Weinberger's Computers, rigidity, and moduli: the large-scale fractal geometry of Riemannian moduli space contains several apparently useful references on pages 93-4, in the notes section of the chapter on designer homology spheres (which you may also find of interest). Weinberger mentions "the algorithmic nature of simply connected homotopy theory" and cites the paper of Brown that Mike mentioned before going on to cite Sullivan's "Infinitesimal computations in topology." Pub. Math. IHÉS, 47 269 (1977), Griffiths and Morgan's Rational Homotopy Theory and Differential Forms, Halperin's "Lectures on minimal models." Mém. Soc. Math. France, Sér. 2, 9-10 1 (1983), and Dwyer's "Tame Homotopy Theory." Topology 18 321 (1979).

Weinberger's Computers, rigidity, and moduli: the large-scale fractal geometry of Riemannian moduli space contains several apparently useful references on pages 93-4, in the notes section of the chapter on designer homology spheres (which you may also find of interest). Weinberger mentions "the algorithmic nature of simply connected homotopy theory" and cites the paper of Brown that Mike mentioned before going on to cite Sullivan's "Infinitesimal computations in topology." Pub. Math. IHÉS, 47 269 (1977), Griffiths and Morgan's Rational Homotopy Theory and Differential Forms, Halperin's "Lectures on minimal models." Mém. Soc. Math. France, Sér. 2, 9-10 1 (1983), and Dwyer's "Tame Homotopy Theory." Topology 18 321 (1979).

The practical upshot of these later references seems to be the calculation of $\pi_k(S^n) \otimes \mathbb{Q}$, or in the case of tame homotopy theory the analogous object involving a finite number of primes (which number increases with dimension).