On the Capacity of Eilenberg-MacLane and Moore Spaces

TL;DR

This paper investigates the capacity of Moore and Eilenberg-MacLane spaces, providing explicit calculations for various constructions and answering some of Borsuk's longstanding questions on the properties of capacity in compacta.

Contribution

It offers new explicit formulas for the capacity of Moore and Eilenberg-MacLane spaces, including wedge sums and products, advancing understanding of their homotopy domination properties.

Findings

Capacity of Moore spaces $M(A,n)$ computed

Capacity of Eilenberg-MacLane spaces $K(G,n)$ determined

Capacity of wedge sums and products of these spaces calculated

Abstract

K. Borsuk in 1979, in the Topological Conference in Moscow, introduced the concept of the capacity of a compactum and asked some questions concerning properties of the capacity of compacta. In this paper, we give partial positive answers to three of these questions in some cases. In fact, by describing spaces homotopy dominated by Moore and Eilenberg-MacLane spaces, we obtain the capacity of a Moore space $M(A,n)$ and an Eilenberg-MacLane space $K(G,n)$. Also, we compute the capacity of the wedge sum of finitely many Moore spaces of different degrees and the capacity of the product of finitely many Eilenberg-MacLane spaces of different homotopy types. In particular, we give exact capacity of the wedge sum of finitely many spheres of the same or different dimensions.