Abstract
This PhD thesis consists of two research papers, background material and
perspectives for future research. In G-equivariant homotopy theory, there are many
possible 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 of
the idempotent summands of certain equivariant commutative ring spectra in terms
of the subgroup lattice and conjugation in G. Algebraically, my results characterize
the extent to which multiplicative induction on the level of homotopy groups is compatible
with the idempotent splitting. Here, G always denotes a finite group.
In the first paper “Multiplicativity of the idempotent splittings of the Burnside ring
and the G-sphere spectrum”, the above program is carried out for the G-equivariant
sphere spectrum. As an application, I obtain an explicit description of the multiplicativity
of the idempotent splitting of the equivariant stable homotopy category.
In the second paper “Idempotent characters and equivariantly multiplicative splittings
of K-theory”, the above is established in the case of G-equivariant topological
K-theory. The main new ingredient is a classification of the primitive idempotents of
the p-local complex representation ring. It implies that all of these idempotents come
from primitive idempotents of the Burnside ring, which is used to reduce the solution
for K-theory to that for the sphere given in the first paper.
perspectives for future research. In G-equivariant homotopy theory, there are many
possible 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 of
the idempotent summands of certain equivariant commutative ring spectra in terms
of the subgroup lattice and conjugation in G. Algebraically, my results characterize
the extent to which multiplicative induction on the level of homotopy groups is compatible
with the idempotent splitting. Here, G always denotes a finite group.
In the first paper “Multiplicativity of the idempotent splittings of the Burnside ring
and the G-sphere spectrum”, the above program is carried out for the G-equivariant
sphere spectrum. As an application, I obtain an explicit description of the multiplicativity
of the idempotent splitting of the equivariant stable homotopy category.
In the second paper “Idempotent characters and equivariantly multiplicative splittings
of K-theory”, the above is established in the case of G-equivariant topological
K-theory. The main new ingredient is a classification of the primitive idempotents of
the p-local complex representation ring. It implies that all of these idempotents come
from primitive idempotents of the Burnside ring, which is used to reduce the solution
for K-theory to that for the sphere given in the first paper.
| Original language | English |
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| Publisher | Department of Mathematical Sciences, Faculty of Science, University of Copenhagen |
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| Publication status | Published - 2018 |