The linear isometries operad in Lie--Tate homology

Bimonoidal operad-actions and the product in negative Tate-cohomology

We study a certain construction designed to bring together the following two topics: $i$) Dyer--Lashof-operations in negative Tate-cohomology, $ii$) the description of negative Tate-cohomology in terms of joins. It has the merit of making (some) sense in a more general context: where the loop-space of the space under study no longer has to be of compact homotopy-type. The exposition given here is a streamlined version of a previous version of mine, available here at the Arxiv.

Characterizations of equivariant Steiner and linear isometries operads

We study the indexing systems that correspond to equivariant Steiner and linear isometries operads. When $G$ is a finite abelian group, we prove that a $G$-indexing system is realized by a Steiner operad if and only if it is generated by cyclic $G$-orbits. When $G$ is a finite cyclic group, whose order is either a prime power or a product of two distinct primes greater than $3$, we prove that a $G$-indexing system is realized by a linear isometries operad if and only if it satisfies Blumberg and Hill's horn-filling condition. We also repackage the data in an indexing system as a certain kind o…

通过线性等距 $N_\infty$-operads 实现阶为 $p^nq^m$ 的循环群上的饱和传递系统

Realization of saturated transfer systems on cyclic groups of order $p^nq^m$ by linear isometries $N_\infty$-operads

本文证明了Rubin饱和猜想的一个新情形，即阶为 $p^nq^m$ 的有限循环群上的 $G$-传递系统可通过线性等距 $N_\infty$-operads 实现。

Equivariant operads, string topology, and Tate cohomology

From an operad C with an action of a group G, we construct new operads using the homotopy fixed point and orbit spectra. These new operads are shown to be equivalent when the generalized G-Tate cohomology of C is trivial. Applying this theory to the little disk operad C_2 (which is an S^1 operad) we obtain variations on Getzler's gravity operad, which we show governs the Chas-Sullivan string bracket.

通过转移系统的等变线性等距与无限小圆盘操作代数

Equivariant linear isometries and infinite little discs operads via transfer systems

利用转移系统理论研究有限群\(G\)的\(G -\)等变线性等距与无限小圆盘操作代数的关系，将拓扑问题转化为离散问题，分类了特定同伦等价的\(G -\)全域，还给出相关代数结构见解及条件。

阿贝尔群上的等变线性等距算子

Equivariant linear isometries operads over Abelian groups

本文利用代数和极值组合技术，确定了所有使线性等距算子分类有效的有限阿贝尔群，解决了Blumberg和Hill提出的分类问题在有限阿贝尔群情形下的有效性判定。