The Singer transfer for infinite real projective space

Abstract

Abstract The article is devoted to studying the Singer transfer. The image of the Singer transfer Tr * ℝ ⁢ ℙ ∞ {\operatorname{Tr}_{*}^{\mathbb{R}\mathbb{P}^{\infty}}} for the infinite real projective space is proved to be a module over the image of the transfer Tr * {\operatorname{Tr}_{*}} for the sphere. Further, the algebraic Kahn–Priddy homomorphism is shown to be an epimorphism from Im ⁡ Tr * ℝ ⁢ ℙ ∞ {\operatorname{Im}\operatorname{Tr}_{*}^{\mathbb{R}\mathbb{P}^{\infty}}} onto Im ⁡ Tr * {\operatorname{Im}\operatorname{Tr}_{*}} in positive stems. The indecomposable elements h ^ i {\widehat{h}_{i}} for i ≥ 1 {i\geq 1} and c ^ i , d ^ i , e ^ i , f ^ i , p ^ i {\widehat{c}_{i},\widehat{d}_{i},\widehat{e}_{i},\widehat{f}_{i},\widehat{p}_{% i}} for i ≥ 0 {i\geq 0} are in the image of the Singer transfer Tr * ℝ ⁢ ℙ ∞ {\operatorname{Tr}_{*}^{\mathbb{R}\mathbb{P}^{\infty}}} , whereas the ones g ^ i {\widehat{g}_{i}} for i ≥ 1 {i\geq 1} and D ^ 3 ⁢ ( i ) , p ^ i ′ {\widehat{D}_{3}(i),\widehat{p}^{\prime}_{i}} for i ≥ 0 {i\geq 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 ∞ {\operatorname{Tr}_{*}^{\mathbb{R}\mathbb{P}^{\infty}}} 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.