Cubical Blakers-Massey Theorem for CDGA

Abstract

We state and prove an algebraic version of the Blakers-Massey Theorem. The Blakers-Massey Theorem is a classical result in homotopy theory that measures the obstruction to homotopy excision. It also measures how far a homotopy pushout square of topological spaces is from being a homotopy pullback. This theorem can be generalized to higher-dimensional cubical diagrams of topological spaces, where it measures how far a cubical homotopy colimit is from being a homotopy limit. This work is inspired by Quillen's rational homotopy theory, where commutative di erential graded algebras (or CDGAs for short) over the eld Q of rational numbers are algebraic models. We construct the Blakers-Massey Theorem for n-cubes of (simply connected) CDGAs, measuring how far an n-cube of CDGAs which is a homotopy limit is from being a homotopy colimit.