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Homological computations in the universal Steenrod algebra

A. Ciampella, L. A. Lomonaco (2004)

Fundamenta Mathematicae

We study the (bigraded) homology of the universal Steenrod algebra Q over the prime field ₂, and we compute the groups $H_{s, s} (Q)$ , s ≥ 0, using some ideas and techniques of Koszul algebras developed by S. Priddy in [5], although we presently do not know whether or not Q is a Koszul algebra. We also provide an explicit formula for the coalgebra structure of the diagonal homology $D ⁎ (Q) = ⨁_{s \geq 0} H_{s, s} (Q)$ and show that D⁎(Q) is isomorphic to the coalgebra of invariants Γ introduced by W. Singer in [6].

Homology Comodules.

R.M. Switzer (1973)

Inventiones mathematicae

Homology exponents for $H$ -spaces.

A. Cément, J. Scherer (2008)

Revista Matemática Iberoamericana

Hopf algebra structure on topological Hochschild homology.

Angeltveit, Vigleik, Rognes, John (2005)

Algebraic & Geometric Topology

Indecomposable A-module summands in H* V which are unstable algebras.

Hanns-Werner Henn, Lionel Schwartz (1990)

Mathematische Zeitschrift

Intersection homology operations.

R. Mark Goresky (1984)

Commentarii mathematici Helvetici

Isoparametric Hypersurfaces with Four or Six Distinct Principal Curvatures. Necessary Conditions on the Multiplicities.

U. Abresch (1983)

Mathematische Annalen

La conjecture de Sullivan

Lionel Schwartz (1984/1985)

Séminaire Bourbaki

Lannes' T functor on Hopf algebras over the Steenrod algebra with applications to Carlsson modules and twisted algebras.

R. James Shank (1992)

Mathematische Zeitschrift

Le foncteur $V \mapsto 𝔽_{2} {[V]}^{\otimes 3}$ entre $𝔽_{2}$ -espaces vectoriels est noethérien

Aurélien Djament (2009)

Annales de l’institut Fourier

Les foncteurs entre espaces vectoriels, ou représentations génériques des groupes linéaires d’après Kuhn, interviennent en topologie algébrique et en $K$ -théorie comme en théorie des représentations. Nous présentons ici une nouvelle méthode pour aborder les problèmes de finitude et la dimension de Krull dans ce contexte.Plus précisément, nous démontrons que, dans la catégorie $ℱ$ des foncteurs entre espaces vectoriels sur $𝔽_{2}$ , le produit tensoriel entre $P^{\otimes 3}$ , où $P$ désigne le foncteur projectif $V \mapsto 𝔽_{2} [V]$ , et un foncteur...

Linking first occurrence polynomials over $𝔽_{p}$ by Steenrod operations.

Minh, Pham Anh, Walker, Grant (2002)

Algebraic & Geometric Topology

Localisation of Unstable Ap-Algebras and Smith Theory.

Steve Hutt (1996)

Mathematica Scandinavica

Localizations of unstable A-modules and equivariant mod p cohomology.

H.-W. Henn, J. Lannes, L. Schwartz (1995)

Mathematische Annalen

Modular invariant theory and the iterated total power operation

A. Ciampella (2000)

Bollettino dell'Unione Matematica Italiana

L'operazione coomologica totale iterata in coomologia ordinaria a coefficienti in $Z / p$ ha una sua espressione a seconda della base fissata nell'algebra di Steenrod $A_{p}$ . Fissato un primo $p$ dispari, vengono qui calcolati i coefficienti dell'operazione totale doppia iterata quando si sceglie in $A_{p}$ la base dei monomi ammissibili. Si fornisce inoltre una dimostrazione alternativa di una versione normalizzata di un teorema di Mùi, ottenuta considerando una particolare successione di funzioni, in analogia al...

Monomial bases in the mod- $p$ Steenrod algebra

Ismet Karaca (2005)

Czechoslovak Mathematical Journal

In this paper we study sets of some special monomials which form bases for the mod- $p$ Steenrod algebra $𝒜$ .

Motivic cohomology with $𝐙 / 2$ -coefficients

Vladimir Voevodsky (2003)

Publications Mathématiques de l'IHÉS

Multiplicative operations in the Steenrod algebra for Brown–Peterson cohomology

Michael Slack (1999)

Fundamenta Mathematicae

A family of multiplicative operations in the BP Steenrod algebra is defined which is related to the total Steenrod power operation from the mod p Steenrod algebra. The main result of the paper links the BP versions of the total Steenrod power with the formal group approach to multiplicative BP operations by identifying the p-typical curves (power series) which correspond to these operations. Some relations are derived from this identification, and a short proof of the Hopf invariant one theorem...

Nil-Localization of Unstable Algebras over the Steenrod Algebra.

C. Broto, S. Zarati (1988)

Mathematische Zeitschrift

Noetherian loop spaces

Natàlia Castellana, Juan Crespo, Jérôme Scherer (2011)

Journal of the European Mathematical Society

The class of loop spaces of which the mod $p$ cohomology is Noetherian is much larger than the class of $p$ -compact groups (for which the mod $p$ cohomology is required to be finite). It contains Eilenberg–Mac Lane spaces such as $ℂ P^{\infty}$ and 3-connected covers of compact Lie groups. We study the cohomology of the classifying space $B X$ of such an object and prove it is as small as expected, that is, comparable to that of $B ℂ P^{\infty}$ . We also show that $B$ X differs basically from the classifying space of a $p$ -compact group...

On non-realization results and conjectures of N. Kuhn

Nguyen The Cuong, Gérald Gaudens, Geoffrey Powell, Lionel Schwartz (2016)

Fundamenta Mathematicae

We discuss two extensions of results conjectured by Nick Kuhn about the non-realization of unstable algebras as the mod-p singular cohomology of a space, for p a prime. The first extends and refines earlier work of the second and fourth authors, using Lannes’ mapping space theorem. The second (for the prime 2) is based on an analysis of the -1 and -2 columns of the Eilenberg-Moore spectral sequence, and of the associated extension. In both cases, the statements and proofs use the relationship between...

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