Equivariant linear isometries and infinite little discs operads via transfer systems
Euan Aitken
TL;DR
This work uses the transfer-system framework to relate equivariant linear isometries and infinite little discs operads across finite groups. It establishes a complete classification: a zig-zag between $ ext{D}( rak{U})$ and $ ext{L}( rak{U})$ exists precisely when the universe is of the form $ oldsymbol{R}[G/N]^ olinebreak olinebreak^ olinebreak olinebreak$ for a normal subgroup $N rianglelefteq G$, tying operadic equivalence to discrete representation-theoretic data. It introduces a closure operator on universes and proves that the multiplicative hull of a disc-like transfer system corresponds to a linear-isometries operad on a closed/augmented universe, yielding maximal compatible pairs and realizability results for abelian groups. The paper further shows that many non-abelian groups (e.g., symmetric and Hamiltonian groups) are not saturated, underscoring restrictions on when $ ext{L}( rak{U})$ can realize all saturated transfer systems. Together, these results connect combinatorial transfer-system theory with classical equivariant operad theory, advancing the understanding of when algebraic transfer data can be realized by familiar $N_ olinebreak_ olinebreak ext{infty}$-operads and how to construct compatible operad pairs in the abelian setting.
Abstract
In this article, we apply the recently developed theory of transfer systems to study the relationship between $G$-equivariant linear isometries and infinite little discs operads, for a finite group $G$. This framework allows us to reduce involved topological problems to discrete problems regarding the subgroup structure and representation theory of the group $G$. Our main result is an example of this: we classify the $G$-universes $\mathcal{U}$ for which the linear isometries operad $\mathcal{L}(\mathcal{U})$ and the infinite little discs operad $\mathcal{D}(\mathcal{U})$ are homotopically equivalent. To achieve this, we use ideas that originate from the work of Balchin-Barnes-Roitzheim on the combinatorics of transfer systems on a total order. Additionally, the use of transfer systems gives us insight into the algebraic structures that arise from equivariant homotopy theory. Compatible pairs of transfer systems provide rules for when multiplicative transfer maps can be paired with additive transfer maps. In the case that the group $G$ is abelian, we provide conditions for when the pair $(\mathcal{L}(\mathcal{U}),\mathcal{D}(\mathcal{U}))$ defines a maximally compatible pair of transfer systems. As a consequence, we contribute to a recent conjecture about equivariant operad pairs.