Spherical classes and the algebraic transfer

Author:
Nguyẽn H. V. Hu’ng

Journal:
Trans. Amer. Math. Soc. **349** (1997), 3893-3910

MSC (1991):
Primary 55P47, 55Q45, 55S10, 55T15

DOI:
https://doi.org/10.1090/S0002-9947-97-01991-0

Erratum:
Trans. Amer. Math. Soc. **355** (2003), 3841-3842.

MathSciNet review:
1433119

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study a weak form of the classical conjecture which predicts that there are no spherical classes in $Q_0S^0$ except the elements of Hopf invariant one and those of Kervaire invariant one. The weak conjecture is obtained by restricting the Hurewicz homomorphism to the homotopy classes which are detected by the algebraic transfer. Let $P_k=\mathbb {F}_2[x_1,\ldots ,x_k]$ with $|x_i|=1$. The general linear group $\mathrm {GL}_k=GL(k,\mathbb {F}_2)$ and the (mod 2) Steenrod algebra $\mathcal {A}$ act on $P_k$ in the usual manner. We prove that the weak conjecture is equivalent to the following one: The canonical homomorphism $j_k:\mathbb {F}_2 \underset {\mathcal {A}}{\otimes } (P_k^{\mathrm {GL}_k})\to (\mathbb {F}_2 \underset {\mathcal {A}}{\otimes } P_k)^{\mathrm {GL}_k}$ induced by the identity map on $P_k$ is zero in positive dimensions for $k>2$. In other words, every Dickson invariant (i.e. element of $P_k^{\mathrm {GL}_k}$) of positive dimension belongs to $\mathcal {A}^+ \cdot P_k$ for $k>2$, where $\mathcal {A} ^+$ denotes the augmentation ideal of $\mathcal {A}$. This conjecture is proved for $k=3$ in two different ways. One of these two ways is to study the squaring operation $Sq^0$ on $P(\mathbb {F}_2 \underset {GL_k}{\otimes } P_k^*)$, the range of $j_k^*$, and to show it commuting through $j_k^*$ with Kameko’s $Sq^0$ on $\mathbb {F}_2 \underset {GL_k}{\otimes } P(P_k^*)$, the domain of $j_k^*$. We compute explicitly the action of $Sq^0$ on $P(\mathbb {F}_2 \underset {GL_k}{\otimes } P_k^*)$ for $k \leq 4$.

