Rational homotopy theory

1 The Sullivan model 1.1 Rational homotopy theory of spaces We will restrict our attention to simply-connected spaces. Much of this goes through with nilpotent spaces, but this will keep things easier and less technical. Definition 1. A 1-connected space X is said to be rational if either of the following equivalent conditions holds: 1. π∗X forms a graded Q-vector space. 2. H̃∗X forms a graded Q-vector space. (If no mention of coefficients is made, we are using integral coefficients.) Remark 1. The equivalence of the conditions can be proved first as follows. If p is prime, we haveH∗(K(Q, 1);Fp) ∼= H∗(pt;Fp). An inductive spectral sequence argument shows that this holds for all K(Q, n). Then for arbitrary X, we consider a Postnikov decomposition and argue inductively up the tower. Definition 2. A rationalization of a 1-connected space X is a map φ : X → X0 such that X0 is rational and π∗φ⊗Q : π∗X ⊗Q→ π∗X0 ⊗Q ∼= π∗X0 is an isomorphism. Rationalizations always exist. For a CW-complex X, there is an explicit construction. It is easy to see how to rationalize a sphere by a telescope construction, and then we build X0 out of rationalized spheres and discs corresponding to those in the decomposition of X. Theorem. For any 1-connected space X, there is a relative CW-complex (X0, X) with no 0or 1-cells where X0 is 1-connected and rational, such that the inclusion j : X ↪→ X0 is a rationalization. Moreover, this has the following universal property: if Y is 1-connected and rational, then there is a map f̃ : X0 → Y which is unique up to homotopy that makes the diagram