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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Subgroup conditions for groups acting freely on products of spheres


Author: Judith H. Silverman
Journal: Trans. Amer. Math. Soc. 334 (1992), 153-181
MSC: Primary 55M35; Secondary 55N91
DOI: https://doi.org/10.1090/S0002-9947-1992-1100700-4
MathSciNet review: 1100700
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Abstract: Let $ d$ and $ h$ be integers such that either $ d \geq 2$ and $ h = {2^d} - 1$, or $ d = 4$ and $ h = 5$. Suppose that the group $ \mathcal{G}$ contains an elementary-abelian $ 2$-subgroup $ {E_d}$ of rank $ d$ with an element $ \sigma $ of order $ h$ in its normalizer. We show that if $ \mathcal{G}$ admits a free and $ {{\mathbf{F}}_2}$-cohomologically trivial action on $ {({S^n})^d}$, then some nontrivial power of $ \sigma $ centralizes $ {E_d}$.

The cohomology ring $ {H^{\ast} }({E_d};{{\mathbf{F}}_2}) \simeq {{\mathbf{F}}_2}[{y_1}, \ldots ,{y_d}]$ is a module over the Steenrod algebra $ \mathcal{A}(2)$. Let $ \theta \in {{\mathbf{F}}_2}[{y_1}, \ldots ,{y_d}]$, and let $ c \geq d - 2$ be an integer. We show that $ \theta $ divides $ S{q^{{2^i}}}(\theta )$ in the polynomial ring for $ 0 \leq i \leq c \Leftrightarrow \theta = {\tau ^{{2^{c - d + 3}}}}\pi $ , where $ \tau $ divides $ S{q^{{2^i}}}(\tau )$ for $ 0 \leq i \leq d - 3$ and $ \pi $ is a product of linear forms.


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DOI: https://doi.org/10.1090/S0002-9947-1992-1100700-4
Keywords: Steenrod algebra, free group actions
Article copyright: © Copyright 1992 American Mathematical Society