286 research outputs found
Mod-two cohomology of symmetric groups as a Hopf ring
We compute the mod-2 cohomology of the collection of all symmetric groups as
a Hopf ring, where the second product is the transfer product of Strickland and
Turner. We first give examples of related Hopf rings from invariant theory and
representation theory. In addition to a Hopf ring presentation, we give
geometric cocycle representatives and explicitly determine the structure as an
algebra over the Steenrod algebra. All calculations are explicit, with an
additive basis which has a clean graphical representation. We also briefly
develop related Hopf ring structures on rings of symmetric invariants and end
with a generating set consisting of Stiefel-Whitney classes of regular
representations v2. Added new results on varieties which represent the
cocycles, a graphical representation of the additive basis, and on the Steenrod
algebra action. v3. Included a full treatment of invariant theoretic Hopf
rings, refined the definition of representing varieties, and corrected and
clarified references.Comment: 31 pages, 6 figure
Kervaire Invariant One [after M. A. Hill, M. J. Hopkins, and D. C. Ravenel]
The question of when the Kervaire invariant is nontrivial was the only
question left unresolved by Kervaire and Milnor in their 1963 study of the
relationship between groups of homotopy spheres and stable homotopy groups. In
2009, Mike Hill, Mike Hopkins, and Doug Ravenel resolved this question except
in one dimension, by a highly innovative attack using large amounts of
equivariant stable homotopy theory and small amounts of computation. The
present paper is a Seminaire Bourbaki report on this work.Comment: This is a the submitted Seminaire Bourbaki report. 30 page
Normalized Rewriting: an Alternative to Rewriting modulo a Set of Equations
AbstractIn the first part of this paper, we introducenormalized rewriting, a new rewrite relation. It generalizes former notions of rewriting modulo a set of equationsE, dropping some conditions onE. For example,Ecan now be the theory of identity, idempotence, the theory of Abelian groups or the theory of commutative rings. We give a new completion algorithm for normalized rewriting. It contains as an instance the usual AC completion algorithm, but also the well-known Buchberger algorithm for computing Gröbner bases of polynomial ideals. In the second part, we investigate the particular case of completion of ground equations. In this case we prove by a uniform method that completion moduloEterminates, for some interesting theoriesE. As a consequence, we obtain the decidability of the word problem for some classes of equational theories, including the AC-ground case (a result known since 1991), the ACUI-ground case (a new result to our knowledge), and the cases of ground equations modulo the theory of Abelian groups and commutative rings, which is already known when the signature contains only constants, but is new otherwise. Finally, we give implementation results which show the efficiency of normalized completion with respect to completion modulo AC
On projective spaces over local fields
Let be the set of points of a finite-dimensional projective
space over a local field , endowed with the topology naturally
induced from the canonical topology of . Intuitively, continuous incidence
abelian group structures on are abelian group structures on
preserving both the topology and the incidence of lines
with points. We show that the real projective line is the only
finite-dimensional projective space over an Archimedean local field which
admits a continuous incidence abelian group structure. The latter is unique up
to isomorphism of topological groups. In contrast, in the non-Archimedean case
we construct continuous incidence abelian group structures in any dimension . We show that if and the characteristic of does not
divide , then there are finitely many possibilities up to topological
isomorphism and, in any case, countably many.Comment: 22 page
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