arXiv subjects

Manuel Rivera

Publications and source records attributed to Manuel Rivera.

28 records · Page 2Linked to original sources

arXiv2021

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

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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arXiv2020

Rational homotopy equivalences and singular chains

Bousfield and Kan's $\mathbb{Q}$-completion and fiberwise $\mathbb{Q}$-completion of spaces lead to two different approaches to the rational homotopy theory of non-simply connected spaces. In the first approach, a map is a weak equivalence if it induces an isomorphism on rational homology. In the second, a map of connected and pointed spaces is a weak equivalence if it induces an isomorphism between fundamental groups and higher rationalized homotopy groups; we call these maps $π_1$-rational homotopy equivalences. In this paper, we compare these two notions and show that $π_1$-rational homotopy equivalences correspond to maps that induce $Ω$-quasi-isomorphisms on the rational singular chains, i.e. maps that induce a quasi-isomorphism after applying the cobar functor to the dg coassociative coalgebra of rational singular chains. This implies that both notions of rational homotopy equivalence can be deduced from the rational singular chains by using different algebraic notions of weak equivalences: quasi-isomorphism and $Ω$-quasi-isomorphisms. We further show that, in the second approach, there are no dg coalgebra models of the chains that are both strictly cocommutative and coassociative.

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arXiv2020

Singular chains and the fundamental group

We show that the natural algebraic structure of the singular chains on a path connected topological space determines the fundamental group functorially. Moreover, we describe a notion of weak equivalence for the relevant algebraic structure under which the data of the fundamental group is preserved.

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arXiv2019

The colimit of an $\infty$-local system as a twisted tensor product

We describe several equivalent models for the infinity-category of infinity-local systems of chain complexes over a space using the framework of quasi-categories. We prove that the given models are equivalent as infinity-categories by exploiting the relationship between the differential graded nerve functor and the cobar construction. We use one of these models to calculate the quasi-categorical colimit of an infinity-local system in terms of a twisted tensor product.

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arXiv2019

The functor of singular chains detects weak homotopy equivalences

The normalized singular chains of a path connected pointed space $X$ may be considered as a connected $E_{\infty}$-coalgebra $\mathbf{C}_*(X)$ with the property that the $0^{\text{th}}$ homology of its cobar construction, which is naturally a cocommutative bialgebra, has an antipode, i.e. it is a cocommutative Hopf algebra. We prove that a continuous map of path connected pointed spaces $f: X\to Y$ is a weak homotopy equivalence if and only if $\mathbf{C}_*(f): \mathbf{C}_*(X)\to \mathbf{C}_*(Y)$ is an $\mathbfΩ$-quasi-isomorphism, i.e. a quasi-isomorphism of dg algebras after applying the cobar functor $\mathbfΩ$ to the underlying dg coassociative coalgebras. The proof is based on combining a classical theorem of Whitehead together with the observation that the fundamental group functor and the data of a local system over a space may be described functorially from the algebraic structure of the singular chains.

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arXiv2018

Cubical rigidification, the cobar construction, and the based loop space

We prove the following generalization of a classical result of Adams: for any pointed and connected topological space $(X,b)$, that is not necessarily simply connected, the cobar construction of the differential graded (dg) coalgebra of normalized singular chains in $X$ with vertices at $b$ is weakly equivalent as a differential graded associative algebra (dga) to the singular chains on the Moore based loop space of $X$ at $b$. We deduce this statement from several more general categorical results of independent interest. We construct a functor $\mathfrak{C}_{\square_c}$ from simplicial sets to categories enriched over cubical sets with connections which, after triangulation of their mapping spaces, coincides with Lurie's rigidification functor $\mathfrak{C}$ from simplicial sets to simplicial categories. Taking normalized chains of the mapping spaces of $\mathfrak{C}_{\square_c}$ yields a functor $Λ$ from simplicial sets to dg categories which is the left adjoint to the dg nerve functor. For any simplicial set $S$ with $S_0=\{x\}$, $Λ(S)(x,x)$ is a dga isomorphic to $ΩQ_Δ(S)$, the cobar construction on the dg coalgebra $Q_Δ(S)$ of normalized chains on $S$. We use these facts to show that $Q_Δ$ sends categorical equivalences between simplicial sets to maps of connected dg coalgebras which induce quasi-isomorphisms of dga's under the cobar functor.

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arXiv2018

A combinatorial model for the path fibration

We introduce the abstract notion of a necklical set in order to describe a functorial combinatorial model of the path fibration over the geometric realization of a path connected simplicial set. In particular, to any path connected simplicial set $X$ we associate a necklical set $\widehat{\mathbfΩ}X$ such that its geometric realization $|\widehat{\mathbfΩ}X|$, a space built out of gluing cubical cells, is homotopy equivalent to the based loop space on $|X|$ and the differential graded module of chains $C_*(\widehat{\mathbfΩ}X)$ is a differential graded associative algebra generalizing Adams' cobar construction.

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arXiv2018

A combinatorial model for the free loop fibration

We introduce the abstract notion of a closed necklical set in order to describe a functorial combinatorial model of the free loop fibration $ΩY\rightarrow ΛY\rightarrow Y$ over the geometric realization $Y=|X|$ of a path connected simplicial set $X.$ In particular, to any path connected simplicial set $X$ we associate a closed necklical set $\widehat{\mathbfΛ}X$ such that its geometric realization $|\widehat{\mathbfΛ}X|$, a space built out of gluing "freehedrical" and "cubical" cells, is homotopy equivalent to the free loop space $ΛY$ and the differential graded module of chains $C_*(\widehat{\mathbfΛ}X)$ generalizes the coHochschild chain complex of the chain coalgebra $C_\ast(X).$

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arXiv2018

Singular Hochschild cohomology and algebraic string operations

Given a differential graded (dg) symmetric Frobenius algebra $A$ we construct an unbounded complex $\mathcal{D}^{*}(A,A)$, called the Tate-Hochschild complex, which arises as a totalization of a double complex having Hochschild chains as negative columns and Hochschild cochains as non-negative columns. We prove that the complex $\mathcal{D}^*(A,A)$ computes the singular Hochschild cohomology of $A$. We construct a cyclic (or Calabi-Yau) $A$-infinity algebra structure, which extends the classical Hochschild cup and cap products, and an $L$-infinity algebra structure extending the classical Gerstenhaber bracket, on $\mathcal{D}^*(A,A)$. Moreover, we prove that the cohomology algebra $H^*(\mathcal{D}^*(A,A))$ is a Batalin-Vilkovisky (BV) algebra with BV operator extending Connes' boundary operator. Finally, we show that if two Frobenius algebras are quasi-isomorphic as dg algebras then their Tate-Hochschild cohomologies are isomorphic and we use this invariance result to relate the Tate-Hochschild complex to string topology.

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arXiv2012

Automorphism Groups of Countably Categorical Linear Orders are Extremely Amenable

We show that the automorphism groups of countably categorical linear orders are extremely amenable. Using methods of Kechris, Pestov, and Todorcevic, we use this fact to derive a structural Ramsey theorem for certain families of finite ordered structures with finitely many partial equivalence relations with convex classes.

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