On the Equivariant Lazard Ring and Tom Dieck's Equivariant Cobordism Ring

TL;DR

This paper investigates the relationship between the equivariant Lazard ring and Tom Dieck's equivariant cobordism ring for torus groups, establishing surjectivity and injectivity properties in specific cases.

Contribution

It proves the surjectivity of the canonical map from the equivariant Lazard ring to the cobordism ring for rank 1 tori and shows injectivity of the completion map, extending results under certain algebraic assumptions.

Findings

The map L_G d7 MU_G is surjective for rank 1 torus groups.

The completion map MU_G d7 d7 MU_G = MU(BG) is injective.

Results extend to higher ranks under specific algebraic conditions.

Abstract

For a torus G of rank r = 1, we showed that the canonical ring homomorphism L_G \to MU_G, where L_G is the equivariant Lazard ring and MU_G is the equivariant cobordism ring introduced by Tom Dieck, is surjective. We also showed that the completion map MU_G \to \hat{MU}_G = MU(BG) is injective. Moreover, we showed that the same results hold if we assume a certain algebraic property holds in L_G when r \geq 2.