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In this paper, we study the algebraic loop structures on the set of Lie algebra comultiplications. More specifically, we investigate the fundamental concepts of algebraic loop structures and the set of Lie algebra comultiplications which have inversive, power-associative and Moufang properties depending on the Lie algebra comultiplications up to all the possible quadratic and cubic Lie algebra comultiplications. We also apply those notions to the rational cohomology of Hopf spaces.

We prove a necessary condition for the existence of an [math] –structure on [math] spaces, and also derive a simple proof for the finiteness of the number of [math] [math] –spaces of given rank. As a direct application, we compute a list of possible types of rank [math] [math] homotopy associative [math] –spaces.

We study the existence of essential phantom maps into co-H-spaces, motivated by Iriye's observation that every suspension space $Y$ of finite type with $H_i(Y;\QQ)\neq 0$ for some $i>1$ is the target of essential phantom maps. We show that Iriye's observation can be extended to the collection of nilpotent, finite type co-H-spaces. This work hinges on an enhanced understanding of the connections between homotopy decompositions of looped co-H-spaces and coalgebra decompositions of tensor algebras due to Grbi\`{c}, Theriault, and Wu., Comment: A conjecture left open in an earlier version of this manuscript has been resolved in this updated version

2S2nC1 n ! S 1 E 2 !2S2nC1 is the .p /th power map. This construction was carried out for any p > 3 and r > 1. We also proved in [3, Theorem 4.3] that there is an H –space structure on T2n 1 such that (1) is an H –fibration. The proof of 4.3 involved induction over the skeleton of T2n 1 and cycled through 14 steps ..a/; : : : ; .n//. The argument given for the proof of [3, Theorem 4.3(l)] contained an incorrect statement and is not valid. The purpose of this note is to supply a correct proof for 4.3(l). We will abbreviate T2n 1 as T and write T m for the m skeleton of T . Recall that W a is the collection of all spaces that are of the homotopy type of a simply connected locally finite wedge of mod p Moore spaces for a s b .

Let $p$ be an odd prime, and let $\lambda\in\mathbb{Z}$ . Consider the loop space $Y_{t}=S^{2t-1}_{(p)}$ for $t\ge1$ with $t|(p-1)$ . Then we first determine the condition for the power map $\varPhi_{\lambda}$ on $Y_{t}$ to be an $A_{p}$ -map. We next assume that $X$ is a simply connected $\mathbb{F}_{p}$ -finite $A_{p}$ -space and that $\lambda$ is a primitive $(p-1)$ st root of unity mod $p$ . Our results show that if the reduced power operations $\{\mathscr{P}^{i}\}_{i\ge1}$ act trivially on the indecomposable module $QH^{*}(X;\mathbb{F}_{p})$ and the power map $\varPhi_{\lambda}$ on $X$ is an $A_{n}$ -map with $n\gt (p-1)/2$ , then $X$ is $\mathbb{F}_{p}$ -acyclic.

We prove fibrewise versions of classical theorems of Hopf and Leray-Samelson. Our results imply the fibrewise H-triviality after rationalization of a certain class of fibrewise H-spaces. They apply, in particular, to universal adjoint bundles. From this, we may retrieve a result of Crabb and Sutherland [Proc. London Math. Soc. (2000), 747-768], which is used there as a crucial step in establishing their main finiteness result., Comment: 21 pages

Let $X$ be a 2-connected $p$-local finite $H$-space with a single cell in dimension three. We give a simple cohomological criterion which distinguishes when the inclusion i: $S^3 \underset {\longrightarrow}{i} X$ has the property that the loop of its three-connected cover is null homotopic. In particular, such a null homotopy implies that $\pi_m(i )= 0$ for $m \geq 4$. Applications are made to Harper's rank 2 finite $H$-space and simple, simply-connected, compact Lie groups.

