Homotopy type and v1-periodic homotopy groups of p-compact groups

TL;DR

This paper computes the v1-periodic homotopy groups of all irreducible p-compact groups, revealing their structure and relation to p-completed Lie groups, especially in complex modular cases.

Contribution

It provides a complete determination of v1-periodic homotopy groups for irreducible p-compact groups, including explicit calculations in modular cases.

Findings

v1-periodic homotopy groups are explicitly determined for all irreducible p-compact groups

In odd p cases, these groups are homotopy equivalent to products of spaces related to p-completed Lie groups

The approach connects invariant polynomials to homotopy groups in complex modular scenarios

Abstract

We determine the v1-periodic homotopy groups of all irreducible p-compact groups (BX,X). In the most difficult, modular, cases, we follow a direct path from their associated invariant polynomials to these homotopy groups. We show that, if p is odd, every irreducible p-compact group has X of the homotopy type of a product of explicit spaces related to p-completed Lie groups.