Relations Between Kernels and Images of Reduced Powers for Some Right ${𝒜}_p$-Modules

Theodore Popelensky

Abstract: We investigate the right action of the mod $p$ Steenrod algebra ${𝒜}_p$ on the homology $H_{*}(L^{\wedge s},{\mathbb{Z}}_p)$ where $L=B{\mathbb{Z}}_p$ is the lens space. Following ideas of Ault and Singer we investigate the relation between intersection of kernels of the reduced powers $P^{p^i}$ and Bockstein element $\beta$ and the intersection of images of $P^{p^{i+1}-1}$ and of $\beta$. Namely one can check that ${\bigcap }_{i=0}^{k}im{P}^{p^{i+1}-1}\subset {\bigcap }_{i=0}^{k}ker{P}^{p^i}$ and ${\bigcap }_{i=0}^{k}im{P}^{p^{i+1}-1}\cap im\beta \subset {\bigcap }_{i=0}^{k}ker{P}^{p^i}\cap ker\beta$. We generalize Ault’s homotopy systems to $p>2$ and examine when the reverse inclusions are true.

Keywords: Steenrod algebra, reduced powers

Classification (MSC2000): 55S10; 55R40; 57T25

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