Some perspectives on homotopy obstructions

Abstract

For a projective A-module P, with $n=rank(P)\ge 2$, the Homotopy obstruction sets $\pi _0\left( {{\mathcal {L}}{\mathcal {O}}}(P)\right)$ were defined, in [6], to detect whether P has a free direct summand or not. These sets have a structure of an abelian monoid, under suitable regularity and other conditions. In this article, we provide some further perspective on these sets $\pi _0\left( {{\mathcal {L}}{\mathcal {O}}}(P)\right)$. In particular, under similar regularity and other conditions, we prove that if P, Q are two projective A-modules, with $rank(P)=rank(Q)=d$ and $\det (P) \cong \det Q$, then $\pi _0\left( {{\mathcal {L}}{\mathcal {O}}}(Q)\right) \cong \pi _0\left( {{\mathcal {L}}{\mathcal {O}}}(P)\right)$.