  We study the class of independence complexes of claw-free graphs. The main
theorem give good bounds on the connectivity of these complexes, given bounds
for a few subcomplexes of the same class. Two applications are presented.
Firstly, we show that the independence complex of a claw-free graph with n
vertices and maximal degree d is (cn/d+epsilon)-connected, where c=2/3. This
can be compared with the result of Szabo and Tardos that c=1/2 is optimal with
no restrictions on the graphs. Secondly, we calculate the connectivity of a
family of complexes used in Babson and Kozlov's proof of Lovasz conjecture.
