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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.
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The Simplicial Coalgebra of Chains Under Three Different Notions of Weak Equivalence
Abstract We study the simplicial coalgebra of chains on a simplicial set with respect to three notions of weak equivalence. To this end, we construct three model structures on the category of reduced simplicial sets for any commutative ring $$R$$. The weak equivalences are given by: (1) an $$R$$-linearized version of categorical equivalences, (2) maps inducing an isomorphism on fundamental groups and an $$R$$-homology equivalence between universal covers, and (3) $$R$$-homology equivalences. Analogously, for any field $${\mathbb{F}}$$, we construct three model structures on the category of connected simplicial cocommutative $${\mathbb{F}}$$-coalgebras. The weak equivalences in this context are (1′) maps inducing a quasi-isomorphism of dg algebras after applying the cobar functor, (2′) maps inducing a quasi-isomorphism of dg algebras after applying a localized version of the cobar functor, and (3′) quasi-isomorphisms. Building on a previous work of Goerss in the context of (3)–(3′), we prove that, when $${\mathbb{F}}$$ is algebraically closed, the simplicial $${\mathbb{F}}$$-coalgebra of chains defines a homotopically full and faithful left Quillen functor for each one of these pairs of model categories. More generally, when $${\mathbb{F}}$$ is a perfect field, we compare the three pairs of model categories in terms of suitable notions of homotopy fixed points with respect to the absolute Galois group of $${\mathbb{F}}$$.
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- Award ID(s):
- 2105544
- PAR ID:
- 10547401
- Publisher / Repository:
- Oxford Academic
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2024
- Issue:
- 16
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- 11766 to 11811
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnae031