Let \zeta be an n-dimensional complex matrix bundle over a compact metric space X and let A_\zeta denote the C*-algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UA_\zeta, the group of unitaries of A_\zeta. The answer turns out to be independent of the bundle \zeta and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X., Comment: Final version. To appear in J. of Topology and Analysis. Garbled text in abstract removed

We study a universal property of $Sp(2)$ in the category of $3$-local homotopy associative, homotopy commutative $H$-spaces. We show that while $Sp(2)$ fails to be universal in the full category, there is a subcategory in which it is universal for its $7$-skeleton.

In this note we prove that the [math] –connected component sheaf [math] of an [math] –group [math] is [math] –invariant.

We study what happens when coherence fails. Categories with a tensor product and a natural associativity isomorphism that does not necessarily satisfy the pentagon coherence requirements (called associative categories) are considered. Categorical versions of associahedra where naturality squares commute and pentagons do not, are constructed (called Catalan groupoids, $\An$). These groupoids are used in the construction of the free associative category. They are also used in the construction of the theory of associative categories (given as a 2-sketch). Generators and relations are given for the fundamental group, $\pi(\An)$, of the Catalan groupoids -- thought of as a simplicial complex. These groups are shown to be more than just free groups. Each associative category, $B$, has related fundamental groups $\pi(\bf B_n)$ and homomorphisms $\pi(P_n):\pi(\An) \longrightarrow \pi(\bf B_n)$. If the images of the $\pi(P_n)$ are trivial, i.e. there is only one associativity path between any two objects, then the category is coherent. Otherwise the images of $\pi(P_n)$ are obstructions to coherence. Some progress is made in classifying noncoherence of associative categories., Comment: 51 pages; LaTeX with XY-Pic diagrams

In this note we show that, for a map of (weak or ordinary) $k$-fold homotopy coalgebras and under certain dimension and connectivity restrictions, the homotopy coalgebrastructure is inherited by the homotopy cofibre of the given map.

Associated to a discrete group $G$, one has the topological category of finite dimensional (unitary) $G$-representations and (unitary) isomorphisms. Block sums provide this category with a permutative structure, and the associated $K$-theory spectrum is Carlsson's deformation $K$-theory of G. The goal of this paper is to examine the behavior of this functor on free products. Our main theorem shows the square of spectra associated to $G*H$ (considered as an amalgamated product over the trivial group) is homotopy cartesian. The proof uses a general result regarding group completions of homotopy commutative topological monoids, which may be of some independent interest., 32 pages, 1 figure. Final version: The title has changed, and the paper has been substantially revised to improve clarity

Cohen, Moore, and Neisendorfer's work on the odd primary homotopy theory of spheres and Moore spaces, as well as the first author's work on the secondary suspension, predicted the existence of a p-local fibration S^2n-1 --> T --> ��S^2n+1 whose connecting map is degree p^r. In a long and complex monograph, Anick constructed such a fibration for p>= 5 and r>= 1. Using new methods we give a much more conceptual construction which is also valid for p=3 and r>= 1. We go on to establish several properties of the space T., 30 pages

For a based space $X$ , we consider the group $\mathscr{E}_{\# n}(X)$ of all self homotopy classes $\alpha$ of $X$ such that $\alpha_{\#} = \mathrm{id}:\pi_i(X) \to \pi_i(X)$ , for all $i\le n$ , where $n \le \infty$ , and the group $\mathscr{E}_{\Omega}(X)$ of all $\alpha $ such that $\Omega \alpha = \mathrm{id}$ . Analogously, we study the semigroups $\mathscr{Z}_{\# n}(X)$ and $\mathscr{Z}_{\varOmega}(X)$ defined by replacing ' $\mathrm{id}$ ' by ' $0$ ' above. There is a chain of containments of the $\mathscr{E}$ -groups and the $\mathscr{Z}$ -semigroups, and we discuss examples for which the containment is proper. We then obtain various conditions on $X$ which ensure that the $\mathscr{E}$ -groups and the $\mathscr{Z}$ -semigroups are equal. When $X$ is a group-like space, we derive lower bounds for the order of these groups and their localizations. In the last section we make specific calculations for the $\mathscr{E}$ -groups and $\mathscr{Z}$ -groups of certain low dimensional Lie groups.

