##
**Global group laws and equivariant bordism rings.**
*(English)*
Zbl 07583002

D. Quillen [Bull. Am. Math. Soc. 75, 1293–1298 (1969; Zbl 0199.26705)] proved that the formal group law associated to the complex orientation of complex bordism \(MU\) is the universal one. In particular, the complex bordism ring is isomorphic to the Lazard ring. This result became one of the main tools in understanding the structure of the stable homotopy category.

In view of the importance of Quillen’s theorem much work has been put into obtaining a similar understanding of equivariant bordism rings and their characterization in terms of formal group law data. For an abelian compact Lie group \(A\), the notion of an \(A\)-equivariant formal group law was introduced by M. Cole et al. [Proc. Lond. Math. Soc. (3) 81, No. 2, 355–386 (2000; Zbl 1030.55004)]. As in the non-equivariant case there exists a universal \(A\)-equivariant formal group law over an \(A\)-equivariant Lazard ring \(L_A\) and a map \[ \varphi_A:L_A\rightarrow \pi_*^A(MU_A) \] classifying the Euler class of a product of \(A\)-equivariant complex line bundles. Here \(MU_A\) denotes the homotopical \(A\)-equivariant bordism spectrum.

It has been conjectured by Greenlees that \(\varphi_A\) is an isomorphism. This conjecture has only been proved in case that \(A\) has order two by B. Hanke and M. Wiemeler [Adv. Math. 340, 48–75 (2018; Zbl 1409.55012)]. The paper under review now proves the conjecture in full generality.

It also proves a similar result for homotopy theoretic \(A\)-equivariant unoriented bordism \(MO_A\) in the case that \(A\) is an elementary abelian \(2\)-group. In that case one considers the ring \(L_A^{2\text{-tor}}\) carrying the universal \(A\)-equivariant \(2\)-torsion formal group law. Here the result is:

The map \[ \varphi_A^{2\text{-tor}}:L_A^{2\text{-tor}}\rightarrow \pi_*^A(MO_A) \] classifying the formal group law induced from the real orientation of \(MO_A\) is an isomorphism.

The proof of these results goes as follows. First it is noted that in the complex case it suffices to prove the result for tori \(A\). Then global group laws and global \(2\)-torsion group laws are introduced. These are functors \[ (\text{tori})^{\text{op}}\rightarrow \text{commutative rings} \] and \[ (\text{elementary abelian two-groups})^{\text{op}}\rightarrow \text{commutative rings} \] which have a coordinate, respectively. Examples of such global group laws are given by \(\mathbf{MU}\) and \(\mathbf{MO}\) which assign to a group its equivariant bordism ring. It is shown that in the categories \(GL_{gl}\) and \(GL_{gl}^{2\text{-tor}}\) of these group laws there exist initial objects \(\mathbf{L}\) and \(\mathbf{L}^{2\text{-tor}}\). Moreover functors \[ (\cdot)^\wedge_A:GL_{gl}\rightarrow FGL_A \] and \[ (\cdot)^\wedge_A:GL_{gl}^{2-\text{tor}}\rightarrow FGL_A^{2\text{-tor}} \] for every torus \(A\) and every elementary abelian \(2\)-group \(A\), respectively, are constructed. Here \(FGL_A\) and \(FGL_A^{2\text{-tor}}\) denote the categories of \(A\)-equivariant (2-torsion) formal group laws. It is shown that these functors have right adjoints. In particular,

\[ (\mathbf{L})_A^\wedge=L_A\quad\quad\quad\text{and}\quad\quad\quad(\mathbf{L}^{2\text{-tor}})_A^\wedge=L_A^{2\text{-tor}}. \]

Using the axioms of global group laws it is deduced that the non-trivial Euler classes in \(L_A\) are regular elements. It was previously known by work of Greenlees [The coefficient ring of equivariant homotopical bordism classifies equivariant formal group laws over Noetherian rings, Preprint, 2000] that \(\varphi_A\) and \(\varphi_A^{2\text{-tor}}\) become isomorphisms after inverting all non-trivial Euler classes. Therefore it follows that these maps are injective. Using Greenlees’s result and the properties of global group laws it is then deduced that \(\varphi_A\) and \(\varphi_A^{2\text{-tor}}\) are also surjective. Hence the main results follow.

As a consequence of this line of reasoning it also follows that \(\mathbf{L}=\mathbf{MU}\) and \(\mathbf{L}^{2\text{-tor}}=\mathbf{MO}\).

