Laitinen conjecture
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Let $G$ be a finite group. A ''real $G$-module'' $V$ is a finite dimensional real vector space with a linear action of $G$, i.e., the action of $G$ on $V$ is given by a representation $G \to GL(V)$. | Let $G$ be a finite group. A ''real $G$-module'' $V$ is a finite dimensional real vector space with a linear action of $G$, i.e., the action of $G$ on $V$ is given by a representation $G \to GL(V)$. | ||
− | {{beginrem|Definition}} Two real $G$-modules $U$ and $V$ are called ''Smith equivalent'' if there exists a smooth action of $G$ on a homotopy sphere $\varSigma$ with exactly two fixed points $x$ and $y$ at which the tangent $G$-modules are isomorphic to $U$ and $V$, respectively, where ''the tangent $G$-modules'' are determined on the tangent spaces $T_x(\varSigma)$ and $T_y(\varSigma)$ at $x$ and $y$ by taking the derivatives at $x$ and $y$ of the diffeomorphisms $\varSigma \to \varSigma$, $z \mapsto gz$ considered for all $g \in G$. | + | {{beginrem|Definition}}\label{def:Smith} Two real $G$-modules $U$ and $V$ are called ''Smith equivalent'' if there exists a smooth action of $G$ on a homotopy sphere $\varSigma$ with exactly two fixed points $x$ and $y$ at which the tangent $G$-modules are isomorphic to $U$ and $V$, respectively, where ''the tangent $G$-modules'' are determined on the tangent spaces $T_x(\varSigma)$ and $T_y(\varSigma)$ at $x$ and $y$ by taking the derivatives at $x$ and $y$ of the diffeomorphisms $\varSigma \to \varSigma$, $z \mapsto gz$ considered for all $g \in G$. |
{{endrem}} | {{endrem}} | ||
* Let $RO(G)$ be the representation ring of $G$, i.e., the Grothendieck ring of the differences $U - V$ of real $G$-modules $U$ and $V$. As a group, $RO(G)$ is a finitely generated free abelian group whose rank is the number of real conjugacy classes of elements $g\in G$. Recall that the ''real conjugacy class'' of $g \in G$ is defined by $(g)^{\pm} := (g) \cup (g^{-1})$. Hereafter, $r_G$ denotes the number of real conjugacy classes $(g)^{\pm}$ represented by elements $g \in G$ not of prime power order. | * Let $RO(G)$ be the representation ring of $G$, i.e., the Grothendieck ring of the differences $U - V$ of real $G$-modules $U$ and $V$. As a group, $RO(G)$ is a finitely generated free abelian group whose rank is the number of real conjugacy classes of elements $g\in G$. Recall that the ''real conjugacy class'' of $g \in G$ is defined by $(g)^{\pm} := (g) \cup (g^{-1})$. Hereafter, $r_G$ denotes the number of real conjugacy classes $(g)^{\pm}$ represented by elements $g \in G$ not of prime power order. | ||
− | * Let $PO(G)$ be the subgroup of $RO(G)$ consisting of the differences $U - V \ | + | * Let $PO(G)$ be the subgroup of $RO(G)$ consisting of the differences $U - V$ of two $\mathcal{P}$-matched real $G$-modules $U$ and $V$, where the ''$\mathcal{P}$-matched condition'' means that $U$ and $V$ are isomorphic when restricted to any prime power order subgroup of $G$. The group $PO(G)$ is trivial, $PO(G) = 0$, if and only if $r_G = 0$. In the case $PO(G) \neq 0$, $PO(G)$ is a finitely generated free abelian group of rank $r_G$, $\operatorname{rank} PO(G) = r_G$. |
We shall make use of the notions of ''Smith set'' and ''pimary Smith set'' of $G$. | We shall make use of the notions of ''Smith set'' and ''pimary Smith set'' of $G$. | ||
− | * The ''Smith set'' of $G$ is the subset $Sm(G)$ of $RO(G)$ consisting of the differences $U - V \in RO(G)$ of Smith equivalent real $G$-modules $U$ and $V$. | + | * The ''Smith set'' of $G$ is the subset $Sm(G)$ of $RO(G)$ consisting of the differences $U - V \in RO(G)$ of two Smith equivalent real $G$-modules $U$ and $V$. |
− | * The ''primary Smith set'' of $G$ is the subset $PSm(G)$ of $RO(G)$ consisting of the differences $U - V \in RO(G)$ of | + | * The ''primary Smith set'' of $G$ is the subset $PSm(G)$ of $RO(G)$ consisting of the differences $U - V \in RO(G)$ of two $\mathcal{P}$-matched and Smith equivalent real $G$-modules $U$ and $V$. |
