Tame Homotopy Theory

space is said to be rational if its homotopy groups are rational vector spaces. Quillen has shown that up to homotopy there is a one-one correspondence between rational spaces and differential graded Lie algebras over 0. Call a two-connected space tame if the divisibility of its homotopy groups increases with dimension just quickly enough to prevent stable k-invariants from appearing. We will show that up to homotopy there is a one-one correspondence between tame spaces and differential graded Lie algebras over Z. In more detail, let r be a fixed positive integer (the connectivity plus one) and assume r ZT 3 (see 1.5). For each k r0 let Tk be the smallest subring of gP containing I/p for each prime p such that 2p-3 5 k; in other words, Tk contains l/p only if the existence of mod p reduced power operations can affect (r-I)-connected homotopy theory in dimension r + k. An (r-I)-connected space X is said to be tame if, for each k 2 0, T,+~X is a module over Tk. There is an analogous notion of tameness for (r-I)-reduced (1.7) differential graded Lie algebras (see 07, where tame = @runt). THEOREM 1.1. The homotopy category of tame (r-l)-connected spaces (r 2 3) is equivalent to an algebraic homotopy category of tame (r-1)-reduced differential graded Lie algebras over Z. It follows from $3 that for any (r-I)-connected space X there is a tame space XT together with a map X-+XT inducing isomorphisms nr+kxT = nr+kx @ Tk (k 2 0). If X has only a finite member of non-zero homotopy groups, this map X +XT is an equivalence away from an explicit finite collection of primes, so 1.1 implies that, away from these primes, the homotopy type of X can be specified by giving a differential graded Lie algebra. how if Y is an arbitrary (r-I)-connected finite dimensional complex, the homotopy type of Y is determined by the homotopy type of any Postnikov stage P,Y for n 2 dim Y + 1, and these spaces P,,Y have only a finite number of non-zero homotopy groups. In this way it follows from 1.1 that the homotopy type of any 2-connected fir&e-dimensional complex Y can be specified, away from an explicit finite collection of primes that depends only on the difference between the dimension of Y and its connectivity, by a differential graded Lie algebra. In a later note we …