# COMPLETION THEOREMS IN EQUIVARIANT BORDISM AND THE IDEAL J(G; n)

Lead Research Organisation:
University of Warwick

Department Name: Mathematics

### Abstract

During the first two years of my PhD, I have been focusing in trying to understand better the structure of tom Dieck's homotopical equivariant bordism MU_G ([tD70]) for a compact Lie group G. MU_G is the universal equivariant complex oriented theory and many structure theorems are known when G is an abelian compact Lie group. For example, if A is an abelian compact Lie group:

- MU_A is a free module over MU and concentrated in even degrees ([Com96]);

- MU_A carries the universal A-equivariant formal group law ([Hau19]).

Building upon recent work of Schwede ([Sch20]) I proved a conjecture established by Greenlees and May in [GM97, Conjecture 1.2]. The paper containing the proof of [GM97, Conjecture 1.2] will appear in the journal "Geometry and Topology".

This completion theorem led to the question of equivariant formal group law when the equivariance comes from a not necessarily abelian compact Lie group G.

In fact, following the outline definition 15.1 of [Gre01] if G is non abelian, a G-equivariant formal group law should consists of a collection of algebras which are complete.

Moreover, the main objects to study for better understand what a G-equivariant formal group law should be are the G-equivariant E-cohomology theory of the G-equivariant classifying space of the unitary group U(n), where E is a complex oriented theory. More concretely, we can choose the n-dimensional grassmanians of a complete G-universe U_G as a model for the G-equivariant classifying space of U(n) .

The question whether the G-equivariant E-cohomology theory of the n-dimensional grassmanian of U_G is a (derived) completion of E_G is still open and following [Hau19, Prop 5.27] requires the understanding of the kernel of the restriction from G x U(n) to a graph subgroup.

In the case E is K-theory and G is finite we know that J(G,1) (the intersection of the ideals above) is generated by the Euler class of rv_1, where r is the regular representation of G and v_1 is the tautological U(1) representation.

I am currently investigating the relation between J (G, 2) and J (G, 1) under the restriction homomorphism from G x U(2) to G x U(1) and the splitting homomorphism of this restriction which is defined by Schwede in [Sch20] and we denote by S.

The claim is the following:

- J(G,2)=(e(rv_2), S(e(rv_1)))

Using the fact that the restriction to the maximal torus is injective in K-theory we also have

a desctription of S(e(rv_1 )) in terms of Adams operations of v_2 and the exterior power of r. Hopefully, this is the first step of an inductive argument that will give a description of the ideal J(G,n) for every n in K-theory when G is finite. Building up on this I will be able to conjecture a description of J(G,n) also for equivariant bordism.

References

[Com96] Gustavo Comezana. Calculations in complex equivariant bordism. Equi- variant Homotopy and Cohomology Theory, 91:333-352, 1996.

[GM97] J. P. C. Greenlees and J. P. May. Localization and completion theorems for MU-module spectra. Annals of Mathematics, 146(3):509-544, 1997.

[Gre01] JPC Greenlees. Equivariant formal group laws and complex oriented co- homology theories. Homology, Homotopy and applications, 3(2):225-263, 2001.

[Hau19] Markus Hausmann. Global group laws and equivariant bordism rings. arXiv preprint arXiv:1912.07583, 2019.

[Sch20] Stefan Schwede. Splittings of global Mackey functors and regularity of equivariant Euler classes, 2020. arXiv:2006.09435.

[tD70] Tammo tom Dieck. Bordism of G-manifolds and integrality theorems. Topology, 9(4):345-358, 1970.

- MU_A is a free module over MU and concentrated in even degrees ([Com96]);

- MU_A carries the universal A-equivariant formal group law ([Hau19]).

Building upon recent work of Schwede ([Sch20]) I proved a conjecture established by Greenlees and May in [GM97, Conjecture 1.2]. The paper containing the proof of [GM97, Conjecture 1.2] will appear in the journal "Geometry and Topology".

This completion theorem led to the question of equivariant formal group law when the equivariance comes from a not necessarily abelian compact Lie group G.

In fact, following the outline definition 15.1 of [Gre01] if G is non abelian, a G-equivariant formal group law should consists of a collection of algebras which are complete.

Moreover, the main objects to study for better understand what a G-equivariant formal group law should be are the G-equivariant E-cohomology theory of the G-equivariant classifying space of the unitary group U(n), where E is a complex oriented theory. More concretely, we can choose the n-dimensional grassmanians of a complete G-universe U_G as a model for the G-equivariant classifying space of U(n) .

The question whether the G-equivariant E-cohomology theory of the n-dimensional grassmanian of U_G is a (derived) completion of E_G is still open and following [Hau19, Prop 5.27] requires the understanding of the kernel of the restriction from G x U(n) to a graph subgroup.

In the case E is K-theory and G is finite we know that J(G,1) (the intersection of the ideals above) is generated by the Euler class of rv_1, where r is the regular representation of G and v_1 is the tautological U(1) representation.

I am currently investigating the relation between J (G, 2) and J (G, 1) under the restriction homomorphism from G x U(2) to G x U(1) and the splitting homomorphism of this restriction which is defined by Schwede in [Sch20] and we denote by S.

The claim is the following:

- J(G,2)=(e(rv_2), S(e(rv_1)))

Using the fact that the restriction to the maximal torus is injective in K-theory we also have

a desctription of S(e(rv_1 )) in terms of Adams operations of v_2 and the exterior power of r. Hopefully, this is the first step of an inductive argument that will give a description of the ideal J(G,n) for every n in K-theory when G is finite. Building up on this I will be able to conjecture a description of J(G,n) also for equivariant bordism.

References

[Com96] Gustavo Comezana. Calculations in complex equivariant bordism. Equi- variant Homotopy and Cohomology Theory, 91:333-352, 1996.

[GM97] J. P. C. Greenlees and J. P. May. Localization and completion theorems for MU-module spectra. Annals of Mathematics, 146(3):509-544, 1997.

[Gre01] JPC Greenlees. Equivariant formal group laws and complex oriented co- homology theories. Homology, Homotopy and applications, 3(2):225-263, 2001.

[Hau19] Markus Hausmann. Global group laws and equivariant bordism rings. arXiv preprint arXiv:1912.07583, 2019.

[Sch20] Stefan Schwede. Splittings of global Mackey functors and regularity of equivariant Euler classes, 2020. arXiv:2006.09435.

[tD70] Tammo tom Dieck. Bordism of G-manifolds and integrality theorems. Topology, 9(4):345-358, 1970.

### Organisations

## People |
## ORCID iD |

John Greenlees (Primary Supervisor) | |

Marco La Vecchia (Student) |

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/V520226/1 | 30/09/2020 | 31/10/2025 | |||

2443924 | Studentship | EP/V520226/1 | 04/10/2020 | 04/10/2024 | Marco La Vecchia |