Lusternik-Schnirelmann category and minimal coverings with contractible sets (Q1101996)

Let gcat(X) be the smallest number of contractible sets which cover a space X. The authors show that gcat(X) is not a homotopy invariant and the difference between gcat(X) and Lyusternik-Shnirelman category of X may be arbitrarily large. Theorem. For every positive integer n there are spaces \(K_ n\), \(X_ n\) and \(Y_ n\) with two points \(y_ 1,y_ 2\in Y_ n\) such that a) \(gcat(K_ n)-cat(K_ n)\geq n,\) b) \(gcat(X_ n)-gcat(X_ n\times [0,1])\geq n,\) c) \(gcat(Y_ n/\{y_ 1,y_ 2\})-gcat(Y_ n)\geq n\).