Homotopic invariance of dihedral homologies for 𝐴_{∞}-algebras with involution

It is established that the dihedral homologies of involutive A ∞ A_{\infty } -algebras are homotopically invariant with respect to the homotopy equivalences of involutive A ∞ A_{\infty } -algebras. As a consequence, it is shown that over any field, the dihedral homologies of a topological space are isomorphic to the dihedral homologies of the involutive A ∞ A_{\infty } -algebra of homologies for the simplicial group of Kan loops of the original topological space.