Lawson, Tyler \(E_n\)-spectra and Dyer-Lashof operations. (English) Zbl 1476.55028 Miller, Haynes (ed.), Handbook of homotopy theory. Boca Raton, FL: CRC Press. CRC Press/Chapman Hall Handb. Math. Ser., 793-849 (2020). The Handbook of homotopy theory has been reviewed in [Zbl 1468.55001].Summary: Cohomology operations are absolutely essential in making cohomology an effective tool for studying spaces. In ordinary algebra, commutativity is an extremely useful property possessed by certain monoids and algebras. A multicategory encodes the structure of a category where functions have multiple input objects. They serve as a useful way to encode many multilinear structures in stable homotopy theory: multiplications, module structures, graded rings, and coherent structures on categories. The chapter considers compatibility relations between operations on different homotopy degrees. Every Adem relation between Dyer-Lashof operations produces secondary Dyer-Lashof operations. Secondary operations are part of the homotopy theory of \(\mathcal{C}\), and there is typically no method to determine secondary operations purely in terms of the homotopy category. Topological André-Quillen homology and cohomology are invariants of ring spectra developed by I. Kriz and Maria Basterra.For the entire collection see [Zbl 1468.55001]. Cited in 1 ReviewCited in 1 Document MSC: 55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) 55S12 Dyer-Lashof operations 55S20 Secondary and higher cohomology operations in algebraic topology 55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology Keywords:cohomology operations; compatibility; Adem relation; secondary operations Citations:Zbl 1468.55001 PDFBibTeX XMLCite \textit{T. Lawson}, in: Handbook of homotopy theory. Boca Raton, FL: CRC Press. 793--849 (2020; Zbl 1476.55028) Full Text: DOI