A CONTRIBUTION TO THE THEOREM ON LEVEL HOMOTOPY EQUIVALENCES

Abstract

In [7] it is proved the following result. If the members of an strictly commutative inverse sequence \(\underline{X}\) of topological spaces are replaced by homotopy equivalent spaces, the new spaces can be organized as a coherent inverse system. Moreover, the two inverse sequences are isomorphic in the coherent category. In this paper this result is strengthened, by replacing the commutative inverse sequence \(\underline{X}\) by an arbitrary coherent inverse sequence. The following result is proved also: For arbitrary coherent inverse sequencess \((X_m,\underline{p}_m)\) and \((Y_m,\underline{q}_m)\), if the maps \(f_m:X_m\to Y_m\) and \(f'_m:X_m\to Y_m\) are homotopic, and \(f:(X_m,\underline{p}_m)\to(Y_m,\underline{q}_m)\) is a coherent map for maps \(f_m:X_m\to Y_m\), then there exists a coherent map \(f'\) for the maps \(f'_m:X_m\to Y_m\) such that \(f\) and \(f'\) are coherently homotopic.