Algebraic invariants for homotopy types

We define inductively a sequence of purely algebraic invariants - namely, classes in the Quillen cohomology of the Pi-algebra \pi_* X - for distinguishing between different homotopy types of spaces. Another sequence of such cohomology classes allows one to decide whether a given abstract Pi-algebra can be realized as the homotopy Pi-algebra of a space in the first place. The paper is written for a relatively general "resolution model category", so it also applies, for example, to rational homotopy types

A Framework for Analyzing Tariffs and Subsidies in Water Provision to Urban Households in Developing Countries

This paper aims to present a basic conceptual framework for understanding the main practical issues and challenges relating to tariffs and subsidies in the water sector in developing countries. The paper introduces the basic economic notions relevant to the water sector; presents an analytical framework for assessing the need for and evaluating subsidies; and discusses the recent evidence on the features and performance of water tariffs and subsidies in various regions, with a special focus on Africa. The discussion is limited to the provision of drinking water to urban households in developing countries.water, access to water, tariffs, subsidies, urban development

Loop spaces and homotopy operations

The question of whether a given H-space X is, up to homotopy, a loop space has been studied from a variety of viewpoints. Here we address this question from the aspect of homotopy operations, in the classical sense of operations on homotopy groups. First, we show how an H-space structure on X can be used to define the action of the primary homotopy operations on the shifted homotopy groups \pi_{*-1} X (which are isomorphic to \pi_* Y, if X=\Omega\Y. This action will behave properly with respect to composition of operations if X is homotopy-associative, and will lift to a topological action of the monoid of all maps between spheres if and only if X is a loop space. The obstructions to having such a topological action may be formulated in the framework of an obstruction theory for realizing \Pi-algebras, which is simplified here by showing that any (suitable) \Delta-simplicial space may be made into a full simplicial space (a result which may be useful in other contexts)

On realizing diagrams of Pi-algebras

Given a diagram of Pi-algebras (graded groups equipped with an action of the primary homotopy operations), we ask whether it can be realized as the homotopy groups of a diagram of spaces. The answer given here is in the form of an obstruction theory, of somewhat wider application, formulated in terms of generalized Pi-algebras. This extends a program begun in [J. Pure Appl. Alg. 103 (1995) 167-188] and [Topology 43 (2004) 857-892] to study the realization of a single Pi-algebra. In particular, we explicitly analyze the simple case of a single map, and provide a detailed example, illustrating the connections to higher homotopy operations.Comment: This is the version published by Algebraic & Geometric Topology on 21 June 200