THE SIMPLICIAL COALGEBRA OF CHAINS DETERMINES HOMOTOPY TYPES RATIONALLY AND ONE PRIME AT A TIME

Manuel Rivera, Felix Wierstra, Mahmoud Zeinalian

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Pages (from-to)3267-3303
Number of pages37
JournalTransactions of the American Mathematical Society
Volume375
Issue number5
DOIs
StatePublished - 2022
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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