On the computability of the Steenrod squares

Transactions of the American Mathematical Society, 1973

We define two kinds of Steenrod operations on the spectral sequence of a bisimplicial coalgebra. We show these operations compatible with the differentials of the spectral sequence, and with the Steenrod squares defined on the cohomology of the total complex. We give a general rule for computing the operations on E-. who consider the related problem of defining Steentod squares on the spectral sequence of a "mixed" bisimplicial object. .. , i.e., a functor from Ox G that is contravariant in one vatiable and covariant in the other. Rector and Smith obtain only operations of type (0.1). Whethet opetations of type (0.2) can be defined for mixed bisimplicial objects remain an open problem.

Appalachian Set Theory 2006–2012

2021

Let Pk = H((RP )) be the modulo-2 cohomology algebra of the direct product of k copies of infinite dimensional real projective spaces RP . Then, Pk is isomorphic to the graded polynomial algebra F2[x1, . . . , xk] of k variables, in which each xj is of degree 1, and let GLk be the general linear group over the prime field F2 which acts naturally on Pk. Here the cohomology is taken with coefficients in the prime field F2 of two elements. We study the hit problem, set up by Frank Peterson, of finding a minimal set of generators for the polynomial algebra Pk as a module over the mod-2 Steenrod algebra, A. In this Note, we explicitly compute the hit problem for k = 5 and the degree 5(2 − 1) + 24.2 with s an arbitrary non-negative integer. These results are used to study the Singer algebraic transfer which is a homomorphism from the homology of the mod-2 Steenrod algebra, TorAk,k+n(F2, F2), to the subspace of F2 ⊗A Pk consisting of all the GLk-invariant classes of degree n. We show that ...

arXiv (Cornell University), 2024

We describe the action of the mod 2 Steenrod algebra on the cohomology of various polyhedral products and related spaces. We carry this out for Davis-Januszkiewicz spaces and their generalizations, for moment-angle complexes as well as for certain polyhedral joins. By studying the combinatorics of underlying simplicial complexes, we deduce some consequences for the lowest cohomological dimension in which non-trivial Steenrod operations can appear. We present a version of cochain-level formulas for Steenrod operations on simplicial complexes. We explain the idea of "propagating" such formulas from a simplicial complex K to polyhedral joins over K and we give examples of this process. We tie the propagation of the Steenrod algebra actions on polyhedral joins to those on moment-angle complexes. Although these are cases where one can understand the Steenrod action via a stable homotopy decomposition, we anticipate applying this method to cases where there is no such decomposition.

1993

Let F be a unimodular r • s matrix with entries being n-variate polynomials over an infinite field K. Denote by deg(F) the maximum of the degrees of the entries of F and let d = 1 + deg(F). We describe an algorithm which computes a unimodular 8 x s matrix M

arXiv: Algebraic Topology, 2020

Steenrod defined in 1947 the Steenrod squares on the mod 2 cohomology of spaces using explicit cochain formulae for the cup-$i$ products; a family of coherent homotopies derived from the broken symmetry of Alexander-Whitney's chain approximation to the diagonal. Later, following a viewpoint developed by Adem, Steenrod defined his homonymous operations for all primes using the homology of symmetric groups. In recent years, thanks to the development of new applications of cohomology -- most notably in Applied Topology and Quantum Field Theory -- having a definition of Steenrod operations that can be effectively computed in specific examples has become a key issue. This article provides such definition, at all primes, for spaces expressed via simplicial or cubical cell structures. Two further concrete contributions of this work are the first explicit description of an $E_\infty$-structure on cubical chains, and an open-source implementation of all constructions used.