− | If two real $G$-modules $U$ and $V$ with $\dim U^G = \dim V^G = 0$ are isomorphic, then $U$ and $V$ are Smith equivalent (the sphere $\varSigma = S(V \oplus \mathbb{R})$ admits the required action of $G$, where $G$ acts trivially on $\mathbb{R}$ and diagonally on $V \oplus \mathbb{R}$). Therefore, the sets $PSm(G)$ and $Sm(G)$ both contain the zero $0 = V - V$ of $RO(G)$. The following problems are motivated by the question of Paul A. Smith posed in 1960, in the article {{cite|Smith1960|the footenote on p. | + | If two real $G$-modules $U$ and $V$ with $\dim U^G = \dim V^G = 0$ are isomorphic, then $U$ and $V$ are Smith equivalent (the sphere $\varSigma = S(V \oplus \mathbb{R})$ admits the required action of $G$, where $G$ acts trivially on $\mathbb{R}$ and diagonally on $V \oplus \mathbb{R}$). Therefore, the sets $PSm(G)$ and $Sm(G)$ both contain the zero $0 = V - V$ of $RO(G)$. The following problems are motivated by the question of Paul A. Smith posed in 1960, in the article {{cite|Smith1960|the footenote on p. 406}}. |
* ''Is it true that any two Smith equivalent real $G$-modules are isomorphic'', i.e., ''is it true that $Sm(G) = 0$''? | * ''Is it true that any two Smith equivalent real $G$-modules are isomorphic'', i.e., ''is it true that $Sm(G) = 0$''? | ||
− | * ''Is it true that any two | + | * ''Is it true that any two $\mathcal{P}$-matched and Smith equivalent real $G$-modules are isomorphic'', i.e., ''is it true that $PSm(G) = 0$''? |
Let $d^G \colon RO(G) \to \mathbb{Z}$ be the dimension homomorphism, i.e., $d^G(U-V) = \dim U^G - \dim V^G$ for any two real $G$-modules $U$ and $V$. | Let $d^G \colon RO(G) \to \mathbb{Z}$ be the dimension homomorphism, i.e., $d^G(U-V) = \dim U^G - \dim V^G$ for any two real $G$-modules $U$ and $V$. | ||
{{beginthm|Lemma|(see {{cite|Laitinen&Pawałowski1999}})}} \label{lem:rank} For a finite group $G$, the following two statements are true. | {{beginthm|Lemma|(see {{cite|Laitinen&Pawałowski1999}})}} \label{lem:rank} For a finite group $G$, the following two statements are true. | ||
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* If $r_G \geq 2$, $PO(G) \cap \Ker \, d^G$ is a finitely generated free abelian group of rank $r_G - 1$, $\operatorname{rank}(PO(G) \cap \Ker \, d^G) = r_G - 1$. | * If $r_G \geq 2$, $PO(G) \cap \Ker \, d^G$ is a finitely generated free abelian group of rank $r_G - 1$, $\operatorname{rank}(PO(G) \cap \Ker \, d^G) = r_G - 1$. | ||
{{endthm}} | {{endthm}} | ||
− | + | As $PSm(G) = PO(G) \cap Sm(G) = PO(G) \cap \Ker \, d^G \cap Sm(G)$, Lemma \ref{lem:rank} yields the following corollary. | |
{{beginthm|Corollary|}}\label{cor:rank} | {{beginthm|Corollary|}}\label{cor:rank} | ||
− | Let $G$ be a finite group with $r_G = 0 $ or $1$. Then any two | + | Let $G$ be a finite group with $r_G = 0 $ or $1$. Then any two $\mathcal{P}$-matched and Smith equivalent real $G$-modules are isomorphic, i.e., $PSm(G) = 0$. |
{{endthm}} | {{endthm}} | ||
{{beginrem|Problem|(Primary problem)}}\label{prob:primary} | {{beginrem|Problem|(Primary problem)}}\label{prob:primary} | ||
For which finite groups $G$, the following statement is true? | For which finite groups $G$, the following statement is true? | ||
− | * ''Any two | + | * ''Any two $\mathcal{P}$-matched and Smith equivalent real $G$-modules are isomorphic if and only if $r_G = 0$ or $1$.'' |
{{endrem}} | {{endrem}} | ||
− | In order to answer the question in Problem \ref{prob:primary}, one shall check for which finite groups $G$ with $r_G \geq 2$, there exist two | + | In order to answer the question in Problem \ref{prob:primary}, one shall check for which finite groups $G$ with $r_G \geq 2$, there exist two $\mathcal{P}$-matched and Smith equivalent real $G$-modules that are not isomorphic (cf. Corollary \ref{cor:rank}). For a systematic discussion on the Smith equivalence of real $G$-modules, we refer to the page ''Smith equivalence of reresentations''. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. {{cite|Laitinen&Pawałowski1999|Appendix}}). |
</wikitex> | </wikitex> | ||
== Laitinen conjecture == | == Laitinen conjecture == | ||
<wikitex>; | <wikitex>; | ||
− | {{beginrem|Definition}} | + | {{beginrem|Definition}}\label{def:Laitinen} |
− | Let $G$ be a finite group. Two real $G$-modules $U$ and $V$ are called ''Laitinen-Smith equivalent'' if | + | Let $G$ be a finite group. Two real $G$-modules $U$ and $V$ are called ''Laitinen-Smith equivalent'' if there exists a smooth action of $G$ on a homotopy sphere $\varSigma$ with exactly two fixed points $x$ and $y$ at which the tangent $G$-modules are isomorphic to $U$ and $V$, respectively (cf. Definition \ref{def:Smith}), and the action of $G$ on $\varSigma$ satisfies the ''Laitinen condition'' asserting that for any element $g \in G$ of order $2^a$ for $a \geq 3$, the fixed point set $\varSigma^g = \{z \in \varSigma \ |\ gz = z\}$ is connected. |
− | $ | + | |
{{endrem}} | {{endrem}} | ||
