We uncover a somewhat surprising connection between spaces of multiplicative maps between $A_\infty$-ring spectra and topological Hochschild cohomology. As a consequence we show that such spaces become infinite loop spaces after looping only once. We also prove that any multiplicative cohomology operation in complex cobordisms theory $MU$ canonically lifts to an $A_\infty$-map $MU\to MU$. This implies, in particular, that the Brown-Peterson spectrum $BP$ splits off $MU$ as an $A_\infty$-ring spectrum.
|Translated title of the contribution||Spaces of multiplicative maps between highly structured ring spectra|
|Title of host publication||Categorical Decomposition Techniques in Algebraic Topology, International Conference in Algebraic Topology, Isle of Skye, Scotland, June 2001|
|Editors||Arone , G.; Hubbuck, J.; Levi, M R.; Weiss|
|Pages||237 - 259|
|Publication status||Published - 2004|