The simplicial coalgebra of chains determines homotopy types rationally and one prime at a time

TL;DR

This paper demonstrates that the algebraic structure of simplicial chains on a space uniquely determines its rational homotopy type and prime localizations, including fundamental group and local system homology, without restrictions on the fundamental group.

Contribution

It introduces a new algebraic framework using simplicial cocommutative coalgebras to recover homotopy types rationally and prime-by-prime, extending previous results to spaces with arbitrary fundamental groups.

Findings

Homotopy type is determined by the coalgebra of chains rationally and at each prime.

Fundamental group and local system homology are algebraically recoverable from chain structures.

The approach applies to spaces with arbitrary fundamental groups, without restrictions.

Abstract

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.