Subgroup conditions for groups acting freely on products of spheres
HTML articles powered by AMS MathViewer
- by Judith H. Silverman
- Trans. Amer. Math. Soc. 334 (1992), 153-181
- DOI: https://doi.org/10.1090/S0002-9947-1992-1100700-4
- PDF | Request permission
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.References
- Alejandro Adem and William Browder, The free rank of symmetry of $(S^n)^k$, Invent. Math. 92 (1988), no.Β 2, 431β440. MR 936091, DOI 10.1007/BF01404462
- J. F. Adams and C. W. Wilkerson, Finite $H$-spaces and algebras over the Steenrod algebra, Ann. of Math. (2) 111 (1980), no.Β 1, 95β143. MR 558398, DOI 10.2307/1971218
- Gunnar Carlsson, On the nonexistence of free actions of elementary abelian groups on products of spheres, Amer. J. Math. 102 (1980), no.Β 6, 1147β1157. MR 595008, DOI 10.2307/2374182
- Gunnar Carlsson, Some restrictions on finite groups acting freely on $(S^{n})^{k}$, Trans. Amer. Math. Soc. 264 (1981), no.Β 2, 449β457. MR 603774, DOI 10.1090/S0002-9947-1981-0603774-2
- H. E. A. Campbell and P. S. Selick, Polynomial algebras over the Steenrod algebra, Comment. Math. Helv. 65 (1990), no.Β 2, 171β180. MR 1057238, DOI 10.1007/BF02566601
- Robert Oliver, Free compact group actions on products of spheres, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Lecture Notes in Math., vol. 763, Springer, Berlin, 1979, pp.Β 539β548. MR 561237 J.-P. Sene, AlgΓ¨bre locale. MultiplicitΓ©s, Springer-Verlag, 1965.
- Jean-Pierre Serre, Sur la dimension cohomologique des groupes profinis, Topology 3 (1965), 413β420 (French). MR 180619, DOI 10.1016/0040-9383(65)90006-6
- R. M. W. Wood, Splitting $\Sigma (\textbf {C}\textrm {P}^\infty \times \cdots \times \textbf {C}\textrm {P}^\infty )$ and the action of Steenrod squares $\textrm {Sq}^i$ on the polynomial ring $F_2[x_1,\cdots ,x_n]$, Algebraic topology, Barcelona, 1986, Lecture Notes in Math., vol. 1298, Springer, Berlin, 1987, pp.Β 237β255. MR 928837, DOI 10.1007/BFb0083014
Bibliographic Information
- © Copyright 1992 American Mathematical Society
- 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