  This paper deals with links and braids up to link-homotopy, studied from the
viewpoint of Habiro's clasper calculus. More precisely, we use clasper homotopy
calculus in two main directions. First, we define and compute a faithful linear
representation of the homotopy braid group, by using claspers as geometric
commutators. Second, we give a geometric proof of Levine's classification of
4-component links up to link-homotopy, and go further with the classification
of 5-component links in the algebraically split case.
