Connected Cover Isomorphism

Martin Frankland (UIUC) Moduli spaces CMS 2011 3 / 22 Π-algebras Π-algebra ≈ graded group with additional structure which looks like the homotopy groups of a space. Martin Frankland (UIUC) Moduli spaces CMS 2011 4 / 22 Π-algebras Π-algebra ≈ graded group with additional structure which looks like the homotopy groups of a space. Definition Π: = full subcategory of the homotopy category of pointed spaces consisting of finite wedges of spheres ∨Sni, ni ≥ 1. Π-algebra: = product-preserving functor A: Πop → Set∗. Martin Frankland (UIUC) Moduli spaces CMS 2011 4 / 22 Π-algebras Π-algebra ≈ graded group with additional structure which looks like the homotopy groups of a space. Definition Π: = full subcategory of the homotopy category of pointed spaces consisting of finite wedges of spheres ∨Sni, ni ≥ 1. Π-algebra: = product-preserving functor A: Πop → Set∗