete intersection ideals; this answers a question of S. Bhatwadekar and R. Sridharan.

The Euler class group of a polynomial algebra

Let $R$ be a commutative Noetherian ring of dimension~$d$ and \[ B=R[X_1,\ldots ,X_m,Y_1^{\pm 1},\ldots ,Y_n^{\pm 1}] \] a Laurent polynomial ring over $R$. If $A=B[Y,f^{-1}]$ for some $f\in R[Y]$, then we prove the following results: (i) if $f$ is a monic polynomial, then the Serre dimension of $A$ is $\leq d$. The case $n=0$ is due to Bhatwadekar, without the condition that $f$ is a monic polynomial. (ii) The $p$th Euler class group $E^p(A)$ of $A$, defined by Bhatwadekar and Sridharan, is trivial for $p\geq \max \{d+1,\dim A -p+3\}$. The case $m=n=0$ is due to Mandal and Parker.

Vanishing powers of the Euler class

Zero cycles and the Euler class groups of smooth real affine varieties

The mapping class group Γ g 1 \Gamma _g^1 of a closed orientable surface of genus g ≥ 1 g \geq 1 with one marked point can be identified, by the Nielsen action, with a subgroup of the group of orientation preserving homeomorphisms of the circle. This inclusion pulls back the powers of the discrete universal Euler class producing classes E n ∈ H 2 n ( Γ g 1 ; Z ) E^n \in H^{2n}(\Gamma _g^1;\mathbb {Z}) for all n ≥ 1 n\geq 1 . In this paper we study the power n = g n=g , and prove: E g E^g is a torsion class which generates a cyclic subgroup of H 2 g ( Γ g 1 ; Z ) H^{2g}(\Gamma _g^1; \mathbb Z) whose order is a positive integer multiple of 4 g ( 2 g + 1 ) ( 2 g − 1 ) 4g(2g+1)(2g-1) .