− | {{ | + | {{beginrem|Definition}}\label{def:2-matched} |
− | Let $G$ be a finite group. | + | Let $G$ be a finite group. Two real $G$-modules $U$ and $V$ are said to be ''$\mathcal{P}_2$-matched'' if $U$ and $V$ are isomorphic when restricted to any cyclic subgroup of $G$ of order $2^a$ for $a \geq 3$. |
− | {{ | + | {{endrem}} |
− | In | + | In Definition \ref{def:Laitinen}, the Laitinen condition implies that the two Smith equivalent real $G$-modules $U$ and $V$ are $\mathcal{P}_2$-matched. As any two $\mathcal{P}_2$-matched and Smith equivalent real $G$-modules $U$ and $V$ are $\mathcal{P}$-matched, Corollary \ref{cor:rank} yields the next corollary. |
{{beginthm|Corollary}}\label{cor:isomorphic} | {{beginthm|Corollary}}\label{cor:isomorphic} | ||
Let $G$ be a finite group with $r_G = 0$ or $1$. Then any two Laitinen-Smith equivalent real $G$-modules are isomorphic. | Let $G$ be a finite group with $r_G = 0$ or $1$. Then any two Laitinen-Smith equivalent real $G$-modules are isomorphic. | ||
{{endthm}} | {{endthm}} | ||
− | + | In 1996, Erkki Laitinen ({{cite|Laitinen&Pawałowski1999|Appendix}}) posed the following conjecture (cf. Problem \ref{prob:primary}). | |
{{beginrem|Problem|(Laitinen conjecture)}}\label{prob:Laitinen} For a finite Oliver group $G$, any two Laitinen-Smith equivalent real $G$-modules are isomorphic if and only if $r_G = 0$ or $1$. | {{beginrem|Problem|(Laitinen conjecture)}}\label{prob:Laitinen} For a finite Oliver group $G$, any two Laitinen-Smith equivalent real $G$-modules are isomorphic if and only if $r_G = 0$ or $1$. | ||
{{endrem}} | {{endrem}} | ||
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Summarizing the results of {{cite|Laitinen&Pawałowski1999}}, {{cite|Pawałowski&Solomon2002}}, {{cite|Morimoto2008}}, {{cite|Morimoto2010}}, and {{cite|Pawałowski&Sumi2010}}, one obtains the following theorem. | Summarizing the results of {{cite|Laitinen&Pawałowski1999}}, {{cite|Pawałowski&Solomon2002}}, {{cite|Morimoto2008}}, {{cite|Morimoto2010}}, and {{cite|Pawałowski&Sumi2010}}, one obtains the following theorem. | ||
{{beginthm|Theorem}} For a finite non-solvable group $G$ not isomorphic to $\Aut(A_6)$, the following two statements are true. | {{beginthm|Theorem}} For a finite non-solvable group $G$ not isomorphic to $\Aut(A_6)$, the following two statements are true. | ||
− | * Any two | + | * Any two $\mathcal{P}$-matched and Smith equivalent real $G$-modules are isomorphic if and only if $r_G = 0$ or $1$ (cf. Problem \ref{prob:primary}). |
* Any two Laitinen-Smith equivalent real $G$-modules are isomorphic if and only if $r_G = 0$ or $1$ (cf. Problem \ref{prob:Laitinen}). | * Any two Laitinen-Smith equivalent real $G$-modules are isomorphic if and only if $r_G = 0$ or $1$ (cf. Problem \ref{prob:Laitinen}). | ||
{{endthm}} | {{endthm}} |
Latest revision as of 21:57, 30 December 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
[edit] 1 Primary problem
Let be a finite group. A real -module is a finite dimensional real vector space with a linear action of , i.e., the action of on is given by a representation .
Definition 1.1. Two real -modules and are called Smith equivalent if there exists a smooth action of on a homotopy sphere with exactly two fixed points and at which the tangent -modules are isomorphic to and , respectively, where the tangent -modules are determined on the tangent spaces and at and by taking the derivatives at and of the diffeomorphisms , considered for all .
- Let be the representation ring of , i.e., the Grothendieck ring of the differences of real -modules and . As a group, is a finitely generated free abelian group whose rank is the number of real conjugacy classes of elements . Recall that the real conjugacy class of is defined by . Hereafter, denotes the number of real conjugacy classes represented by elements not of prime power order.
- Let be the subgroup of consisting of the differences of two -matched real -modules and , where the -matched condition means that and are isomorphic when restricted to any prime power order subgroup of . The group is trivial, , if and only if . In the case , is a finitely generated free abelian group of rank , .
We shall make use of the notions of Smith set and pimary Smith set of .
- The Smith set of is the subset of consisting of the differences of two Smith equivalent real -modules and .
- The primary Smith set of is the subset of consisting of the differences of two -matched and Smith equivalent real -modules and .
