# Higher homotopy associativity in the Harris decomposition of Lie groups

## TL;DR

This paper investigates how the Harris decomposition of certain Lie groups preserves higher homotopy associativity at various primes, extending classical homotopy group relations to a richer algebraic structure.

## Contribution

It extends Harris's classical homotopy group results to show preservation of higher homotopy associativity in the p-local decomposition of specific Lie groups.

## Key findings

- Homotopy equivalence $G o H imes G/H$ at primes $p$
- Preservation of higher homotopy associativity in the decomposition
- Extension of classical Harris results to p-local homotopy theory

## Abstract

Let $(G,H)=(SU(2n+1),SO(2n+1)),\,(SU(2n),Sp(n)),\,(SO(2n),SO(2n-1)),\,(E_6,F_4),\,(Spin(8),G_2)$, and let $p$ be any prime $\ge 5$ for $(G,H)=(E_6,F_4)$, any prime $p\ne 3$ for $(G,H)=(Spin(8),G_2)$, and any odd prime otherwise. The classical result of Harris on the relation between the homotopy groups of $G$ and $H$ is reinterpreted as a $p$-local homotopy equivalence $G\simeq_{(p)}H\times G/H$, which yields a projection $G_{(p)}\to H_{(p)}$. We show how much this projection preserves the higher homotopy associativity.