The Singer transfer for infinite real projective space

Nguyễn H. V. Hưng; Lưu X. Trường

doi:10.1515/forum-2021-0245

Article

The Singer transfer for infinite real projective space

Nguyễn H. V. Hưng and Lưu X. Trường

Published/Copyright: August 30, 2022

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From the journal Forum Mathematicum Volume 34 Issue 6

Abstract

The article is devoted to studying the Singer transfer. The image of the Singer transfer Tr * ℝ ⁢ ℙ ∞ for the infinite real projective space is proved to be a module over the image of the transfer Tr * for the sphere. Further, the algebraic Kahn–Priddy homomorphism is shown to be an epimorphism from Im ⁡ Tr * ℝ ⁢ ℙ ∞ onto Im ⁡ Tr * in positive stems. The indecomposable elements h ^ i for i ≥ 1 and c ^ i , d ^ i , e ^ i , f ^ i , p ^ i for i ≥ 0 are in the image of the Singer transfer Tr * ℝ ⁢ ℙ ∞ , whereas the ones g ^ i for i ≥ 1 and D ^ 3 ⁢ ( i ) , p ^ i ′ for i ≥ 0 are not in its image. This transfer is shown to be not monomorphic in every positive homological degree. The transfer behavior is also investigated near “critical elements”. The squaring operation on the domain of Tr * ℝ ⁢ ℙ ∞ is proved to be eventually isomorphic. This phenomenon leads to the so-called “stability” of the Singer transfer for the infinite real projective space under the iterated squaring operation.

Keywords: Steenrod algebra; Adams spectral sequences; Singer transfer; lambda algebra; invariant theory; Dickson algebra

MSC 2010: 55T15; 55Q45; 55S10; 55P47

Dedicated to the memory of Nguy~n Thị Thanh Bình

Communicated by Jan Bruinier

Funding source: National Foundation for Science and Technology Development

Award Identifier / Grant number: 101.04-2019.300

Funding statement: This research was carried out when the authors recently visited the Vietnam Institute for Advanced Study in Mathematics (VIASM), Hanoi. They would like to express their warmest thanks to the VIASM for the hospitality and for the wonderful working conditions. This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2019.300.

Acknowledgements

The authors would like to express their warm thanks to R. Bruner for helpful discussions on the squaring operation. The authors are grateful to the referees for their careful reading of the paper, giving many helpful suggestions and comments, which have led to the improvement of the paper’s exposition.

References

[1] J. F. Adams, On the structure and applications of the Steenrod algebra, Comment. Math. Helv. 32 (1958), 180–214. 10.1142/9789814401319_0007Search in Google Scholar

[2] J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20–104. 10.2307/1970147Search in Google Scholar

[3] J. F. Adams, Operations of the nth kind in K-theory, and what we don’t know about R ⁢ P ∞ , New Developments in Topology, London Math. Soc. Lecture Note Ser. 11, Cambridge University, London (1974), 1–9. 10.1017/CBO9780511662607.002Search in Google Scholar

[4] J. M. Boardman, Modular representations on the homology of powers of real projective space, Algebraic Topology (Oaxtepec 1991), Contemp. Math. 146, American Mathematical Society, Providence (1993), 49–70. 10.1090/conm/146/01215Search in Google Scholar

[5] A. K. Bousfield, E. B. Curtis, D. M. Kan, D. G. Quillen, D. L. Rector and J. W. Schlesinger, The mod - p lower central series and the Adams spectral sequence, Topology 5 (1966), 331–342. 10.1016/0040-9383(66)90024-3Search in Google Scholar

[6] R. R. Bruner, L. M. Hà and N. H. V. Hưng, On the behavior of the algebraic transfer, Trans. Amer. Math. Soc. 357 (2005), no. 2, 473–487. 10.1090/S0002-9947-04-03661-XSearch in Google Scholar

[7] T.-W. Chen, Determination of Ext 𝒜 5 , * ⁢ ( ℤ / 2 , ℤ / 2 ) , Topology Appl. 158 (2011), no. 5, 660–689. 10.1016/j.topol.2011.01.002Search in Google Scholar

[8] R. L. Cohen, W. H. Lin and M. E. Mahowald, The Adams spectral sequence of the real projective spaces, Pacific J. Math. 134 (1988), no. 1, 27–55. 10.2140/pjm.1988.134.27Search in Google Scholar

[9] D. M. Davis, An infinite family in the cohomology of the Steenrod algebra, J. Pure Appl. Algebra 21 (1981), no. 2, 145–150. 10.1016/0022-4049(81)90003-7Search in Google Scholar

[10] L. E. Dickson, A fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans. Amer. Math. Soc. 12 (1911), no. 1, 75–98. 10.1090/S0002-9947-1911-1500882-4Search in Google Scholar

[11] L. M. Hà, Sub-Hopf algebras of the Steenrod algebra and the Singer transfer, Proceedings of the School and Conference in Algebraic Topology, Geom. Topol. Monogr. 11, Vietnam National University, Hanoi (2007), 101–124. Search in Google Scholar

[12] N. H. V. Hưng, Spherical classes and the algebraic transfer, Trans. Amer. Math. Soc. 349 (1997), no. 10, 3893–3910. 10.1090/S0002-9947-97-01991-0Search in Google Scholar

[13] N. H. V. Hưng, The weak conjecture on spherical classes, Math. Z. 231 (1999), no. 4, 727–743. 10.1007/PL00004750Search in Google Scholar

[14] N. H. V. Hưng, Spherical classes and the lambda algebra, Trans. Amer. Math. Soc. 353 (2001), no. 11, 4447–4460. 10.1090/S0002-9947-01-02766-0Search in Google Scholar

