Bordism groups of Poincare $E_\infty$-coalgebras and symmetric $L$-groups

A Poincare $E_\infty$-coalgebra construction over involutive algebras is introduced in this paper. Various types of bordism between Poincare $E_\infty$-coalgebras are defined and the relations between the corresponding bordism groups are studied. It is shown in particular that the Thom bordism groups of closed non-oriented smooth manifolds and the rational Wall groups of a unitary group have a common algebraic origin, that is, they are obtained by the same construction considered over the fields $\mathbb Z/2$ and $\mathbb Q$, respectively.