Group actions on Euclidean spaces
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==== History ==== | ==== History ==== | ||
+ | ==== Definitions ==== | ||
+ | <wikitex>; | ||
+ | Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$. | ||
+ | The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes. | ||
+ | *$\mathcal{A}=\{G\;\colon\; G$ has a $pq$-dihedral subquotient $\}$. | ||
+ | *$\mathcal{B}=\{G\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$. | ||
+ | *$\mathcal{C}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$. | ||
+ | *$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$. | ||
+ | *$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$. | ||
+ | *$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$. | ||
+ | |||
+ | Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternion $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$: | ||
+ | $$\widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$$ and the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$: | ||
+ | $$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ | ||
+ | |||
+ | For a finitely generated abelian group $A$, denote by $\operatorname{Tor}(A)$ the torsion part of $A$. | ||
+ | </wikitex> | ||
==== Results ==== | ==== Results ==== | ||
+ | <wikitex>; | ||
+ | Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Oliver \cite{Oliver1975}, \cite{Oliver1976} has defined an integer $n_G\geq 0$, which we refer to as [[Group_actions_on_disks#Oliver_number|the Oliver number]] of $G$. Recall that $n_G=n_{G/G_0}$ when $G_0$ is abelian, and otherwise $n_G=1$. | ||
+ | |||
+ | {{beginthm|Theorem|(\cite{Oliver1975}, \cite{Oliver1976})}} | ||
+ | Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite CW-complex. Then the following three statements are equivalent. | ||
+ | *The Euler-Poincaré characteristic $\chi(F)\equiv 1 \pmod{n_G}$. | ||
+ | *There exist a finite contractible $G$-CW-complex $X$ such that the fixed point set $X^G$ is homeomorphic to $F$. | ||
+ | *There exists a smooth action of $G$ on a disk $D$ such that the fixed point set $D^G$ is homotopy equivalent to $F$. | ||
+ | |||
+ | {{endthm}} | ||
+ | |||
+ | {{beginthm|Theorem|(\cite{Oliver1996})}} | ||
+ | Let $G$ be a finite group not of prime power order. Let $F$ be a compact smooth manifold. Then there exists a smooth action of $G$ on some disk $D$ such that the fixed point $D^G$ is diffeomorphic to $F$ if and only if $\chi(F)\equiv 1\pmod{n_G}$ and the class $[\tau_F]\in \widetilde{K}O(F)$ satisfies one of the following condition. | ||
+ | *If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$. | ||
+ | *If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{Tor}(\widetilde{K}U(F))$. | ||
+ | *If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{Tor}(\widetilde{K}O(F))$. | ||
+ | *If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex. | ||
+ | *If $G\in\mathcal{E}$, then $[\tau_F]\in\text{Tor}(\widetilde{K}O(F))$. | ||
+ | *If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{Tor}(\widetilde{K}U(F)))$. | ||
+ | {{endthm}} | ||
+ | </wikitex> | ||
== References == | == References == |
Revision as of 16:29, 27 November 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Topological actions
...
2 Smooth actions
2.1 Fixed point free
2.1.1 History
The question whether contractible manifolds (e.g. Euclidean spaces) admit smooth fixed point free actions of compact Lie groups has been discussed for the first time by Paul Althaus Smith [Smith1938], [Smith1939], [Smith1941], and [Smith1945]. Conner and Montgomery [Conner&Montgomery1962] have constructed smooth fixed point free actions of on Euclidean spaces, for . By generalizing their construction, Hsiang and Hsiang [Hsiang&Hsiang1967] have shown that any non-abelian compact connected Lie group admits such actions.
For for two relatively primes integers , the construction of Conner and Floyd [Conner&Floyd1959], modified and improved by Kister [Kister1961] and [Kister1963], yields a smooth fixed point free actions on Euclidean spaces (see [Bredon1972, pp. 58-61]). For more general groups such that there exist a surjection and an injection , such actions have been obtained by Edmonds and Lee [Edmonds&Lee1976].
2.1.2 Results
The results of [Conner&Montgomery1962], [Hsiang&Hsiang1967], [Conner&Floyd1959], [Kister1961], and [Edmonds&Lee1976] yield the following theorem.
Theorem 2.1. A compact Lie group has smooth fixed point free action on some Euclidean space if and only if at least one of the following conditions holds.
- The identity connected component of is non-abelian.
- The quotient is not of prime power order.
