  We prove that the simplicial cocommutative coalgebra of singular chains on a
connected topological space determines the homotopy type rationally and one
prime at a time, without imposing any restriction on the fundamental group. In
particular, the fundamental group and the homology groups with coefficients in
arbitrary local systems of vector spaces are completely determined by the
natural algebraic structure of the chains. The algebraic structure is presented
as the class of the simplicial cocommutative coalgebra of chains under a notion
of weak equivalence induced by a functor from coalgebras to algebras coined by
Adams as the cobar construction. The fundamental group is determined by a
quadratic equation on the zeroth homology of the cobar construction of the
normalized chains which involves Steenrod's chain homotopies for
cocommutativity of the coproduct. The homology groups with local coefficients
are modeled by an algebraic analog of the universal cover which is invariant
under our notion of weak equivalence. We conjecture that the integral homotopy
type is also determined by the simplicial coalgebra of integral chains, which
we prove when the universal cover is of finite type.
