In the paper ``Homology and cohomology for topological algebras'' [Adv. Math. 9, 137-182 (1972; Zbl 0271.46040)] \textit{J. L. Taylor} developed important homological methods in the general framework of topological algebras and obtained one result, which has the following meaning: a contractible Arens-Michael algebra is topologically isomorphic to the direct sum of the topological cartesian product of a certain family of full matrix algebras and of some algebra, which in the commutative case is always zero. This result initiates a more general problem: whether an arbitrary contractible Arens-Michael algebra is topologically isomorphic to the topological cartesian product of a certain family of full matrix algrebras? This question was considered by the school of A. Helemskii with successful answers in several case, and A. Helemskii conjectured that Taylor's result holds for all contractible locally \(C^*\)-algebras. This paper is devoted to the details of the proof of this conjecture. Namely, it is proved that a locally \(C^*\)-algebra is contractible iff it is topologically isomorphic to the topological cartesian product of a certain family of full matrix algebras.