Module Structure on Lie Powers and Natural Coalgebra-Split Sub Hopf Algebras of Tensor Algebras

Lie coalgebras and rational homotopy theory II: Hopf invariants

We give a new solution of the "homotopy periods" problem, as highlighted by Sullivan, which places explicit geometrically meaningful formulae first dating back to Whitehead in the context of Quillen's formalism for rational homotopy theory and Koszul-Moore duality. Geometrically, we show that homotopy groups are rationally given by "generalized linking/intersection invariants" of cochain data. Moreover, we give a method for determining when two maps from $S^n$ to $X$ are homotopic after allowing for multiplication by some integer. For applications, we investigate wedges of spheres and homogene…

The $mod2$ Steenrod and Dyer-Lashof algebras as quotients of a free algebra

A non-connected neither of finite type Hopf algebra $\mathcal{F}_{0}$ is defined over $\mathbb{Z}/ 2\mathbb{Z}$ and its hom dual turns out to be a tensor product of polynomial algebras. Certain quotient Hopf algebras include the Steenrod and Dyer-Lashof algebras. This setting provides a map between the Steenrod coalgebra and a direct limit of Dyer-Lashof coalgebras.

Homotopy Algebra Structures on Twisted Tensor Products and String Topology Operations

Given a $C_\infty$ coalgebra $C_*$, a strict dg Hopf algebra $H_*$, and a twisting cochain $τ:C_* \rightarrow H_*$ such that $Im(τ) \subset Prim(H_*)$, we describe a procedure for obtaining an $A_\infty$ coalgebra on $C_* \otimes H_*$. This is an extension of Brown's work on twisted tensor products. We apply this procedure to obtain an $A_\infty$ coalgebra model of the chains on the free loop space $LM$ based on the $C_\infty$ coalgebra structure of $H_*(M)$ induced by the diagonal map $M \rightarrow M \times M$ and the Hopf algebra model of the based loop space given by $T(H_*(M)[-1])$. When …

Koszul duality in algebraic topology - an historical perspective

We survey the topology which led to the original bar and cobar constructions, for both associative algebras and coalgebras and for Lie algebras and commutative coalgebras. These constructions are often viewed as part of the larger theory of Koszul duality of operads, so this survey is meant to offer an historical perspective on the most prominent cases of that theory. We also explain recent work which shows that Hopf/linking invariants for homotopy are at the heart of the duality between commutative algebras and Lie coalgebras.

Applications of combinatorial groups to Hopf invariant and the exponent problem

Combinatorial groups together with the groups of natural coalgebra transformations of tensor algebras are linked to the groups of homotopy classes of maps from the James construction to a loop space. This connection gives rise to applications to homotopy theory. The Hopf invariants of the Whitehead products are studied and a rate of exponent growth for the strong version of the Barratt Conjecture is given.

同伦纤维闭路空间同调的代数模型

An algebraic model for the loop space homology of a homotopy fiber

设F表示2-约化单纯集映射f：K->L的同伦纤维.以K和L的链复形的强同伦余代数结构为输入数据，构造了一个小的显式链代数，它的同调同构于F上的单纯（Kan）环群GF的同调为分次代数.为了构建这个模型，我们开发了机器的链霍普夫代数的态射的同伦纤维建模。我们的建设是必不可少的是一个推广的操作描述范畴DCSH链余代数和强同伦余代数映射（数学。AT/0505559）强同伦态射的余模在霍普夫代数。这种操作性的描述表示在一个一般的理论monoidal结构的范畴与态射集参数化的co-rings，我们在这里阐述。