Recent zbMATH articles in MSC 55Shttps://zbmath.org/atom/cc/55S2022-07-25T18:03:43.254055ZWerkzeugA computer algebra system for the study of commutativity up to coherent homotopieshttps://zbmath.org/1487.550012022-07-25T18:03:43.254055Z"Medina-Mardones, Anibal M."https://zbmath.org/authors/?q=ai:medina-mardones.anibal-mSummary: The Python package ComCH, is a lightweight specialized computer algebra system that provides models for well-known objects, the surjection and Barratt-Eccles operads, parameterizing the product structure of algebras that are commutative in a derived sense. The primary examples of such algebras treated by ComCH, are the cochain complexes of spaces, for which it provides effective constructions of Steenrod cohomology operations at all primes.Cochain level May-Steenrod operationshttps://zbmath.org/1487.550262022-07-25T18:03:43.254055Z"Kaufmann, Ralph M."https://zbmath.org/authors/?q=ai:kaufmann.ralph-m"Medina-Mardones, Anibal M."https://zbmath.org/authors/?q=ai:medina-mardones.anibal-mIn this paper, the authors provide effective constructions for Steenrod operations at the cochain level, based on May's approach [\textit{J. P. May}, Lect. Notes Math. 168, 153--231 (1970; Zbl 0242.55023)] via May-Steenrod structures (in the terminology of this paper).
The authors fix a commutative ring \(R\) (usually \(\mathbb{Z}\) or \(\mathbb{F}_p\), for \(p\) a prime) and work in chain complexes over \(R\). They choose an \(E_\infty\)-operad \(\mathcal{R}\) (in particular, \(\mathcal{R} (0) = R\) and, for \(t \geq 0\), \(\mathcal{R}(t)\) is a free \(R[S_t]\)-resolution of the trivial \(S_t\)-module \(R\), where \(S_t\) denotes the symmetric group); \(\mathcal{W}(t)\) denotes the usual `minimal' free \(C_t\)-resolution of the trivial \(C_t\)-module \(R\), where \(C_t \subset S_t\) is the cyclic group of order \(t\).
As the authors recall, the key step in constructing Steenrod operations via \(\mathcal{R}\) is to give a May-Steenrod structure. This boils down to specifying, for each \(t \geq 0\), a quasi-isomorphism
\[
\mathcal{W} (t) \stackrel{\simeq}{\rightarrow} \mathcal{R} (t)
\]
of complexes over \(R[C_t]\), where the codomain is given the restricted structure via \(C_t \subset S_t\).
The main contribution of the paper is to exhibit such structures on some standard combinatorial \(E_\infty\)-operads, including the Barratt-Eccles operad and the surjection operad.
The results are applied to construct a natural May-Steenrod structure on the normalized cochains of any simplicial (respectively cubical) set. This can be implemented to give the effective computation of Steenrod operations.
Reviewer: Geoffrey Powell (Angers)A cochain level proof of Adem relations in the mod 2 Steenrod algebrahttps://zbmath.org/1487.550272022-07-25T18:03:43.254055Z"Brumfiel, Greg"https://zbmath.org/authors/?q=ai:brumfiel.greg"Medina-Mardones, Anibal"https://zbmath.org/authors/?q=ai:medina-mardones.anibal-m"Morgan, John"https://zbmath.org/authors/?q=ai:morgan.john-c-ii|morgan.john-d-iii|morgan.john-o|morgan.john-c-morgan-ii|morgan.john-p|morgan.john-w|morgan.john-aThe purpose of this article is to revisit the proof of the Adem relations for the Steenrod reduced square cohomology operations with \(\mathbb{F}_2\)-coefficients by making explicit cochain homotopies that determine this relation. The authors rely on N. Steenrod's original definition of the reduced square operations in terms of the higher cup-products \(\mathrm{Sq}^{k-i}(\alpha) = \alpha\smile_i\alpha\), for \(\alpha\) a cocycle of degree \(k\) in the normalized cochain complex \(N^*(X)\) of a simplicial set \(X\), and on the explicit definition of the higher cup-products \(\smile_i: N^*(X)\otimes N^*(X)\rightarrow N^*(X)\), given in \textit{N. E. Steenrod}'s paper [Ann. Math. (2) 48, 290--320 (1947; Zbl 0030.41602)].
