Operadic multiplications in equivariant spectra, norms, and transfers

By Andrew Justin Blumberg, M. Hill. 2013

154 citations recorded by the source corpus.

We study homotopy-coherent commutative multiplicative structures on equivariant spaces and spectra. We define N-infinity operads, equivariant generalizations of E-infinity operads. Algebras in equivariant spectra over an N-infinity operad model homotopically commutative equivariant ring spectra that only admit certain collections of Hill-Hopkins-Ravenel norms, determined by the operad. Analogously, algebras in equivariant spaces over an N-infinity operad provide explicit constructions of certain transfers. This characterization yields a conceptual explanation of the structure of equivariant infinite loop spaces. To explain the relationship between norms, transfers, and N-infinity operads, we discuss the general features of these operads, linking their properties to families of finite sets with group actions and analyzing their behavior under norms and geometric fixed points. A surprising consequence of our study is that in stark contract to the classical setting, equivariantly the little disks and linear isometries operads for a general universe $U$ need not determine the same algebras. Our work is motivated by the need to provide a framework to describe the flavors of commutativity seen in recent work of the second author and Hopkins on localization of equivariant commutative ring spectra.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e324fd98f3ed34292541831b43099946c7a0950a

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Citations in the corpus

The C2–spectrum Tmf1(3) and its invertible modules (2015, background): “[G: H] = m, then there is a canonical homotopy class of maps NG Hi HX!Sym m(X) coming from any inclusion of G m= into E G m, where is the graph of the homomorphism G! m dening G=Has a G-set [HHR09], [BH15]. In particular, if ˆG H is the representation IndG H R and if ˆG H is the quotient of ˆ G H by”
Genuine equivariant operads (2017, methodology): “as cartesian ﬁxed points. Theorem III is proven in §6.4. Condition (vi), which is not needed in either of Theorems I,II is discussed in §6.2. Lastly, our techniques also verify the main conjecture of [3], which we discuss in §6.5. Moreover, we note that our models for NF-operads are given by explici”
Incomplete Tambara functors (2016, background): “Much of the richness and subtlety of equivariant stable homotopy theory arises from the complexity of the notion of a commutative ring spectrum (i.e., multiplicative cohomology theory) in this context.”
Model structures on commutative monoids in general model categories (2014, background): “For equivariant orthogonal (hence for orthogonal, taking G to be trivial) this is contained in the appendix of [6] among other places.”