Assume that all spaces and maps are localised at a fixed prime $p$. We study the possibility of generating a universal space $U(X)$ from a space $X$ which is universal in the category of homotopy associative, homotopy commutative H-spaces in the sense that any map f:X->Y to a homotopy associative, homotopy commutative H-space extends to a uniquely determined H-map F: U(X)->Y. Developing a method for recognising certain universal spaces, we show the existence of the universal space F_2(n) of a certain three-cell complex L. Using this specific example, we derive some consequences for the calculation of the unstable homotopy groups of spheres, namely, we obtain a formula for the d_1-differential of the EHP-spectral sequence valid in a certain range., 29 pages

In [BG], it is proved that the Whitehead length of a space $Z$ is less than or equal to the nilpotency of $\Omega Z$ . As for rational spaces, those two invariants are equal. We show this for a 1-connected rational space $Z$ by giving a way to calculate those invariants from a minimal model for $Z$ . This also gives a way to calculate the nilpotency of an homotopy associative rational $H$ -space.

Stasheff showed that if a map between H-spaces is an H-map, then the suspension of the map is extendable to a map between cprojective planes of the H-spaces. Stahseff also proved the converse under the assumption that the multiplication of the target space of the map is homotopy associative. We show by giving an example that the assumption of homotopy associativity of the multiplication of the target space is necessary to show the converse. We also show an analogous fact for maps between higher homotopy associative H-spaces., 6 pages

We characterize Hopf spaces with finitely generated cohomology as an algebra over the Steenrod algebra. We "deconstruct" the original space into an H-space Y with finite mod p cohomology and a finite number of p-torsion Eilenberg-Mac Lane spaces. We give a precise description of homotopy commutative H-spaces in this setting., 17 pages, Section 6 replaces an incorrect proof of former Proposition 6.2

If [math] is a collection of topological spaces, then a homotopy class [math] in [math] is called [math] –trivial if ¶ α ∗ = 0 : [ A , X ] → [ A , Y ] ¶ for all [math] . In this paper we study the collection [math] of all [math] –trivial homotopy classes in [math] when [math] , the collection of spheres, [math] , the collection of Moore spaces, and [math] , the collection of suspensions. Clearly ¶ Z Σ ( X , Y ) ⊆ Z ℳ ( X , Y ) ⊆ Z § ( X , Y ) , ¶ and we find examples of finite complexes [math] and [math] for which these inclusions are strict. We are also interested in [math] , which under composition has the structure of a semigroup with zero. We show that if [math] is a finite dimensional complex and [math] , [math] or [math] , then the semigroup [math] is nilpotent. More precisely, the nilpotency of [math] is bounded above by the [math] –killing length of [math] , a new numerical invariant which equals the number of steps it takes to make [math] contractible by successively attaching cones on wedges of spaces in [math] , and this in turn is bounded above by the [math] –cone length of X. We then calculate or estimate the nilpotency of [math] when [math] , [math] or [math] for the following classes of spaces: (1) projective spaces (2) certain Lie groups such as [math] and [math] . The paper concludes with several open problems.

In this paper we derive a formula about the action of $\mathscr{P}^{1}$ on the odd generators of the mod 3 cohomology of a finite simply connected mod 3 $H$-space with associative mod 3 homology. This formula will be used in a subsequent paper to classify all possible cohomology rings that can occur as the mod 3 cohomology of such $H$-spaces.

We prove that if $X$ is aconnected $H$ -space with at most three cells of positive dimension, then the self homotopy set of $X$ becomes a group relative to the binary operation induced from any multiplication on $X$ , and we determine it's group structure in some cases.

Given an $H$-map $i : Y \to X$, we say that $i$ is mod $p$ homotopy normal if the commutator map from $X_{(p)}\times Y_{(9)}$ to $X_{(p)}$ can be deformed into $Y_{(p)}$. In this paper, we study necessary conditions of mod $p$ homotopy normality for the cases that $X$ are exceptional Lie groups with odd torsion in the cohomology, by using the Morava K-theory.