2.2 Fixed point sets
2.2.1 History
2.2.2 Definitions
Let be a finite group. For two distinct primes and , a -element of is an element of order . One says that has -dihedral subquotient if contains two subgroups and such that is isomorphic to the dihedral group of order . Denote by a -Sylow subgroup of .
The class of finite groups not of prime power order divides into the following six mutually disjoint classes.
- has a -dihedral subquotient .
- has no -dihedral subquotient, has a -element conjugate to its inverse .
- has no -element conjugate to its inverse, has a -element, .
- has no -element conjugate to its inverse, has a -element, .
- has no -element, .
- has no -element, .
Let be a compact smooth manifold. Between the reduced real, complex, and quaternion -theory groups , , and , respectively, consider the induction (complexification and quaternization) homomorphisms and :
For a finitely generated abelian group , denote by the torsion part of .
2.2.3 Results
Let be a compact Lie group such that the identity connected component of is non-abelian, or the quotient is not of prime power order. Oliver [Oliver1975], [Oliver1976] has defined an integer , which we refer to as the Oliver number of . Recall that when is abelian, and otherwise .
Theorem 2.2 ([Oliver1975], [Oliver1976]). Let be a compact Lie group such that the identity connected component of is non-abelian, or the quotient is not of prime power order. Let be a finite CW-complex. Then the following three statements are equivalent.
- The Euler-Poincaré characteristic .
- There exist a finite contractible -CW-complex such that the fixed point set is homeomorphic to .
- There exists a smooth action of on a disk such that the fixed point set is homotopy equivalent to .
Theorem 2.3 ([Oliver1996]). Let be a finite group not of prime power order. Let be a compact smooth manifold. Then there exists a smooth action of on some disk such that the fixed point is diffeomorphic to if and only if and the class satisfies one of the following condition.
- If , then there is no restriction on .
- If , then .
- If , then .
- If , then , i.e., is stably complex.
- If , then .
- If , then .
3 References
- [Bredon1972] G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972. MR0413144 (54 #1265) Zbl 0484.57001
- [Conner&Floyd1959] P. E. Conner and E. E. Floyd, On the construction of periodic maps without fixed points, Proc. Amer. Math. Soc. 10 (1959), 354–360. MR0105115 (21 #3860) Zbl 0092.39701
- [Conner&Montgomery1962] P. Conner and D. Montgomery, An example for , Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 1918–1922. MR0148795 (26 #6300) Zbl 0107.16604
- [Edmonds&Lee1976] A. L. Edmonds and R. Lee, Compact Lie groups which act on Euclidean space without fixed points, Proc. Amer. Math. Soc. 55 (1976), no.2, 416–418. MR0420672 (54 #8684) Zbl 0326.57011
- [Hsiang&Hsiang1967] W. Hsiang and W. Hsiang, Differentiable actions of compact connected classical groups. I, Amer. J. Math. 89 (1967), 705–786. MR0217213 (36 #304) Zbl 0205.53902
- [Kister1961] J. M. Kister, Examples of periodic maps on Euclidean spaces without fixed points. , Bull. Amer. Math. Soc. 67 (1961), 471–474. MR0130929 (24 #A783) Zbl 0101.15602
- [Kister1963] J. M. Kister, Differentiable periodic actions on without fixed points, Amer. J. Math. 85 (1963), 316–319. MR0154278 (27 #4227) Zbl 0119.18801
- [Oliver1975] R. Oliver, Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155–177. MR0375361 (51 #11556) Zbl 0304.57020
- [Oliver1976] R. Oliver, Smooth compact Lie group actions on disks, Math. Z. 149 (1976), no.1, 79–96. MR0423390 (54 #11369) Zbl 0334.57023
- [Oliver1996] B. Oliver, Fixed point sets and tangent bundles of actions on disks and Euclidean spaces, Topology 35 (1996), no.3, 583–615. MR1396768 (97g:57059) Zbl 0861.57047
- [Smith1938] P. A. Smith, Transformations of finite period, Ann. of Math. (2) 39 (1938), no.1, 127–164. MR1503393 Zbl 0063.07093
- [Smith1939] P. A. Smith, Transformations of finite period. II, Ann. of Math. (2) 40 (1939), 690–711. MR0000177 (1,30c) Zbl 0063.07093
- [Smith1941] P. A. Smith, Transformations of finite period. III. Newman's theorem, Ann. of Math. (2) 42 (1941), 446–458. MR0004128 (2,324c)
- [Smith1945] P. A. Smith, Transformations of finite period. IV. Dimensional parity, Ann. of Math. (2) 46 (1945), 357–364. MR0013304 (7,136e) Zbl 0063.07093