If two real -modules and with are isomorphic, then and are Smith equivalent (the sphere admits the required action of , where acts trivially on and diagonally on ). Therefore, the sets and both contain the zero of . The following problems are motivated by the question of Paul A. Smith posed in 1960, in the article [Smith1960, the footenote on p. 406].
- Is it true that any two Smith equivalent real -modules are isomorphic, i.e., is it true that ?
- Is it true that any two -matched and Smith equivalent real -modules are isomorphic, i.e., is it true that ?
Let be the dimension homomorphism, i.e., for any two real -modules and .
Lemma 1.2 (see [Laitinen&Pawałowski1999]). For a finite group , the following two statements are true.
- The group is trivial, , if and only if or .
- If , is a finitely generated free abelian group of rank , .
As , Lemma 1.2 yields the following corollary.
Corollary 1.3. Let be a finite group with or . Then any two -matched and Smith equivalent real -modules are isomorphic, i.e., .
Problem 1.4 (Primary problem). For which finite groups , the following statement is true?
- Any two -matched and Smith equivalent real -modules are isomorphic if and only if or .
In order to answer the question in Problem 1.4, one shall check for which finite groups with , there exist two -matched and Smith equivalent real -modules that are not isomorphic (cf. Corollary 1.3). For a systematic discussion on the Smith equivalence of real -modules, we refer to the page Smith equivalence of reresentations. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. [Laitinen&Pawałowski1999, Appendix]).
[edit] 2 Laitinen conjecture
Definition 2.1. Let be a finite group. Two real -modules and are called Laitinen-Smith equivalent if there exists a smooth action of on a homotopy sphere with exactly two fixed points and at which the tangent -modules are isomorphic to and , respectively (cf. Definition 1.1), and the action of on satisfies the Laitinen condition asserting that for any element of order for , the fixed point set is connected.
Definition 2.2. Let be a finite group. Two real -modules and are said to be -matched if and are isomorphic when restricted to any cyclic subgroup of of order for .
In Definition 2.1, the Laitinen condition implies that the two Smith equivalent real -modules and are -matched. As any two -matched and Smith equivalent real -modules and are -matched, Corollary 1.3 yields the next corollary.
Corollary 2.3. Let be a finite group with or . Then any two Laitinen-Smith equivalent real -modules are isomorphic.
In 1996, Erkki Laitinen ([Laitinen&Pawałowski1999, Appendix]) posed the following conjecture (cf. Problem 1.4).
Problem 2.4 (Laitinen conjecture). For a finite Oliver group , any two Laitinen-Smith equivalent real -modules are isomorphic if and only if or .
In order to prove that the Laitinen conjecture holds for a finite Oliver group , it sufficies to restrict attention to the case where , and to check that there exist two Laitinen-Smith equivalent real -modules that are not isomorphic (cf. Corollary 2.3).
[edit] 3 Results so far
- Laitinen and Pawałowski [Laitinen&Pawałowski1999] prove that the Laitinen conjecture holds for any finite (non-trivial) perfect group .
- Pawałowski and Solomon [Pawałowski&Solomon2002] prove that the Laitinen conjecture holds under either of the following condition:
- is a finite Oliver group of odd order (where always ).
- is a finite Oliver group with a quotient isomorphic to for two distinct odd primes and (where always ).
- is a finite non-solvable gap group not isomorphic to , the splitting extension of by the group of automorphism of the field .
- Morimoto [Morimoto2008] obtains the first counterexample to the Laitinen conjecture by proving that the Smith set and for .
- Pawałowski and Sumi [Pawałowski&Sumi2009] compute the primary Smith set for some finite solvable Oliver groups, to the effect that:
- and for , confirming the Latinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , and they prove that any element of is the difference of two Laitinen-Smith equivalent real -modules, confirming the Laitinen conjecture.
- Morimoto [Morimoto2010] checks that for , where , and he proves that any element of is the difference of two Laitinen-Smith equivalent real -modules, confirming the Laitinen conjecture.
- Pawałowski and Sumi [Pawałowski&Sumi2010] confirm the Laitinen conjecture for any finite non-solvable group not isomorphic to or , and more generally, for any finite Oliver group satisfying the Nil-Condition.
[edit] 4 Further discussion
Summarizing the results of [Laitinen&Pawałowski1999], [Pawałowski&Solomon2002], [Morimoto2008], [Morimoto2010], and [Pawałowski&Sumi2010], one obtains the following theorem.
Theorem 4.1. For a finite non-solvable group not isomorphic to , the following two statements are true.
