The Chow variety of effective algebraic p-cycles of degree d in complex projective n-space sits inside the space of all effective p-cycles, and that inclusion turns out to be 2d-connected: below degree 2d the two spaces have the same homotopy groups, so bounding the degree of a cycle costs nothing topologically until dimension 2d. A student learns how a codimension estimate on a family of divisors lets one deform any sphere of cycles into a subspace where suspension can be inverted, and how the resulting isomorphism reads the first 2d+1 homotopy and homology groups of the Chow variety off the known homotopy type of the full cycle space as a product of Eilenberg-MacLane spaces. The same argument survives base change to an algebraically closed field of any characteristic once the analytic topology is replaced by the etale topological type, giving l-adic etale homotopy groups in the same range.
The Chow variety C_{p,d}(P^n) is the compact algebraic set whose points are the effective algebraic p-dimensional cycles of degree d in complex projective n-space; adding a fixed p-plane embeds each …