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[Music] welcome to the course organized by the international center of applied and pure mathematics the course of the lecture is titled semi-free resolution and application the aims of this course is focus on the proof of the johns theorem where this denote the hostile chromology of the normalized angular cushion x and this denote the pre-loops chromology space indeed what we want to prove here is to one what one want to do here is to give a new proof of this to them by doing the by using the semi-free resolution material so it is well known that the hostile chromology that for a given x a to political space a topological space if x please denote the normalize you normalize singular cushion and l x the free loop space then the hostile chromology russian homology sorry of nx is isomorphic to the island motor of it is also known that the free loops is isomorphic tools and x as n a module and this is the island by motor of nx as n times n of x times x so what we are going to prove is to