[edit] 5 References
- [Laitinen&Pawałowski1999] E. Laitinen and K. Pawałowski, Smith equivalence of representations for finite perfect groups, Proc. Amer. Math. Soc. 127 (1999), no.1, 297–307. MR1468195 (99b:57070) Zbl 0914.57025
- [Morimoto2008] M. Morimoto, Smith equivalent -representations are isomorphic, Proc. Amer. Math. Soc. 136 (2008), no.10, 3683–3688. MR2415055 (2009c:57054)
- [Morimoto2010] M. Morimoto, Nontrivial -matched -related pairs for finite gap Oliver groups, J. Math. Soc. Japan 62 (2010), no.2, 623–647. MR2662855 ()
- [Pawałowski&Solomon2002] K. Pawałowski and R. Solomon, Smith equivalence and finite Oliver groups with Laitinen number 0 or 1, Algebr. Geom. Topol. 2 (2002), 843–895 (electronic). MR1936973 (2003j:57057) Zbl 1022.57019
- [Pawałowski&Sumi2009] K. Pawałowski and T. Sumi, The Laitinen conjecture for finite solvable Oliver groups, Proc. Amer. Math. Soc. 137 (2009), no.6, 2147–2156. MR2480297 (2009k:57052) Zbl 1173.57015
- [Pawałowski&Sumi2010] K. Pawałowski and T. Sumi, Smith equivalence of representations and the Nil-Condition, to appear in the Proceedings of the Edinburgh Mathematical Society.
- [Smith1960] P. A. Smith, New results and old problems in finite transformation groups, Bull. Amer. Math. Soc. 66 (1960), 401–415. MR0125581 (23 #A2880) Zbl 0096.37501
Definition 1.1. Two real -modules and are called Smith equivalent if there exists a smooth action of on a homotopy sphere with exactly two fixed points and at which the tangent -modules are isomorphic to and , respectively, where the tangent -modules are determined on the tangent spaces and at and by taking the derivatives at and of the diffeomorphisms , considered for all .
- Let be the representation ring of , i.e., the Grothendieck ring of the differences of real -modules and . As a group, is a finitely generated free abelian group whose rank is the number of real conjugacy classes of elements . Recall that the real conjugacy class of is defined by . Hereafter, denotes the number of real conjugacy classes represented by elements not of prime power order.
- Let be the subgroup of consisting of the differences of two -matched real -modules and , where the -matched condition means that and are isomorphic when restricted to any prime power order subgroup of . The group is trivial, , if and only if . In the case , is a finitely generated free abelian group of rank , .
We shall make use of the notions of Smith set and pimary Smith set of .
- The Smith set of is the subset of consisting of the differences of two Smith equivalent real -modules and .
- The primary Smith set of is the subset of consisting of the differences of two -matched and Smith equivalent real -modules and .
If two real -modules and with are isomorphic, then and are Smith equivalent (the sphere admits the required action of , where acts trivially on and diagonally on ). Therefore, the sets and both contain the zero of . The following problems are motivated by the question of Paul A. Smith posed in 1960, in the article [Smith1960, the footenote on p. 406].
- Is it true that any two Smith equivalent real -modules are isomorphic, i.e., is it true that ?
- Is it true that any two -matched and Smith equivalent real -modules are isomorphic, i.e., is it true that ?
Let be the dimension homomorphism, i.e., for any two real -modules and .
Lemma 1.2 (see [Laitinen&Pawałowski1999]). For a finite group , the following two statements are true.
- The group is trivial, , if and only if or .
- If , is a finitely generated free abelian group of rank , .
As , Lemma 1.2 yields the following corollary.
Corollary 1.3. Let be a finite group with or . Then any two -matched and Smith equivalent real -modules are isomorphic, i.e., .
Problem 1.4 (Primary problem). For which finite groups , the following statement is true?
- Any two -matched and Smith equivalent real -modules are isomorphic if and only if or .
In order to answer the question in Problem 1.4, one shall check for which finite groups with , there exist two -matched and Smith equivalent real -modules that are not isomorphic (cf. Corollary 1.3). For a systematic discussion on the Smith equivalence of real -modules, we refer to the page Smith equivalence of reresentations. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. [Laitinen&Pawałowski1999, Appendix]).
[edit] 2 Laitinen conjecture
Definition 2.1. Let be a finite group. Two real -modules and are called Laitinen-Smith equivalent if there exists a smooth action of on a homotopy sphere with exactly two fixed points and at which the tangent -modules are isomorphic to and , respectively (cf. Definition 1.1), and the action of on satisfies the Laitinen condition asserting that for any element of order for , the fixed point set is connected.
Definition 2.2. Let be a finite group. Two real -modules and are said to be -matched if and are isomorphic when restricted to any cyclic subgroup of of order for .
In Definition 2.1, the Laitinen condition implies that the two Smith equivalent real -modules and are -matched. As any two -matched and Smith equivalent real -modules and are -matched, Corollary 1.3 yields the next corollary.
Corollary 2.3. Let be a finite group with or . Then any two Laitinen-Smith equivalent real -modules are isomorphic.
In 1996, Erkki Laitinen ([Laitinen&Pawałowski1999, Appendix]) posed the following conjecture (cf. Problem 1.4).
Problem 2.4 (Laitinen conjecture). For a finite Oliver group , any two Laitinen-Smith equivalent real -modules are isomorphic if and only if or .
In order to prove that the Laitinen conjecture holds for a finite Oliver group , it sufficies to restrict attention to the case where , and to check that there exist two Laitinen-Smith equivalent real -modules that are not isomorphic (cf. Corollary 2.3).
[edit] 3 Results so far
- Laitinen and Pawałowski [Laitinen&Pawałowski1999] prove that the Laitinen conjecture holds for any finite (non-trivial) perfect group .
