A note on the modular representation on the mathbb Z/2-homology\n groups of the fourth power of real projective space and its application

2021/06/04 by Đặng Võ Phúc, Phuc, Dang Vo · 1 citation
Mathematics · #13A50 #55Q45 #55R12 #55S05 #55S10 #55T15 #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2106.14606

openalex publication_date 2021/06/04 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

One knows that, the connected graded ring P^{\⊗ h}= mathbb Z/2[t₁,\n\…, t_h]= P_n^{\⊗ h} _{n\≥ 0}, which is graded by the degree of\nthe homogeneous terms P^{\⊗ h}_n of degree n in h generators with\nthe degree of each t_i being one, admits a left action of mathcal A as\nwell as a right action of the general linear group GL_h. A central problem of\nhomotopy theory is to determine the structure of the space of\nGL_h-coinvariants, mathbb Z/2\⊗_GLh rm Ann_\ mathcal\nA[P^{\⊗ h}_n]^*. Solving this problem is very difficult and still\nopen for h\≥ 4. In this Note, our intent is of studying the dimension of\n mathbb Z/2\⊗_GLh rm Ann_\ mathcal A[P^\⊗\nh_n]^* for the case h = 4 and the "generic" degrees n of the form\nn_{k, r, s} = k(2^s - 1) + r.2^s, where k, , r, , s are positive\nintegers. Applying the results, we investigate the behaviour of the Singer\ncohomological "transfer" of rank 4. Singer's transfer is a homomorphism from\na certain subquotient of the divided power algebra \Γ(a₁⁽¹⁾, \…,\na_h⁽¹⁾) to mod-2 cohomology groups rm Ext _{mathcal A}^{h, h+*}( mathbb\nZ/2, mathbb Z/2) of the algebra mathcal A. This homomorphism is useful for\ndepicting the Ext groups. Additionally, in higher ranks, by using the results\non mathcal A-generators for P^{\⊗ 5} and P^{\⊗ 6}, we show in\nAppendix that the transfer of rank 5 is an isomorphism in som certain degrees\nof the form n_{k, r, s}, and that the transfer of rank 6 does not detect the\nnon-zero elements h₂²g₁ = h₄Ph₂\∈ rm Ext _{mathcal A}^6, 6+n_{6,\n10, 1}( mathbb Z/2, mathbb Z/2), and D₂\∈ rm Ext _{mathcal A}^6,\n6+n_{6, 10, 2}( mathbb Z/2, mathbb Z/2). Besides, we also probe the behavior\nof the Singer transfer of ranks 7 and 8 in internal degrees \≤ 15.\n

Citations

• On the lambda algebra and Singer's cohomological transfer

Cited by

• The affirmative answer to Singer's conjecture on the algebraic transfer of rank four

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