155 research outputs found
The classification of 2-compact groups
We prove that any connected 2-compact group is classified by its 2-adic root
datum, and in particular the exotic 2-compact group DI(4), constructed by
Dwyer-Wilkerson, is the only simple 2-compact group not arising as the
2-completion of a compact connected Lie group. Combined with our earlier work
with Moeller and Viruel for p odd, this establishes the full classification of
p-compact groups, stating that, up to isomorphism, there is a one-to-one
correspondence between connected p-compact groups and root data over the p-adic
integers. As a consequence we prove the maximal torus conjecture, giving a
one-to-one correspondence between compact Lie groups and finite loop spaces
admitting a maximal torus. Our proof is a general induction on the dimension of
the group, which works for all primes. It refines the
Andersen-Grodal-Moeller-Viruel methods to incorporate the theory of root data
over the p-adic integers, as developed by Dwyer-Wilkerson and the authors, and
we show that certain occurring obstructions vanish, by relating them to
obstruction groups calculated by Jackowski-McClure-Oliver in the early 1990s.Comment: 47 page
The C*-algebra of an affine map on the 3-torus
We study the C*-algebra of an affine map on a compact abelian group and give
necessary and sufficient conditions for strong transitivity when the group is a
torus. The structure of the C*-algebra is completely determined for all
strongly transitive affine maps on a torus of dimension one, two or three
Reduced, tame and exotic fusion systems
We define here two new classes of saturated fusion systems, reduced fusion
systems and tame fusion systems. These are motivated by our attempts to better
understand and search for exotic fusion systems: fusion systems which are not
the fusion systems of any finite group. Our main theorems say that every
saturated fusion system reduces to a reduced fusion system which is tame only
if the original one is realizable, and that every reduced fusion system which
is not tame is the reduction of some exotic (nonrealizable) fusion system
Automorphisms of p-compact groups and their root data
We construct a model for the space of automorphisms of a connected p-compact
group in terms of the space of automorphisms of its maximal torus normalizer
and its root datum. As a consequence we show that any homomorphism to the outer
automorphism group of a p-compact group can be lifted to a group action,
analogous to a classical theorem of de Siebenthal for compact Lie groups. The
model of this paper is used in a crucial way in our paper ``The classification
of 2-compact groups'', where we prove the conjectured classification of
2-compact groups and determine their automorphism spaces.Comment: 24 pages. Introduction restructured and title changed (from
"Automorphisms of root data, maximal torus normalizers, and p-compact
groups"). Various other adjustments mad
Spaces with Noetherian cohomology
Is the cohomology of the classifying space of a p-compact group, with Noetherian twisted coefficients, a Noetherian module? In this paper we provide, over the ring of p-adic integers, such a generalization to p-compact groups of the Evens-Venkov Theorem. We consider the cohomology of a space with coefficients in a module, and we compare Noetherianity over the field with p elements with Noetherianity over the p-adic integers, in the case when the fundamental group is a finite p-grou
The Steenrod problem of realizing polynomial cohomology rings
In this paper we completely classify which graded polynomial R-algebras in
finitely many even degree variables can occur as the singular cohomology of a
space with coefficients in R, a 1960 question of N. E. Steenrod, for a
commutative ring R satisfying mild conditions. In the fundamental case R = Z,
our result states that the only polynomial cohomology rings over Z which can
occur, are tensor products of copies of H^*(CP^\infty;Z) = Z[x_2],
H^*(BSU(n);Z) = Z[x_4,x_6,...,x_{2n}], and H^*(BSp(n):Z) =
Z[x_4,x_8,...,x_{4n}] confirming an old conjecture. Our classification extends
Notbohm's solution for R = F_p, p odd. Odd degree generators, excluded above,
only occur if R is an F_2-algebra and in that case the recent classification of
2-compact groups by the authors can be used instead of the present paper. Our
proofs are short and rely on the general theory of p-compact groups, but not on
classification results for these.Comment: 14 pages. v3: Final version. To appear in Journal of Topolog
Spaces with Noetherian cohomology
Is the cohomology of the classifying space of a p-compact group, with
Noetherian twisted coefficients, a Noetherian module? This note provides, over
the ring of p-adic integers, such a generalization to p-compact groups of the
Evens-Venkov Theorem. We consider the cohomology of a space with coefficients
in a module, and we compare Noetherianity over the field with p elements, with
Noetherianity over the p-adic integers, in the case when the fundamental group
is a finite p-group.Comment: 12 page
A finite loop space not rationally equivalent to a compact Lie group
We construct a connected finite loop space of rank 66 and dimension 1254
whose rational cohomology is not isomorphic as a graded vector space to the
rational cohomology of any compact Lie group, hence providing a counterexample
to a classical conjecture. Aided by machine calculation we verify that our
counterexample is minimal, i.e., that any finite loop space of rank less than
66 is in fact rationally equivalent to a compact Lie group, extending the
classical known bound of 5.Comment: 8 page
Comparison of vaccine-induced antibody neutralization against SARS-CoV-2 variants of concern following primary and booster doses of COVID-19 vaccines
The SARS-CoV-2 pandemic has, as of July 2022, infected more than 550 million people and caused over 6 million deaths across the world. COVID-19 vaccines were quickly developed to protect against severe disease, hospitalization and death. In the present study, we performed a direct comparative analysis of four COVID-19 vaccines: BNT162b2 (Pfizer/BioNTech), mRNA-1273 (Moderna), ChAdOx1 (Oxford/AstraZeneca) and Ad26.COV2.S (Johnson & Johnson/Janssen), following primary and booster vaccination. We focused on the vaccine-induced antibody-mediated immune response against multiple SARS-CoV-2 variants: wildtype, B.1.1.7 (Alpha), B.1.351 (Beta), B.1.617.2 (Delta) and B.1.1.529 (Omicron). The analysis included the quantification of total IgG levels against SARS-CoV-2 Spike, as well as the quantification of antibody neutralization titers. Furthermore, the study assessed the high-throughput ACE2 competition assay as a surrogate for the traditional pseudovirus neutralization assay. The results demonstrated marked differences in antibody-mediated immune responses. The lowest Spike-specific IgG levels and antibody neutralization titers were induced by one dose of the Ad26.COV2.S vaccine, intermediate levels by two doses of the BNT162b2 vaccine, and the highest levels by two doses of the mRNA-1273 vaccine or heterologous vaccination of one dose of the ChAdOx1 vaccine and a subsequent mRNA vaccine. The study also demonstrated that accumulation of SARS-CoV-2 Spike protein mutations was accompanied by a marked decline in antibody neutralization capacity, especially for B.1.1.529. Administration of a booster dose was shown to significantly increase Spike-specific IgG levels and antibody neutralization titers, erasing the differences between the vaccine-induced antibody-mediated immune response between the four vaccines. The findings of this study highlight the importance of booster vaccines and the potential inclusion of future heterologous vaccination strategies for broad protection against current and emerging SARS-CoV-2 variants
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