- J. F. Adams,
*On the non-existence of elements of Hopf invariant one*, Ann. of Math. (2)**72**(1960), 20–104. MR**141119**, DOI https://doi.org/10.2307/1970147 - J. F. Adams,
*Operations of the $n$th kind in $K$-theory, and what we don’t know about $RP^{\infty }$*, New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), Cambridge Univ. Press, London, 1974, pp. 1–9. London Math. Soc. Lecture Note Ser., No. 11. MR**0339178** - Mohamed Ali Alghamdi, M. C. Crabb, and J. R. Hubbuck,
*Representations of the homology of $BV$ and the Steenrod algebra. I*, Adams Memorial Symposium on Algebraic Topology, 2 (Manchester, 1990) London Math. Soc. Lecture Note Ser., vol. 176, Cambridge Univ. Press, Cambridge, 1992, pp. 217–234. MR**1232208**, DOI https://doi.org/10.1017/CBO9780511526312.020 - J. Michael Boardman,
*Modular representations on the homology of powers of real projective space*, Algebraic topology (Oaxtepec, 1991) Contemp. Math., vol. 146, Amer. Math. Soc., Providence, RI, 1993, pp. 49–70. MR**1224907**, DOI https://doi.org/10.1090/conm/146/01215 - William Browder,
*The Kervaire invariant of framed manifolds and its generalization*, Ann. of Math. (2)**90**(1969), 157–186. MR**251736**, DOI https://doi.org/10.2307/1970686 - D. P. Carlisle and R. M. W. Wood,
*The boundedness conjecture for the action of the Steenrod algebra on polynomials*, Adams Memorial Symposium on Algebraic Topology, 2 (Manchester, 1990) London Math. Soc. Lecture Note Ser., vol. 176, Cambridge Univ. Press, Cambridge, 1992, pp. 203–216. MR**1232207**, DOI https://doi.org/10.1017/CBO9780511526312.019 - H. Cartan,
*Puissances divisées*, Séminaire H. Cartan, École Norm. Sup., Paris, 1954/55, Exposé 7. - M. C. Crabb and J. R. Hubbuck,
*Representations of the homology of BV and the Steenrod algebra II,*Algebraic Topology: New Trends in Localization and Periodicity (Sant Feliu de Guixois, 1994; C. Broto et al., eds.), Progr. Math. 136, Birkhäuser, 1996, pp. 143–154. - Edward B. Curtis,
*The Dyer-Lashof algebra and the $\Lambda $-algebra*, Illinois J. Math.**19**(1975), 231–246. MR**377885** - 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), 75–98. - M. Kameko,
*Products of projective spaces as Steenrod modules*, Thesis, Johns Hopkins University 1990. - Jean Lannes and Saïd Zarati,
*Invariants de Hopf d’ordre supérieur et suite spectrale d’Adams*, C. R. Acad. Sci. Paris Sér. I Math.**296**(1983), no. 15, 695–698 (French, with English summary). MR**705694** - Jean Lannes and Saïd Zarati,
*Sur les foncteurs dérivés de la déstabilisation*, Math. Z.**194**(1987), no. 1, 25–59 (French). MR**871217**, DOI https://doi.org/10.1007/BF01168004 - Arunas Liulevicius,
*The factorization of cyclic reduced powers by secondary cohomology operations*, Mem. Amer. Math. Soc.**42**(1962), 112. MR**182001** - Ib Madsen,
*On the action of the Dyer-Lashof algebra in $H_{\ast }(G)$*, Pacific J. Math.**60**(1975), no. 1, 235–275. MR**388392** - J. Peter May,
*A general algebraic approach to Steenrod operations*, The Steenrod Algebra and its Applications (Proc. Conf. to Celebrate N. E. Steenrod’s Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970), Lecture Notes in Mathematics, Vol. 168, Springer, Berlin, 1970, pp. 153–231. MR**0281196** - N. Minami,
*On the Hurewicz image of elementary p groups and an iterated transfer analogue of the new doomsday conjecture*, Preprint. - Nguyễn H. V. Hu’ng and Franklin P. Peterson,
*$\scr A$-generators for the Dickson algebra*, Trans. Amer. Math. Soc.**347**(1995), no. 12, 4687–4728. MR**1316852**, DOI https://doi.org/10.1090/S0002-9947-1995-1316852-X - Nguyễn H. V. Hu’ng and F. P. Peterson,
*Spherical classes and the Dickson algebra*, Math. Proc. Camb. Phil. Soc. (to appear). - William M. Singer,
*The transfer in homological algebra*, Math. Z.**202**(1989), no. 4, 493–523. MR**1022818**, DOI https://doi.org/10.1007/BF01221587 - Robert J. Wellington,
*The unstable Adams spectral sequence for free iterated loop spaces*, Mem. Amer. Math. Soc.**36**(1982), no. 258, viii+225. MR**646741**, DOI https://doi.org/10.1090/memo/0258 - Clarence Wilkerson,
*Classifying spaces, Steenrod operations and algebraic closure*, Topology**16**(1977), no. 3, 227–237. MR**442932**, DOI https://doi.org/10.1016/0040-9383%2877%2990003-9

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
55P47,
55Q45,
55S10,
55T15

Retrieve articles in all journals with MSC (1991): 55P47, 55Q45, 55S10, 55T15

Additional Information

**Nguyẽn H. V. Hu’ng**

Affiliation:
Centre de Recerca Matemàtica, Institut d’Estudis Catalans, Apartat 50, E–08193 Bellaterra, Barcelona, España

Address at time of publication:
Department of Mathematics, University of Hanoi, 90 Nguyẽn Trãi Street, Hanoi, Vietnam

Email:
nhvhung@it-hu.ac.vn

Keywords:
Spherical classes,
loop spaces,
Adams spectral sequences,
Steenrod algebra,
invariant theory,
Dickson algebra,
algebraic transfer

Received by editor(s):
April 7, 1995

Additional Notes:
The research was supported in part by the DGU through the CRM (Barcelona).