Images of the Singer transfers and their possibility to be injective

Abstract

This article is an attempt to investigate the possibility to be injective of the Singer transfer TrM s : F2 ⊗GLs P(H∗Vs ⊗ M∗) → Exts A (Σ-sM, F2) for M being the A -modules F2 = Hå∗S0 or Hå∗RP∞. The existence of a positive stem critical element of Exts,t A (Hå∗RP∞, F2) in the image of the transfer TrRP∞ s is equivalent to the existence of a positive stem critical element of Exts+1,t+1 A (F2, F2) in the image of the transfer Trs+1. If the existences happen, then TrRP∞ s and Trs+1 are not injective. We show that the critical element P h ä2 is not in the image of the fourth transfer, TrRP∞ 4 : F2 ⊗GL4 P(H∗V4 ⊗ Hå∗RP∞)t-4 → Ext4 A,t(Hå∗RP∞, F2). Singer’s conjecture is still open, as we have not known any critical element, which is in the image of the transfer.