Homotopical depth and a conjecture of Grothendieck

Let (X,Y) be a polyhedral pair (see conventions) such that X is second coun table. We give a relation between the homotopical depth of X along Y and th e degree of connectivity of the pair (X, X \ Y) (a conjecture of Grothendie ck). In particular, we obtain a relation between the rectified homotopical depth and the global rectified homotopical depth of a complex analytic spac e. (C) 2000 Editions scientifiques et medicales Elsevier SAS.