Multiplicative Structures on Homotopy Spectral Sequences, Part I

A tower of homotopy fiber sequences gives rise to a spectral sequence on homotopy groups. In modern times such towers are ubiquitous, and most of the familiar spectral sequences in topology can be constructed in this way. A pairing of towers W∗ ∧X∗ → Y∗ consists of maps Wm ∧Xn → Ym+n which commute (on-the-nose) with the maps in the towers. It is a piece of folklore that a pairing of towers gives rise to a pairing of the associated homotopy spectral sequences. This paper gives a careful proof of this general fact, for towers of spaces and towers of spectra.