Cyclicity in homotopy algebras and rational homotopy theory

Abstract

Abstract Kadeishvili proposes a minimal C ∞ {C_{\infty}} -algebra as a rational homotopy model of a space. We discuss a cyclic version of this Kadeishvili C ∞ {C_{\infty}} -model and apply it to classifying rational Poincaré duality spaces. We classify 1-connected minimal cyclic C ∞ {C_{\infty}} -algebras whose cohomology algebras are those of ( S p × S p + 2 ⁢ q - 1 ) ⁢ ♯ ⁢ ( S q × S 2 ⁢ p + q - 1 ) {(S^{p}\times S^{p+2q-1})\sharp(S^{q}\times S^{2p+q-1})} , where 2 ≤ p ≤ q {2\leq p\leq q} . We also include a proof of the decomposition theorem for cyclic A ∞ {A_{\infty}} and C ∞ {C_{\infty}} -algebras.