[15] N. H. V. Hưng, On triviality of Dickson invariants in the homology of the Steenrod algebra, Math. Proc. Cambridge Philos. Soc. 134 (2003), no. 1, 103–113. 10.1017/S0305004102006187Search in Google Scholar

[16] N. H. V. Hưng, The cohomology of the Steenrod algebra and representations of the general linear groups, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4065–4089. 10.1090/S0002-9947-05-03889-4Search in Google Scholar

[17] N. H. V. Hưng and G. Powell, The 𝒜 -decomposability of the Singer construction, J. Algebra 517 (2019), 186–206. 10.1016/j.jalgebra.2018.09.030Search in Google Scholar

[18] N. H. V. Hưng and V. T. N. Quỳnh, The image of Singer’s fourth transfer, C. R. Math. Acad. Sci. Paris 347 (2009), no. 23–24, 1415–1418. 10.1016/j.crma.2009.10.018Search in Google Scholar

[19] N. H. V. Hưng and L. X. Trường, The algebraic transfer for the real projective space, C. R. Math. Acad. Sci. Paris 357 (2019), no. 2, 111–114. 10.1016/j.crma.2019.01.001Search in Google Scholar

[20] N. H. V. Hưng and N. A. Tuấn, The generalized algebraic conjecture on spherical classes, Manuscripta Math. 162 (2020), no. 1–2, 133–157. 10.1007/s00229-019-01117-wSearch in Google Scholar

[21] M. Kameko, Products of projective spaces as Steenrod modules, Thesis, The Johns Hopkins University, 1990. Search in Google Scholar

[22] J. Lannes and S. Zarati, Sur les foncteurs dérivés de la déstabilisation, Math. Z. 194 (1987), no. 1, 25–59. 10.1007/BF01168004Search in Google Scholar

[23] W. H. Lin, Algebraic Kahn–Priddy theorem, Pacific J. Math. 96 (1981), no. 2, 435–455. 10.2140/pjm.1981.96.435Search in Google Scholar

[24] W.-H. Lin, Ext A 4 , ∗ ⁢ ( ℤ / 2 , ℤ / 2 ) and Ext A 5 , ∗ ⁢ ( ℤ / 2 , ℤ / 2 ) , Topology Appl. 155 (2008), no. 5, 459–496. 10.1016/j.topol.2007.11.003Search in Google Scholar

[25] W.-H. Lin and M. Mahowald, The Adams spectral sequence for Minami’s theorem, Homotopy Theory via Algebraic Geometry and Group Representations (Evanston 1997), Contemp. Math. 220, American Mathematical Society, Providence, (1998), 143–177. 10.1090/conm/220/03098Search in Google Scholar

[26] A. Liulevicius, The factorization of cyclic reduced powers by secondary cohomology operations, Mem. Amer. Math. Soc. 42 (1962), 1–112. 10.1090/memo/0042Search in Google Scholar

[27] J. P. May, A general algebraic approach to Steenrod operations, The Steenrod Algebra and its Applications, Lecture Notes in Math. 168, Springer, Berlin (1970), 153–231. 10.1007/BFb0058524Search in Google Scholar

[28] N. Minami, The iterated transfer analogue of the new doomsday conjecture, Trans. Amer. Math. Soc. 351 (1999), no. 6, 2325–2351. 10.1090/S0002-9947-99-02037-1Search in Google Scholar

[29] H. Mui, Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), no. 3, 319–369. Search in Google Scholar

[30] T. N. Nam, Transfert algébrique et action du groupe linéaire sur les puissances divisées modulo 2, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 5, 1785–1837. 10.5802/aif.2399Search in Google Scholar

[31] W. M. Singer, Invariant theory and the lambda algebra, Trans. Amer. Math. Soc. 280 (1983), no. 2, 673–693. 10.1090/S0002-9947-1983-0716844-7Search in Google Scholar

[32] W. M. Singer, The transfer in homological algebra, Math. Z. 202 (1989), no. 4, 493–523. 10.1007/BF01221587Search in Google Scholar

[33] N. E. Steenrod, Cohomology Operations, Ann. of Math. Stud. 50, Princeton University, Princeton, 1962. Search in Google Scholar

[34] M. C. Tangora, On the cohomology of the Steenrod algebra, Math. Z. 116 (1970), 18–64. 10.1007/BF01110185Search in Google Scholar

[35] J. S. P. Wang, On the cohomology of the mod - 2 Steenrod algebra and the non-existence of elements of Hopf invariant one, Illinois J. Math. 11 (1967), 480–490. 10.1215/ijm/1256054570Search in Google Scholar

[36] C. Wilkerson, Classifying spaces, Steenrod operations and algebraic closure, Topology 16 (1977), no. 3, 227–237. 10.1016/0040-9383(77)90003-9Search in Google Scholar

[37] R. M. W. Wood, Steenrod squares of polynomials and the Peterson conjecture, Math. Proc. Cambridge Philos. Soc. 105 (1989), no. 2, 307–309. 10.1017/S0305004100067797Search in Google Scholar

[38] R. M. W. Wood, Problems in the Steenrod algebra, Bull. Lond. Math. Soc. 30 (1998), no. 5, 449–517. 10.1112/S002460939800486XSearch in Google Scholar

Received: 2021-09-21

Revised: 2022-06-28

Published Online: 2022-08-30

Published in Print: 2022-11-01

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Articles in the same Issue

https://doi.org/10.1515/forum-2021-0245

Keywords for this article

Steenrod algebra; Adams spectral sequences; Singer transfer; lambda algebra; invariant theory; Dickson algebra