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

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- by Manuel Rivera, Felix Wierstra and Mahmoud Zeinalian PDF
- Trans. Amer. Math. Soc.
**375**(2022), 3267-3303 Request permission

## 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.## References

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## Additional Information

**Manuel Rivera**- Affiliation: Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, Indiana 47907-2067
- MR Author ID: 1022985
- ORCID: 0000-0002-1817-7619
- Email: manuelr@purdue.edu
**Felix Wierstra**- Affiliation: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 107, Postbus 94248, 1090 GE Amsterdam, The Netherlands
- MR Author ID: 1295763
- Email: f.p.wierstra@uva.nl
**Mahmoud Zeinalian**- Affiliation: Department of Mathematics, City University of New York, Lehman College, 250 Bedford Park Blvd W, Bronx, New York 10468
- MR Author ID: 773273
- Email: mahmoud.zeinalian@lehman.cuny.edu
- Received by editor(s): May 23, 2021
- Received by editor(s) in revised form: September 30, 2021
- Published electronically: February 17, 2022
- Additional Notes: The first author was supported by NSF Grant 210554 and the Karen EDGE Fellowship

The second author was supported by grant number 2019-00536 from the Swedish Research Council - © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 3267-3303 - MSC (2020): Primary 55P15, 57T30, 55P60, 55P62, 55U15
- DOI: https://doi.org/10.1090/tran/8579
- MathSciNet review: 4402661