I' working on this problem, and I got stuck in the middle. Could someone give me a hand? Let be a contractible space. I'd like to show that is simply connected. I already showed that is path connected. I need to show for every , is the trivial group. So, it suffices to show every loop based at is path homotopic to the path defined by for every

Let be a loop based at fixed . is contractible implies there exists a homotopy such that and . I tried to construct a path homotopy from to using H, but I don't think mine works. I define . Then and . But this is only a homotopy, I want . I can't get it with this map.