arXiv subjects

Paul-Eugene Parent

Publications and source records attributed to Paul-Eugene Parent.

5 recordsLinked to original sources

arXiv2026

Homotopical Nilpotency and Homology Nilpotency: A Dimension--Connectivity Bound

Let $X$ be a $(q-1)$-connected rational space of finite type, where $q\ge2$, with homotopical nilpotency $nil_h(X)=n\ge1$. If $H^{>N}(X;Q)=0$ and $N\le q(n+3)-3$, we prove that $Hnil(X)=n$. More precisely, the sufficient bound is $N\le a_n(M_X)+2q-3$, where $a_n(M_X)$ is the first nonzero internal cohomology degree of $(M_X^+)^{n+1}$ in the minimal model. A family of examples proves that the constant $-3$ in this refined bound is optimal; optimality of the uniform bound is not asserted. The proof uses two stages of adjoining primitives to an ideal of the fixed minimal model.

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arXiv2010

Lawrence-Sullivan models for the interval

Two constructions of a Lie model of the interval were performed by R. Lawrence and D. Sullivan. The first model uses an inductive process and the second one comes directly from solving a differential equation. They conjectured that these two models are the same. We prove this conjecture here.

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arXiv2008

CoHochschild homology of chain coalgebras

Generalizing work of Doi and of Idrissi, we define a coHochschild homology theory for chain coalgebras over any commutative ring and prove its naturality with respect to morphisms of chain coalgebras up to strong homotopy. As a consequence we obtain that if the comultiplication of a chain coalgebra $C$ is itself a morphism of chain coalgebras up to strong homotopy, then the coHochschild complex $\cohoch (C)$ admits a natural comultiplicative structure. In particular, if $K$ is a reduced simplicial set and $C_{*}K$ is its normalized chain complex, then $\cohoch (C_{*}K)$ is naturally a homotopy-coassociative chain coalgebra. We provide a simple, explicit formula for the comultiplication on $\cohoch (C_{*}K)$ when $K$ is a simplicial suspension. The coHochschild complex construction is topologically relevant. Given two simplicial maps $g,h:K\to L$, where $K$ and $L$ are reduced, the homology of the coHochschild complex of $C_{*}L$ with coefficients in $C_{*}K$ is isomorphic to the homology of the homotopy coincidence space of the geometric realizations of $g$ and $h$, and this isomorphism respects comultiplicative structure. In particular, there a isomorphism, respecting comultiplicative structure, from the homology of $\cohoch(C_{*}K)$ to $H_{*}\op L|K|$, the homology of the free loops on the geometric realization of $K$.

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arXiv2006

A chain coalgebra model for the James map

Let EK be the simplicial suspension of a pointed simplicial set K. We construct a chain model of the James map, $α_{K} : CK \to ΩCEK$. We compute the cobar diagonal on $ΩCEK$, not assuming that $EK$ is 1-reduced, and show that $α_{K}$ is comultiplicative. As a result, the natural isomorphism of chain algebras $TCK \cong ΩCK$ preserves diagonals. In an appendix, we show that the Milgram map, $Ω(A \otimes B) \to ΩA \otimes ΩB$, where A and B are coaugmented coalgebras, forms part of a strong deformation retract of chain complexes. Therefore, it is a chain equivalence even when A and B are not 1-connected.

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arXiv2006

Co-rings over operads characterize morphisms

Let M be a bicomplete, closed symmetric monoidal category. Let P be an operad in M, i.e., a monoid in the category of symmetric sequences of objects in M, with its composition monoidal structure. Let R be a P-co-ring, i.e., a comonoid in the category of P-bimodules. The co-ring R induces a natural ``fattening'' of the category of P-(co)algebras, expanding the morphism sets while leaving the objects fixed. Co-rings over operads are thus ``relative operads,'' parametrizing morphisms as operads parametrize (co)algebras. Let A denote the associative operad in the category of chain complexes. We define a ``diffracting'' functor that produces A-co-rings from symmetric sequences of chain coalgebras, leading to a multitude of ``fattened'' categories of (co)associative chain (co)algebras. In particular, we obtain a purely operadic description of the categories DASH and DCSH first defined by Gugenheim and Munkholm, via an A-co-ring that has the two-sided Koszul resolution of A as its underlying A-bimodule. The diffracting functor plays a crucial role in enabling us to prove existence of higher, ``up to homotopy'' structure of morphisms via acyclic models methods. It has already been successfully applied in this sense in a number of recent articles and preprints.

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