nLab integral homotopy theory

Idea

Integral homotopy theory is the refinement of rational homotopy theory to integer coefficients.

Just as there are two established approaches in rational homotopy theory for encoding rational homotopy types, those of Quillen and of Sullivan, so there are analog approaches for integral homotopy types. In (Blomquist & Harper 16) chains in ordinary cohomology with rational number coefficients are lifted to chains with integer coefficients. While (Horel 22) and (Yuan 23) both employ cochains.

References

• Jacobson Blomquist, John Harper, An integral chains analog of Quillen’s rational homotopy theory equivalence (arXiv:1611.04157)

Analogs of Sullivan’s rational homotopy theory equivalence are in

• Geoffroy Horel, Binomial rings and homotopy theory (arXiv:2211.02349)

• Allen Yuan, Integral models for spaces via the higher Frobenius. Journal of the American Mathematical Society, 36(1):107–175, 2023, (arXiv:1910.00999).