Spherical classes and the algebraic transfer

Abstract

We study a weak form of the classical conjecture which predicts that there are no spherical classes in Q 0 S 0 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 = F 2 [ x 1 , … , x k ] P_k=\mathbb {F}_2[x_1,\ldots ,x_k] with | x i | = 1 |x_i|=1 . The general linear group G L k = G L ( k , F 2 ) \mathrm {GL}_k=GL(k,\mathbb {F}_2) and the (mod 2) Steenrod algebra A \mathcal {A} act on P k P_k in the usual manner. We prove that the weak conjecture is equivalent to the following one: The canonical homomorphism j k : F 2 ⊗ A ( P k G L k ) → ( F 2 ⊗ A P k ) G L k 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 P_k is zero in positive dimensions for k > 2 k>2 . In other words, every Dickson invariant (i.e. element of P k G L k P_k^{\mathrm {GL}_k} ) of positive dimension belongs to A + ⋅ P k \mathcal {A}^+ \cdot P_k for k > 2 k>2 , where A + \mathcal {A} ^+ denotes the augmentation ideal of A \mathcal {A} . This conjecture is proved for k = 3 k=3 in two different ways. One of these two ways is to study the squaring operation S q 0 Sq^0 on P ( F 2 ⊗ G L k P k ∗ ) P(\mathbb {F}_2 \underset {GL_k}{\otimes } P_k^*) , the range of j k ∗ j_k^* , and to show it commuting through j k ∗ j_k^* with Kameko’s S q 0 Sq^0 on F 2 ⊗ G L k P ( P k ∗ ) \mathbb {F}_2 \underset {GL_k}{\otimes } P(P_k^*) , the domain of j k ∗ j_k^* . We compute explicitly the action of S q 0 Sq^0 on P ( F 2 ⊗ G L k P k ∗ ) P(\mathbb {F}_2 \underset {GL_k}{\otimes } P_k^*) for k ≤ 4 k \leq 4 .