On the Rothenberg–Steenrod spectral sequence for the mod 2 cohomology of classifying spaces of spinor groups

Kameko, Masaki; Mimura, Mamoru

doi:10.2140/gtm.2008.13.261

Cited by 17 publications

(68 citation statements)

References 11 publications

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“…Since we have H 5 (BSpin(n), Z) = 0, the reduction modulo 2 is an isomorphism. To compute the relevant Steenrod square, we can use the result [69,71] that the cohomology with Z 2 coefficients of BSpin(n) can be obtained from that of BSO(n) via the pullback associated to the map f : BSpin(n) → BSO(n). Now, H i (BSO(n), Z 2 ) is a polynomial Z 2 ring generated by the Stiefel-Whitney classes, with w 1 = 0.…”

Section: Spin(n)mentioning

confidence: 99%

Dai-Freed anomalies in particle physics

García-Etxebarria

Montero

2019

J. High Energ. Phys.

158

280

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Anomalies can be elegantly analyzed by means of the Dai-Freed theorem. In this framework it is natural to consider a refinement of traditional anomaly cancellation conditions, which sometimes leads to nontrivial extra constraints in the fermion spectrum. We analyze these more refined anomaly cancellation conditions in a variety of theories of physical interest, including the Standard Model and the SU (5) and Spin(10) GUTs, which we find to be anomaly free. Turning to discrete symmetries, we find that baryon triality has a Z 9 anomaly that only cancels if the number of generations is a multiple of 3. Assuming the existence of certain anomaly-free Z 4 symmetry we relate the fact that there are 16 fermions per generation of the Standard model -including right-handed neutrinos -to anomalies under time-reversal of boundary states in four-dimensional topological superconductors. A similar relation exists for the MSSM, only this time involving the number of gauginos and Higgsinos, and it is non-trivially, and remarkably, satisfied for the SU (3) × SU (2) × U (1) gauge group with two Higgs doublets. We relate the constraints we find to the well-known Ibañez-Ross ones, and discuss the dependence on UV data of the construction. Finally, we comment on the (non-)existence of K-theoretic θ angles in four dimensions. A On reduced bordism groups 63 B Tables of bordism groups of a point 64 C Bordism groups for Z k 65 D 3d currents 71 E Alternate generators for Ω Spin 5 (BZ n ) 73 -1 -An important technical point is that (2.4) and (2.5) require regularization as usual in quantum field theory. If a regularization preserving the symmetry G for an arbitrary background gauge field can be found, then (2.4) is not anomalous. In particular, this always happens whenever there is a G-invariant mass term m d d xψψ (2.8) for the fermions. In this case, Pauli-Villars regularization is available [5], which is manifestly gauge invariant.

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Section: Spin(n)mentioning

confidence: 99%

Dai-Freed anomalies in particle physics

García-Etxebarria

Montero

2019

J. High Energ. Phys.

158

280

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“…This problem was first studied by F. Peterson [6], R. Wood [10], W. Singer [8] and S. Priddy [7], who show its relationships to several classical problems in cobordism theory, modular representation theory, the Adams spectral sequence for the stable homotopy of spheres, and the stable homotopy type of classifying spaces of finite groups. The tensor product F 2 ⊗ A P k has explicitly been computed for k 3 (see [5]). It seems unlikely that an explicit description of F 2 ⊗ A P k for general k will appear in the near future.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

The smallest subgroup whose invariants are hit by the Steenrod algebra

Hùng¹,

Luong²

2007

Math. Proc. Camb. Phil. Soc.

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Let V be a k-dimensional ${\mathbb{F}_2}$-vector space and let W be an n-dimensional vector subspace of V. Denote by GL(n, ${\mathbb{F}_2}$) • 1k-n the subgroup of GL(V) consisting of all isomorphisms ϕ:V → V with ϕ(W) = W and ϕ(v) ≡ v (mod W) for every v ∈ V. We show that GL(3, ${\mathbb{F}_2}$) • 1k-3 is, in some sense, the smallest subgroup of GL(V)≅ GL(k, ${\mathbb{F}_2})$, whose invariants are hit by the Steenrod algebra acting on the polynomial algebra, ${\mathbb{F}_2})\cong{\mathbb{F}_2}[x_{1},\ldots,x_{k}]$. The result is some aspect of an algebraic version of the classical conjecture that the only spherical classes inQ0S0are the elements of Hopf invariant one and those of Kervaire invariant one.

