[Paper Review] Power maps in algebra and topology

Kathryn Hess|arXiv (Cornell University)|Jun 23, 2011

Homotopy and Cohomology in Algebraic Topology17 references3 citations

TL;DR

This paper introduces the Hochschild complex of a twisting cochain as a unified algebraic framework that generalizes both Hochschild and coHochschild complexes. It constructs natural power maps on this complex for chain Hopf algebras, showing that when the algebra is cocommutative or arises from a double suspension, these maps model the topological power map on free loop spaces, providing a key algebraic tool for chain models of topological cyclic homology.

ABSTRACT

Given any twisting cochain t:C -->A, where C is a connected, coaugmented chain coalgebra and A is an augmented chain algebra over an arbitrary PID R, we construct a twisted extension of chain complexes A --> H(t) --> C. We show that both the well-known Hochschild complex of an associative algebra and the coHochschild complex of a coassociative coalgebra are special cases of H(t), which we therefore call the Hochschild complex of t. We explore the extent of the naturality of the Hochschild complex construction and apply the results of this exploration to determining conditions under which H(t) admits multiplicative or comultiplicative structure. In particular, we show that the Hochschild complex on a chain Hopf algebra always admits a natural comultiplication. Furthermore, when A is a chain Hopf algebra, we determine conditions under which H(t) admits an rth-power map extending the usual rth-power map on A and lifting the identity on C. As special cases, we obtain that both the Hochschild complex of any cocommutative Hopf algebra and the coHochschild complex of the normalized chain complex of a double suspension admit power maps. We show moreover that if K is a double suspension, then the power map on the coHochschild complex of the normalized chain complex of K is a model of the topological power map on the space of free loops on the realization of K, illustrating the topological relevance of our algebraic construction.

Motivation & Objective

• To unify Hochschild and coHochschild complexes under a single construction via twisting cochains.
• To define and study power maps on the Hochschild complex that extend the classical r-th power map on loop spaces.
• To establish conditions under which the Hochschild complex of a chain Hopf algebra admits natural multiplicative or comultiplicative structures.
• To show that for a double suspension X, the coHochschild complex of its normalized chain complex models the topological power map on the free loop space L(X).

Proposed method

• Construct the Hochschild complex H(t) as a twisted extension of chain complexes from a twisting cochain t:C→A, where C is a coaugmented coalgebra and A an augmented algebra over a PID R.
• Show that H(t) generalizes both the Hochschild complex of an algebra and the coHochschild complex of a coalgebra.
• Prove that when A is a chain Hopf algebra, H(t) naturally admits a comultiplication.
• Establish sufficient conditions for H(t) to carry an r-th power map extending the standard power map on A and lifting the identity on C.
• Define a small chain complex fls*(X) for simply connected double suspensions X, equipped with an endomorphism λ̃_r modeling the topological power map on L(X).
• Use simplicial and homotopical constructions to relate the algebraic power map on fls*(X) to the topological power map on free loops.

Experimental results

Research questions

• RQ1 Under what conditions does the Hochschild complex of a twisting cochain admit a comultiplicative structure?
• RQ2 When does the Hochschild complex support an r-th power map extending the classical power map on the base algebra and lifting the identity on the coalgebra?
• RQ3 How can the algebraic construction of the Hochschild complex model the topological power map on the free loop space of a double suspension?
• RQ4 What is the relationship between the coHochschild complex of the normalized chain complex of a double suspension and the topological power map on its free loop space?
• RQ5 In what sense is the algebraic power map on fls*(X) a chain model of the topological power map on L(X)?

Key findings

• The Hochschild complex H(t) of any twisting cochain t:C→A admits a natural comultiplication when A is a chain Hopf algebra.
• For a cocommutative chain Hopf algebra A, the Hochschild complex H(t) admits an r-th power map extending the standard r-th power map on A and lifting the identity on C.
• When X is a simply connected double suspension, the coHochschild complex of the normalized chain complex of X models the topological power map on the free loop space L(X).
• The endomorphism λ̃_r on the small chain complex fls*(X) induces the same map on homology as the topological r-th power map on L(X), i.e., H*(λ̃_r) ≅ H*(λ_top^r).
• The construction provides a chain-level model of the topological power map, which is essential for building chain models of topological cyclic homology.
• The power map on fls*(X) is compatible with the fibration sequence structure of the free loop space, preserving the homotopical context of the topological construction.