Chains on suspension spectra

Abstract

Some background is needed to understand why this functor deserves attention. There is a much simpler functor called N (normalised simplicial chains) from simplicial sets to integral chain complexes that computes integral homology, and one can just tensor with Q to compute rational homology. There is a dual complex N that calculates integral cohomology. This is equipped with a natural product N .X / N .X /! N .X / which is commutative up to homotopy but not on the nose. The theory of Steenrod operations shows that if we work integrally then neither N .X / nor any reasonable replacement can be given a strictly commutative product (even with the usual signs). Rationally, however, the situation is better: in [10] Sullivan developed a rational and simplicial version of de Rham theory giving a cochain complex  .X / with a strictly commutative product that computes the ordinary rational cohomology of X . This can be used as a starting point for the rich and powerful theory of rational homotopy (originally introduced by Quillen [8] using slightly different machinery). One can then stabilise and consider the category SQ of rational spectra, which makes things considerably simpler: it is well-known that the homotopy category of SQ is equivalent to the category of graded rational vector spaces. However, we can make things harder again by considering rational spectra with a ring structure or a group action. To handle these, we need to improve the homotopy classification of rational spectra to some kind of monoidal Quillen equivalence of SQ with a suitable model category ChQ of rational chain complexes.