- Pawałowski and Solomon [Pawałowski&Solomon2002] prove that the Laitinen conjecture holds under either of the following condition:
- is a finite Oliver group of odd order (where always ).
- is a finite Oliver group with a quotient isomorphic to for two distinct odd primes and (where always ).
- is a finite non-solvable gap group not isomorphic to , the splitting extension of by the group of automorphism of the field .
- Morimoto [Morimoto2008] obtains the first counterexample to the Laitinen conjecture by proving that the Smith set and for .
- Pawałowski and Sumi [Pawałowski&Sumi2009] compute the primary Smith set for some finite solvable Oliver groups, to the effect that:
- and for , confirming the Latinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , and they prove that any element of is the difference of two Laitinen-Smith equivalent real -modules, confirming the Laitinen conjecture.
- Morimoto [Morimoto2010] checks that for , where , and he proves that any element of is the difference of two Laitinen-Smith equivalent real -modules, confirming the Laitinen conjecture.
- Pawałowski and Sumi [Pawałowski&Sumi2010] confirm the Laitinen conjecture for any finite non-solvable group not isomorphic to or , and more generally, for any finite Oliver group satisfying the Nil-Condition.
[edit] 4 Further discussion
Summarizing the results of [Laitinen&Pawałowski1999], [Pawałowski&Solomon2002], [Morimoto2008], [Morimoto2010], and [Pawałowski&Sumi2010], one obtains the following theorem.
Theorem 4.1. For a finite non-solvable group not isomorphic to , the following two statements are true.
[edit] 5 References
- [Laitinen&Pawałowski1999] E. Laitinen and K. Pawałowski, Smith equivalence of representations for finite perfect groups, Proc. Amer. Math. Soc. 127 (1999), no.1, 297–307. MR1468195 (99b:57070) Zbl 0914.57025
- [Morimoto2008] M. Morimoto, Smith equivalent -representations are isomorphic, Proc. Amer. Math. Soc. 136 (2008), no.10, 3683–3688. MR2415055 (2009c:57054)
- [Morimoto2010] M. Morimoto, Nontrivial -matched -related pairs for finite gap Oliver groups, J. Math. Soc. Japan 62 (2010), no.2, 623–647. MR2662855 ()
- [Pawałowski&Solomon2002] K. Pawałowski and R. Solomon, Smith equivalence and finite Oliver groups with Laitinen number 0 or 1, Algebr. Geom. Topol. 2 (2002), 843–895 (electronic). MR1936973 (2003j:57057) Zbl 1022.57019
- [Pawałowski&Sumi2009] K. Pawałowski and T. Sumi, The Laitinen conjecture for finite solvable Oliver groups, Proc. Amer. Math. Soc. 137 (2009), no.6, 2147–2156. MR2480297 (2009k:57052) Zbl 1173.57015
- [Pawałowski&Sumi2010] K. Pawałowski and T. Sumi, Smith equivalence of representations and the Nil-Condition, to appear in the Proceedings of the Edinburgh Mathematical Society.
- [Smith1960] P. A. Smith, New results and old problems in finite transformation groups, Bull. Amer. Math. Soc. 66 (1960), 401–415. MR0125581 (23 #A2880) Zbl 0096.37501
Definition 1.1. Two real -modules and are called Smith equivalent if there exists a smooth action of on a homotopy sphere with exactly two fixed points and at which the tangent -modules are isomorphic to and , respectively, where the tangent -modules are determined on the tangent spaces and at and by taking the derivatives at and of the diffeomorphisms , considered for all .
- Let be the representation ring of , i.e., the Grothendieck ring of the differences of real -modules and . As a group, is a finitely generated free abelian group whose rank is the number of real conjugacy classes of elements . Recall that the real conjugacy class of is defined by . Hereafter, denotes the number of real conjugacy classes represented by elements not of prime power order.
- Let be the subgroup of consisting of the differences of two -matched real -modules and , where the -matched condition means that and are isomorphic when restricted to any prime power order subgroup of . The group is trivial, , if and only if . In the case , is a finitely generated free abelian group of rank , .
We shall make use of the notions of Smith set and pimary Smith set of .
- The Smith set of is the subset of consisting of the differences of two Smith equivalent real -modules and .
- The primary Smith set of is the subset of consisting of the differences of two -matched and Smith equivalent real -modules and .
If two real -modules and with are isomorphic, then and are Smith equivalent (the sphere admits the required action of , where acts trivially on and diagonally on ). Therefore, the sets and both contain the zero of . The following problems are motivated by the question of Paul A. Smith posed in 1960, in the article [Smith1960, the footenote on p. 406].
- Is it true that any two Smith equivalent real -modules are isomorphic, i.e., is it true that ?
- Is it true that any two -matched and Smith equivalent real -modules are isomorphic, i.e., is it true that ?
Let be the dimension homomorphism, i.e., for any two real -modules and .
Lemma 1.2 (see [Laitinen&Pawałowski1999]). For a finite group , the following two statements are true.
- The group is trivial, , if and only if or .
- If , is a finitely generated free abelian group of rank , .
As , Lemma 1.2 yields the following corollary.
Corollary 1.3. Let be a finite group with or . Then any two -matched and Smith equivalent real -modules are isomorphic, i.e., .
