A CONTRIBUTION TO THE THEOREM ON LEVEL HOMOTOPY EQUIVALENCES - Shekutkovski, Nikita (Matematichki Bilten, 2000)
In [7] it is proved the following result. If the members of an strictly commutative inverse sequence \(X\) 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 \(X\) by an arbitrary coherent inverse sequence. The following result is proved also: For arbitrary coherent inverse sequencess \((X_m,p_m)\) and \((Y_m,q_m)\), if the maps \(f_m:X_m→ Y_m\) and \(f’_m:X_m→ Y_m\) are homotopic, and \(f:(X_m,p_m)→(Y_m,q_m)\) is a coherent map for maps \(f_m:X_m→ Y_m\), then there exists a coherent map \(f’\) for the maps \(f’_m:X_m→ Y_m\) such that \(f\) and \(f’\) are coherently homotopic.