A NOTE ON A-ANNIHILATED GENERATORS OF H*QX

ZARE, HADI

Glasgow mathematical journal, Tome 62 (2020) no. 2, pp. 281-295

Résumé

For a path connected space X, the homology algebra $H_*(QX; \mathbb{Z}/2)$ is a polynomial algebra over certain generators QIx. We reinterpret a technical observation, of Curtis and Wellington, on the action of the Steenrod algebra A on the Λ algebra in our terms. We then introduce a partial order on each grading of H*QX which allows us to separate terms in a useful way when computing the action of dual Steenrod operations $Sq^i_*$ on $H_*(QX; \mathbb{Z}/2)$. We use these to completely characterise the A-annihilated generators of this polynomial algebra. We then propose a construction for sequences I so that QIx is A-annihilated. As an application, we offer some results on the form of potential spherical classes in H*QX upon some stability condition under homology suspension. Our computations provide new numerical conditions in the context of hit problem.

Détail

DOI : 10.1017/S0017089519000090

Bibliographie

[1] Asadi-Golmankhaneh, M. A. and Eccles, P. J., Determining the characteristic numbers of self-intersection manifolds, J. Lond. Math. Soc., II. Ser. 62(1) (2000), 278–290. Google Scholar | DOI

[2] Asadi-Golmankhaneh, M. A. and Eccles, P. J., Double point self-intersection surfaces of immersions, Geom. Topol. 4, 149–170, 2000. Google Scholar

[3] Ault, S. V. and Singer, W., On the homology of elementary Abelian groups as modules over the Steenrod algebra, J. Pure Appl. Algebra 215(12) (2011), 2847–2852.10.1016/j.jpaa.2011.04.004 Google Scholar | DOI

[4] Cohen, F. R., Lada, T. J., and May, J. P.. The homology of iterated loop spaces, Lecture Notes in Mathematics, vol. 533 (Springer-Verlag, Berlin, Heidelberg, New York, 1976), vii+490. Google Scholar | DOI

[5] Curtis, E. B., The Dyer-Lashof algebra and the Λ-algebra, Ill. J. Math. 19 (1975), 231–246. Google Scholar | DOI

[6] Galatius, S., Mod p homology of the stable mapping class group, Topology 43(5) (2004), 1105–1132. Google Scholar | DOI

[7] Giambalvo, V. and Peterson, F. P.. A-generators for ideals in the Dickson algebra, J. Pure Appl. Algebra 158(2-3) (2001), 161–182.10.1016/S0022-4049(00)00051-7 Google Scholar | DOI

[8] Zadeh, M. A. H. and Zare, H., An algorithmic search for A-annihilated classes in H Q S 0 (in preparation). Google Scholar

[9] Koschorke, U. and Sanderson, B., Self-intersections and higher Hopf invariants, Topology 17 (1978), 283–290. Google Scholar | DOI

[10] Kuhn, N. J., Adams filtration and generalised Hurewicz maps for infinite loop spaces, arxiv:1403.7501v1. Google Scholar

[11] Kuhn, N. J., The mod p Hurewicz map for infinite loopspaces: theorems, examples, and conjectures. Talk at Princeton University (2015). Google Scholar

[12] Madsen, I., On the action of the Dyer-Lashof algebra in H (G) and H (G/Top), PhD thesis (The University of Chicago, Ann Arbor, MI, 1970). Google Scholar

[13] Madsen, I., On the action of the Dyer-Lashof algebra in H (G), Pac. J. Math. 60(1) (1975), 235–275. Google Scholar | DOI

[14] May, J. P., The geometry of iterated loop spaces, Lecture Notes in Mathematics, vol. 271 (Springer-Verlag, Berlin, Heidelberg, New York, 1972), ix+175. Google Scholar | DOI

[15] Ravenel, D. C., Complex cobordism and stable homotopy groups of spheres, 2nd edition (AMS Chelsea Publishing, Providence, RI, 2004). Google Scholar

[16] Repka, J. and Selick, P.. On the subalgebra of annihilated by Steenrod operations, J. Pure Appl. Algebra 127(3) (1998), 273–288. Google Scholar | DOI

[17] Snaith, V. and Tornehave, J., On and the Arf invariant of framed manifolds, in Symposium on Algebraic Topology in honor of José Adem (Oaxtepec, 1981), Contemporary Mathematics, vol. 12 (American Mathematical Society, Providence, RI, 1982), 299–313. Google Scholar | DOI

[18] Snaith, V. P., Stable homotopy around the Arf-Kervaire invariant (Birkhäuser, Basel, 2009). Google Scholar | DOI

[19] Walker, Grant and Reginald, M. W. Wood. Polynomials and the mod 2 Steenrod algebra. Vol. 1 . The Peterson hit problem, volume 441 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2018. Google Scholar

[20] Walker, G. and Wood, R. M. W., Polynomials and the mod 2 Steenrod algebra, vol. 2. Representations of , London Mathematical Society Lecture Note Series, vol. 442 (Cambridge University Press, Cambridge, 2018).10.1017/9781108304092 Google Scholar | DOI

[21] Wellington, R. J., The A-algebra HΩn+1∑n+1X, the Dyer-Lashof algebra, and the Λ-algebra, PhD Thesis (The University of Chicago, Ann Arbor, MI, 1977). Google Scholar

[22] Wellington, R. J., The unstable Adams spectral sequence for free iterated loop spaces, Mem. Am. Math. Soc. 36(258) (1982), 225. Google Scholar

[23] Wood, R. M. W., Problems in the Steenrod algebra, Bull. Lond. Math. Soc. 30(5) (1998), 449–517.10.1112/S002460939800486X Google Scholar | DOI

[24] Zare, H., The Dyer-Lashof algebra and the hit problems (submitted). Google Scholar

[25] Zare, H., Filtered finiteness of the image of the unstable Hurewicz homomorphism with applications to bordism of immersions, Q. J. Math., hay069, Google Scholar | DOI

[26] Zare, H., On spherical classes in H QSn, PhD Thesis (University of Manchester, 2009). Google Scholar

[27] Zare, H., Spherical classes in some finite loop spaces of spheres, Topol. Appl. 224 (2017) 1–18.10.1016/j.topol.2017.03.013 Google Scholar | DOI