Problem 1.4 (Primary problem). For which finite groups , the following statement is true?
- Any two -matched and Smith equivalent real -modules are isomorphic if and only if or .
In order to answer the question in Problem 1.4, one shall check for which finite groups with , there exist two -matched and Smith equivalent real -modules that are not isomorphic (cf. Corollary 1.3). For a systematic discussion on the Smith equivalence of real -modules, we refer to the page Smith equivalence of reresentations. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. [Laitinen&Pawałowski1999, Appendix]).
[edit] 2 Laitinen conjecture
Definition 2.1. Let be a finite group. Two real -modules and are called Laitinen-Smith equivalent if there exists a smooth action of on a homotopy sphere with exactly two fixed points and at which the tangent -modules are isomorphic to and , respectively (cf. Definition 1.1), and the action of on satisfies the Laitinen condition asserting that for any element of order for , the fixed point set is connected.
Definition 2.2. Let be a finite group. Two real -modules and are said to be -matched if and are isomorphic when restricted to any cyclic subgroup of of order for .
In Definition 2.1, the Laitinen condition implies that the two Smith equivalent real -modules and are -matched. As any two -matched and Smith equivalent real -modules and are -matched, Corollary 1.3 yields the next corollary.
Corollary 2.3. Let be a finite group with or . Then any two Laitinen-Smith equivalent real -modules are isomorphic.
In 1996, Erkki Laitinen ([Laitinen&Pawałowski1999, Appendix]) posed the following conjecture (cf. Problem 1.4).
Problem 2.4 (Laitinen conjecture). For a finite Oliver group , any two Laitinen-Smith equivalent real -modules are isomorphic if and only if or .
In order to prove that the Laitinen conjecture holds for a finite Oliver group , it sufficies to restrict attention to the case where , and to check that there exist two Laitinen-Smith equivalent real -modules that are not isomorphic (cf. Corollary 2.3).
[edit] 3 Results so far
- Laitinen and Pawałowski [Laitinen&Pawałowski1999] prove that the Laitinen conjecture holds for any finite (non-trivial) perfect group .
- Pawałowski and Solomon [Pawałowski&Solomon2002] prove that the Laitinen conjecture holds under either of the following condition:
- is a finite Oliver group of odd order (where always ).
- is a finite Oliver group with a quotient isomorphic to for two distinct odd primes and (where always ).
- is a finite non-solvable gap group not isomorphic to , the splitting extension of by the group of automorphism of the field .
- Morimoto [Morimoto2008] obtains the first counterexample to the Laitinen conjecture by proving that the Smith set and for .
- Pawałowski and Sumi [Pawałowski&Sumi2009] compute the primary Smith set for some finite solvable Oliver groups, to the effect that:
- and for , confirming the Latinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , and they prove that any element of is the difference of two Laitinen-Smith equivalent real -modules, confirming the Laitinen conjecture.
- Morimoto [Morimoto2010] checks that for , where , and he proves that any element of is the difference of two Laitinen-Smith equivalent real -modules, confirming the Laitinen conjecture.
- Pawałowski and Sumi [Pawałowski&Sumi2010] confirm the Laitinen conjecture for any finite non-solvable group not isomorphic to or , and more generally, for any finite Oliver group satisfying the Nil-Condition.
[edit] 4 Further discussion
Summarizing the results of [Laitinen&Pawałowski1999], [Pawałowski&Solomon2002], [Morimoto2008], [Morimoto2010], and [Pawałowski&Sumi2010], one obtains the following theorem.
Theorem 4.1. For a finite non-solvable group not isomorphic to , the following two statements are true.
[edit] 5 References
- [Laitinen&Pawałowski1999] E. Laitinen and K. Pawałowski, Smith equivalence of representations for finite perfect groups, Proc. Amer. Math. Soc. 127 (1999), no.1, 297–307. MR1468195 (99b:57070) Zbl 0914.57025
- [Morimoto2008] M. Morimoto, Smith equivalent -representations are isomorphic, Proc. Amer. Math. Soc. 136 (2008), no.10, 3683–3688. MR2415055 (2009c:57054)
- [Morimoto2010] M. Morimoto, Nontrivial -matched -related pairs for finite gap Oliver groups, J. Math. Soc. Japan 62 (2010), no.2, 623–647. MR2662855 ()
- [Pawałowski&Solomon2002] K. Pawałowski and R. Solomon, Smith equivalence and finite Oliver groups with Laitinen number 0 or 1, Algebr. Geom. Topol. 2 (2002), 843–895 (electronic). MR1936973 (2003j:57057) Zbl 1022.57019
- [Pawałowski&Sumi2009] K. Pawałowski and T. Sumi, The Laitinen conjecture for finite solvable Oliver groups, Proc. Amer. Math. Soc. 137 (2009), no.6, 2147–2156. MR2480297 (2009k:57052) Zbl 1173.57015
- [Pawałowski&Sumi2010] K. Pawałowski and T. Sumi, Smith equivalence of representations and the Nil-Condition, to appear in the Proceedings of the Edinburgh Mathematical Society.