In view of the importance of Quillen’s theorem much work has been put into obtaining a similar understanding of equivariant bordism rings and their characterization in terms of formal group law data. For an abelian compact Lie group \(A\), the notion of an \(A\)-equivariant formal group law was introduced by M. Cole et al. [Proc. Lond. Math. Soc. (3) 81, No. 2, 355–386 (2000; Zbl 1030.55004)]. As in the non-equivariant case there exists a universal \(A\)-equivariant formal group law over an \(A\)-equivariant Lazard ring \(L_A\) and a map \[ \varphi_A:L_A\rightarrow \pi_*^A(MU_A) \] classifying the Euler class of a product of \(A\)-equivariant complex line bundles. Here \(MU_A\) denotes the homotopical \(A\)-equivariant bordism spectrum.

It has been conjectured by Greenlees that \(\varphi_A\) is an isomorphism. This conjecture has only been proved in case that \(A\) has order two by B. Hanke and M. Wiemeler [Adv. Math. 340, 48–75 (2018; Zbl 1409.55012)]. The paper under review now proves the conjecture in full generality.

It also proves a similar result for homotopy theoretic \(A\)-equivariant unoriented bordism \(MO_A\) in the case that \(A\) is an elementary abelian \(2\)-group. In that case one considers the ring \(L_A^{2\text{-tor}}\) carrying the universal \(A\)-equivariant \(2\)-torsion formal group law. Here the result is:

The map \[ \varphi_A^{2\text{-tor}}:L_A^{2\text{-tor}}\rightarrow \pi_*^A(MO_A) \] classifying the formal group law induced from the real orientation of \(MO_A\) is an isomorphism.

The proof of these results goes as follows. First it is noted that in the complex case it suffices to prove the result for tori \(A\). Then global group laws and global \(2\)-torsion group laws are introduced. These are functors \[ (\text{tori})^{\text{op}}\rightarrow \text{commutative rings} \] and \[ (\text{elementary abelian two-groups})^{\text{op}}\rightarrow \text{commutative rings} \] which have a coordinate, respectively. Examples of such global group laws are given by \(\mathbf{MU}\) and \(\mathbf{MO}\) which assign to a group its equivariant bordism ring. It is shown that in the categories \(GL_{gl}\) and \(GL_{gl}^{2\text{-tor}}\) of these group laws there exist initial objects \(\mathbf{L}\) and \(\mathbf{L}^{2\text{-tor}}\). Moreover functors \[ (\cdot)^\wedge_A:GL_{gl}\rightarrow FGL_A \] and \[ (\cdot)^\wedge_A:GL_{gl}^{2-\text{tor}}\rightarrow FGL_A^{2\text{-tor}} \] for every torus \(A\) and every elementary abelian \(2\)-group \(A\), respectively, are constructed. Here \(FGL_A\) and \(FGL_A^{2\text{-tor}}\) denote the categories of \(A\)-equivariant (2-torsion) formal group laws. It is shown that these functors have right adjoints. In particular,

\[ (\mathbf{L})_A^\wedge=L_A\quad\quad\quad\text{and}\quad\quad\quad(\mathbf{L}^{2\text{-tor}})_A^\wedge=L_A^{2\text{-tor}}. \]

Using the axioms of global group laws it is deduced that the non-trivial Euler classes in \(L_A\) are regular elements. It was previously known by work of Greenlees [The coefficient ring of equivariant homotopical bordism classifies equivariant formal group laws over Noetherian rings, Preprint, 2000] that \(\varphi_A\) and \(\varphi_A^{2\text{-tor}}\) become isomorphisms after inverting all non-trivial Euler classes. Therefore it follows that these maps are injective. Using Greenlees’s result and the properties of global group laws it is then deduced that \(\varphi_A\) and \(\varphi_A^{2\text{-tor}}\) are also surjective. Hence the main results follow.

As a consequence of this line of reasoning it also follows that \(\mathbf{L}=\mathbf{MU}\) and \(\mathbf{L}^{2\text{-tor}}=\mathbf{MO}\).

Reviewer: Michael Wiemeler (Münster)

### MSC:

57R85 | Equivariant cobordism |

55N22 | Bordism and cobordism theories and formal group laws in algebraic topology |

55P91 | Equivariant homotopy theory in algebraic topology |

14L05 | Formal groups, \(p\)-divisible groups |

55P42 | Stable homotopy theory, spectra |

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