TY - BOOK

T1 - Equivariant multiplications and idempotent splittings of G-spectra

AU - Böhme, Benjamin

PY - 2018

Y1 - 2018

N2 - This PhD thesis consists of two research papers, background material andperspectives for future research. In G-equivariant homotopy theory, there are manypossible notions of an E¥ ring spectrum, made precise by Blumberg and Hill’s N¥rings. My main results are explicit descriptions of the maximal N¥ ring structures ofthe idempotent summands of certain equivariant commutative ring spectra in termsof the subgroup lattice and conjugation in G. Algebraically, my results characterizethe extent to which multiplicative induction on the level of homotopy groups is compatiblewith the idempotent splitting. Here, G always denotes a finite group.In the first paper “Multiplicativity of the idempotent splittings of the Burnside ringand the G-sphere spectrum”, the above program is carried out for the G-equivariantsphere spectrum. As an application, I obtain an explicit description of the multiplicativityof the idempotent splitting of the equivariant stable homotopy category.In the second paper “Idempotent characters and equivariantly multiplicative splittingsof K-theory”, the above is established in the case of G-equivariant topologicalK-theory. The main new ingredient is a classification of the primitive idempotents ofthe p-local complex representation ring. It implies that all of these idempotents comefrom primitive idempotents of the Burnside ring, which is used to reduce the solutionfor K-theory to that for the sphere given in the first paper.

AB - This PhD thesis consists of two research papers, background material andperspectives for future research. In G-equivariant homotopy theory, there are manypossible notions of an E¥ ring spectrum, made precise by Blumberg and Hill’s N¥rings. My main results are explicit descriptions of the maximal N¥ ring structures ofthe idempotent summands of certain equivariant commutative ring spectra in termsof the subgroup lattice and conjugation in G. Algebraically, my results characterizethe extent to which multiplicative induction on the level of homotopy groups is compatiblewith the idempotent splitting. Here, G always denotes a finite group.In the first paper “Multiplicativity of the idempotent splittings of the Burnside ringand the G-sphere spectrum”, the above program is carried out for the G-equivariantsphere spectrum. As an application, I obtain an explicit description of the multiplicativityof the idempotent splitting of the equivariant stable homotopy category.In the second paper “Idempotent characters and equivariantly multiplicative splittingsof K-theory”, the above is established in the case of G-equivariant topologicalK-theory. The main new ingredient is a classification of the primitive idempotents ofthe p-local complex representation ring. It implies that all of these idempotents comefrom primitive idempotents of the Burnside ring, which is used to reduce the solutionfor K-theory to that for the sphere given in the first paper.

UR - https://soeg.kb.dk/permalink/45KBDK_KGL/fbp0ps/alma99122117812305763

M3 - Ph.D. thesis

BT - Equivariant multiplications and idempotent splittings of G-spectra

PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen

ER -