Transactions of the American Mathematical Society, 2005

Let T r k be the algebraic transfer that maps from the coinvariants of certain GL k-representations to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer tr k : π S * ((BV k) +) → π S * (S 0). It has been shown that the algebraic transfer is highly nontrivial, more precisely, that T r k is an isomorphism for k = 1, 2, 3 and that T r = k T r k is a homomorphism of algebras. In this paper, we first recognize the phenomenon that if we start from any degree d and apply Sq 0 repeatedly at most (k − 2) times, then we get into the region in which all the iterated squaring operations are isomorphisms on the coinvariants of the GL k-representations. As a consequence, every finite Sq 0-family in the coinvariants has at most (k − 2) nonzero elements. Two applications are exploited. The first main theorem is that T r k is not an isomorphism for k ≥ 5. Furthermore, for every k > 5, there are infinitely many degrees in which T r k is not an isomorphism. We also show that if T r detects a nonzero element in certain degrees of Ker(Sq 0), then it is not a monomorphism and further, for each k > , T r k is not a monomorphism in infinitely many degrees. The second main theorem is that the elements of any Sq 0-family in the cohomology of the Steenrod algebra, except at most its first (k − 2) elements, are either all detected or all not detected by T r k , for every k. Applications of this study to the cases k = 4 and 5 show that T r 4 does not detect the three families g, D 3 and p , and that T r 5 does not detect the family {h n+1 g n | n ≥ 1}.

2021

Let Pn := H((RP )) ∼= F2[x1, x2, . . . , xn] be the graded polynomial algebra over the prime field of two elements, F2. We investigate the Peterson hit problem for the polynomial algebra Pn, viewed as a graded left module over the mod-2 Steenrod algebra, A. For n > 4, this problem is still unsolved, even in the case of n = 5 with the help of computers. The purpose of this paper is to continue our study of the hit problem by developing a result in [17] for Pn in the generic degree ks = r(2 − 1) + m.2 with r = n = 5, m = 13, and s an arbitrary non-negative integer. Note that for s = 0, k0 = 5(20 −1)+13.20 = 13, and s = 1, k1 = 5(21 −1)+13.21 = 31, these problems have been studied by Phuc [16], and [17], respectively. Moreover, as an application of these results, we get the dimension result for the graded polynomial algebra in the generic degree d = (n − 1).(2n+u−1 − 1) + l.2n+u−1 with u an arbitrary non-negative integer, l ∈ {23, 67}, and in the case n = 6. One of the major applica...

Journal of Linear and Topological Algebra, 2018

‎The Mod $2$ Steenrod algebra is a Hopf algebra that consists of the primary cohomology operations‎, ‎denoted by $Sq^n$‎, ‎between the cohomology groups with $mathbb{Z}_2$ coefficients of any topological space‎. ‎Regarding to its vector space structure over $mathbb{Z}_2$‎, ‎it has many base systems and some of the base systems can also be restricted to its sub algebras‎. ‎On the contrary‎, ‎in addition to the work of Wood‎, ‎in this paper we define a new base system for the Hopf subalgebras $mathcal{A}(n)$ of the mod $2$ Steenrod algebra which can be extended to the entire algebra‎. ‎The new base system is obtained by defining a new linear ordering on the pairs $(s+t,s)$ of exponents of the atomic squares $Sq^{2^s(2^t-1)}$ for the integers $sgeq 0$ and $tgeq 1$‎.

arXiv (Cornell University), 2014

This note provides a brief guide to the current state of the literature on Tarski's problems with emphasis on features that distinguish the approach based on combinatorial and algorithmic group theory from the topological approach to Tarski's problem. We use this note to provide corrections to some typos and to address some misconceptions from the recent report by Z. Sela about the relations between the concepts and results in the approaches to the Tarski problems. We were forced to read Sela's papers to be able to address some of his comments, and found errors in his papers 6, 3 and 4 on Diophantine Geometry published in GAFA and Israel J. Math. which we mention in Section 4. His proceedings of the ICM 2002 paper also contains wrong Theorem 6 (to make it correct one has to change the definition of non-elementary hyperbolic ω-residually free towers to make them equivalent to our coordinate groups of regular NTQ systems.