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“…On the other hand, as dim(Q ⊗3 n 1 ) >0 = 15 (see Kameko [Kam90]) and dim(Q ⊗4 n 1 ) >0 = 165 (see Sum [Sum15]), we obtain dim(Q ⊗5 n 1 ) 0 = 15. Calculation of (Q ⊗5 n 1 ) ω >0 .…”

Section: The Proof Of Theorem 23mentioning

confidence: 96%

“…The classical "hit problem" for the algebra P m , which is concerned with seeking a minimal set of A -generators for P m , has been initiated in a variety of contexts by Peterson [Pet87], Priddy [Pri90], Singer [Sin91], and Wood [Woo89]. The study of modules over the Steenrod algebra A and related hit problems is a central topic in Algebraic Topology and has received extensive attention from many mathematicians, including Ault and Singer [AS11], Brunetti, Ciampella and Lomonaco [BCL12], Brunetti and Lomonaco [BL14], Crabb and Hubbuck [CH96], Janfada [Jan08], Kameko [Kam90], Mothebe et al [MU15,Mot16,MKR16], Nam [Nam04], Repka and Selick [RS98], Walker and Wood [WW18a,WW18b], the present author [Phu16, Phu20a, Phu20b, Phu21a, Phu21b, Phu22a, Phu22b, Phu22c, Phu22d, Phu23a, Phu23d], etc. We believe that the best references here are the books by Walker and Wood [WW18a,WW18b], where the most popular articles on these subjects are collected.…”

Section: Context and Overviewmentioning

confidence: 99%

“…It must be noted that one of the important parts of the hit problem is to check whether a given polynomial in P m is hit or not. In 2008, Janfada [Jan08] completed the work of Kameko [Kam90] by exhibiting a criterion for monomials of generic degrees in P 3 to be hit. In 2015, Sum conducted a study on the hit problem for four variables in [Sum15].…”

Section: Context and Overviewmentioning

confidence: 99%

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On modules over the mod 2 Steenrod algebra and hit problems

Phúc¹

2023

Preprint

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<p>Let us consider the prime field of two elements, $\mathbb F_2\equiv \mathbb Z_2.$ The classical "hit problem" in Algebraic Topology, which is widely recognized as an important and intriguing open problem, asks for a minimal set of generators for the polynomial algebra, $\mathcal P_m:=\mathbb F_2[x_1, x_2, \ldots, x_m]$, on $m$ variables $x_1, \ldots, x_m$, each of which has degree one, regarded as a connected unstable $\mathscr A$-module. The algebra $\mathcal P_m$ is the cohomology with $\mathbb F_2$-coefficients of the product of $m$ copies of the Eilenberg-MacLan complex $K(\mathbb F_2, 1)$. Despite extensive study over the past three decades, the hit problem remains unresolved for $m\geq 5$. </p> <p>In this article, we develop our previous work [Commun. Korean Math. Soc. 35 (2020), 371-399] on the hit problem for $\mathscr A$-module $\mathcal P_5$ in generic degree $n_s = 5(2^{s}-1) + 18.2^{s}$ with $s$ an arbitrary non-negative integer. As a result, a local version of Kameko's conjecture, which concerns the dimension of the cohit space $\mathbb F_2\otimes_{\mathscr A}\mathcal P_m$ in relation to weight vectors, has been confirmed for the case where $m = 5$ and the degree is $n_s.$ Also, we show that this conjecture also holds for any $m\geq 1$ and degrees $\leq 12.$ This study has two important applications: (1) it establishes the dimension result for the cohit space $\mathbb F_2\otimes_{\mathscr A}\mathcal P_m$ for $m = 6$ in generic degree $5(2^{s+4}-1) + n_1.2^{s+4}$ with $s > 0;$ and (2) it describes the representations of the general linear group of rank $5$ over $\mathbb F_2.$ As a result, we prove that the algebraic transfer, defined by W. Singer [Math. Z. 202 (1989), 493-523], is an isomorphism in bidegrees $(5, 5+n_s)$ with $s\geq 0.$ Besides, we obtain new results on the behavior of this algebra transfer for all homological degrees $m$. Specifically, we demonstrate that Singer's transfer is a trivial isomorphism in bidegree $(m, m+12)$ for any $m > 0$.</p>

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On the Rothenberg–Steenrod spectral sequence for the mod 2 cohomology of classifying spaces of spinor groups

Cited by 17 publications

References 11 publications

Dai-Freed anomalies in particle physics

Dai-Freed anomalies in particle physics

The smallest subgroup whose invariants are hit by the Steenrod algebra

On modules over the mod 2 Steenrod algebra and hit problems

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