$K$-theory of $S^7/Q_8$ and a counterexample to a result of P.M. Akhmet'ev

TL;DR

This paper provides a counterexample in K-theory related to the quotient of the 7-sphere by the quaternion group, challenging a previously used proposition in the context of Hopf and Kervaire invariants.

Contribution

It introduces a specific counterexample in K-theory of $S^7/Q_8$ that refutes a proposition used in prior work on Hopf and Kervaire invariants.

Findings

Counterexample in K-theory of $S^7/Q_8$

Refutation of a proposition used in previous invariants research

Implications for the validity of certain arguments in topology

Abstract

A simple counterexample is presented to a proposition which is used in the arguments given by P. M. Akhmet'ev in his work on the Hopf invariant and Kervaire invariant. The counterexample makes use of the $K$-theory of the quotient of the 7-sphere by the quaternion group of order 8.