In general, one can associate a cochain operation \(\theta_x: N^k(X)\rightarrow N^{kn-i}(X)\) to any element \(x\in N_i(B\Sigma_n)\), where we consider the normalized chain complex of the classifying space of the symmetric group on \(n\) letters \(\Sigma_n\). The authors observe that this correspondence can be made more precise by using the action of the Barratt--Eccles operad \(\mathcal{E}\) on the cochain complex of simplicial sets defined by \textit{C. Berger} and the reviewer in [Math. Proc. Camb. Philos. Soc. 137, No. 1, 135--174 (2004; Zbl 1056.55006)]. We then take the standard simplicial model of the classifying spaces \(B\Sigma_n\) and we use the identity \(N_i(B\Sigma_n) = \mathcal{E}(n)_{\Sigma_n}\), where we take the coinvariants of the components of the Barratt-Eccles operad under the action of the symmetric groups. The Steenrod squares operations are, via the cochain formula \(\mathrm{Sq}^{k-i}(\alpha) = \alpha\smile_i\alpha\), associated to the generators of the \(\mathbb{F}_2\)-vector spaces \(N_i(B\Sigma_2) = \mathbb{F}_2 x_i\).
The composites of the Steenrod square operations that occur in the Adem relations are represented by elements of the complex~\(N_*(B\Sigma_4)\) that come from \(N_*(BD_8)\), where \(D_8\) is the subgroup of \(\Sigma_4\) of order 8 generated by the permutations \((1\ 2)\), \((3\ 4)\) and \((1\ 3)(2\ 4)\). The authors also consider the Klein subgroup \(V_4\subset\Sigma_4\), which satisfies \(V_4\subset D_8\) and is isomorphic to \(\Sigma_2\times\Sigma_2\). They give an expansion of cross product elements \(x_i\times x_j\in N_*(B(\Sigma_2\times\Sigma_2)) = N_*(B V_4)\) in~\(N_*(B\Sigma_4)\) involving a sum of composites of the Steenrod square cochain operations of the form that occur in the Adem relations together with explicitly defined coboundaries. Eventually, they use that the cross product is commutative up to an explicitly defined cochain homotopy to get an explicit representation of coboundaries that give the Adem relations when we take the action of our operations on the normalized cochain complex of a simplicial set.
Reviewer: Benoît Fresse (Villeneuve d'Ascq)Primitive and decomposable elements in homology of \(\Omega \Sigma \mathbb{C} P^{\infty}\)https://zbmath.org/1487.550282022-07-25T18:03:43.254055Z"Lee, Dae-Woong"https://zbmath.org/authors/?q=ai:lee.dae-woongLet \(\mathbb{C}P^\infty\) be the infinite complex projective space. The author follows [\textit{K. Morisugi}, J. Math. Kyoto Univ. 38, No. 1, 151--165 (1998; Zbl 0924.55002)] to define a sequence of maps \(\hat{\varphi}_n : \mathbb{C}P^\infty\to \Omega\Sigma\mathbb{C}P^\infty\) for each positive integer \(n\). Using the adjointness, the sequence of self-maps \(\varphi_n : \Sigma\mathbb{C}P^\infty\to \Sigma\mathbb{C}P^\infty\) of the suspension of \(\mathbb{C}P^\infty\) or the localization of this space at a set of primes which may be an empty set is considered. Furthermore, let \([\varphi_m,\varphi_n] : \Sigma\mathbb{C}P^\infty\to \Sigma\mathbb{C}P^\infty\) be a commutator of \(\varphi_m\) and \(\varphi_n\) for any positive integers \(m\) and \(n\). \par The author shows that the image of the homomorphism \(\widehat{[\varphi_m, \varphi_n]}_\ast\) in homology induced by the adjoint \(\widehat{[\varphi_m, \varphi_n]} : \mathbb{C}P^\infty\to \Omega\Sigma\mathbb{C}P^\infty\) of the commutator \([\varphi_m, \varphi_n]\) is both primitive and decomposable. As a further support of the above statement, an example is provided.
Reviewer: Marek Golasiński (Olsztyn)