- [Smith1960] P. A. Smith, New results and old problems in finite transformation groups, Bull. Amer. Math. Soc. 66 (1960), 401–415. MR0125581 (23 #A2880) Zbl 0096.37501
Definition 1.1. Two real -modules and are called Smith equivalent if there exists a smooth action of on a homotopy sphere with exactly two fixed points and at which the tangent -modules are isomorphic to and , respectively, where the tangent -modules are determined on the tangent spaces and at and by taking the derivatives at and of the diffeomorphisms , considered for all .
- Let be the representation ring of , i.e., the Grothendieck ring of the differences of real -modules and . As a group, is a finitely generated free abelian group whose rank is the number of real conjugacy classes of elements . Recall that the real conjugacy class of is defined by . Hereafter, denotes the number of real conjugacy classes represented by elements not of prime power order.
- Let be the subgroup of consisting of the differences of two -matched real -modules and , where the -matched condition means that and are isomorphic when restricted to any prime power order subgroup of . The group is trivial, , if and only if . In the case , is a finitely generated free abelian group of rank , .
We shall make use of the notions of Smith set and pimary Smith set of .
- The Smith set of is the subset of consisting of the differences of two Smith equivalent real -modules and .
- The primary Smith set of is the subset of consisting of the differences of two -matched and Smith equivalent real -modules and .
If two real -modules and with are isomorphic, then and are Smith equivalent (the sphere admits the required action of , where acts trivially on and diagonally on ). Therefore, the sets and both contain the zero of . The following problems are motivated by the question of Paul A. Smith posed in 1960, in the article [Smith1960, the footenote on p. 406].
- Is it true that any two Smith equivalent real -modules are isomorphic, i.e., is it true that ?
- Is it true that any two -matched and Smith equivalent real -modules are isomorphic, i.e., is it true that ?
Let be the dimension homomorphism, i.e., for any two real -modules and .
Lemma 1.2 (see [Laitinen&Pawałowski1999]). For a finite group , the following two statements are true.
- The group is trivial, , if and only if or .
- If , is a finitely generated free abelian group of rank , .
As , Lemma 1.2 yields the following corollary.
Corollary 1.3. Let be a finite group with or . Then any two -matched and Smith equivalent real -modules are isomorphic, i.e., .
Problem 1.4 (Primary problem). For which finite groups , the following statement is true?
- Any two -matched and Smith equivalent real -modules are isomorphic if and only if or .
In order to answer the question in Problem 1.4, one shall check for which finite groups with , there exist two -matched and Smith equivalent real -modules that are not isomorphic (cf. Corollary 1.3). For a systematic discussion on the Smith equivalence of real -modules, we refer to the page Smith equivalence of reresentations. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. [Laitinen&Pawałowski1999, Appendix]).
[edit] 2 Laitinen conjecture
Definition 2.1. Let be a finite group. Two real -modules and are called Laitinen-Smith equivalent if there exists a smooth action of on a homotopy sphere with exactly two fixed points and at which the tangent -modules are isomorphic to and , respectively (cf. Definition 1.1), and the action of on satisfies the Laitinen condition asserting that for any element of order for , the fixed point set is connected.
Definition 2.2. Let be a finite group. Two real -modules and are said to be -matched if and are isomorphic when restricted to any cyclic subgroup of of order for .
In Definition 2.1, the Laitinen condition implies that the two Smith equivalent real -modules and are -matched. As any two -matched and Smith equivalent real -modules and are -matched, Corollary 1.3 yields the next corollary.
Corollary 2.3. Let be a finite group with or . Then any two Laitinen-Smith equivalent real -modules are isomorphic.
In 1996, Erkki Laitinen ([Laitinen&Pawałowski1999, Appendix]) posed the following conjecture (cf. Problem 1.4).
Problem 2.4 (Laitinen conjecture). For a finite Oliver group , any two Laitinen-Smith equivalent real -modules are isomorphic if and only if or .
In order to prove that the Laitinen conjecture holds for a finite Oliver group , it sufficies to restrict attention to the case where , and to check that there exist two Laitinen-Smith equivalent real -modules that are not isomorphic (cf. Corollary 2.3).
[edit] 3 Results so far
- Laitinen and Pawałowski [Laitinen&Pawałowski1999] prove that the Laitinen conjecture holds for any finite (non-trivial) perfect group .
- Pawałowski and Solomon [Pawałowski&Solomon2002] prove that the Laitinen conjecture holds under either of the following condition:
- is a finite Oliver group of odd order (where always ).
- is a finite Oliver group with a quotient isomorphic to for two distinct odd primes and (where always ).
- is a finite non-solvable gap group not isomorphic to , the splitting extension of by the group of automorphism of the field .
- Morimoto [Morimoto2008] obtains the first counterexample to the Laitinen conjecture by proving that the Smith set and for .
- Pawałowski and Sumi [Pawałowski&Sumi2009] compute the primary Smith set for some finite solvable Oliver groups, to the effect that:
- and for , confirming the Latinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , and they prove that any element of is the difference of two Laitinen-Smith equivalent real -modules, confirming the Laitinen conjecture.
- Morimoto [Morimoto2010] checks that for , where , and he proves that any element of is the difference of two Laitinen-Smith equivalent real -modules, confirming the Laitinen conjecture.
- Pawałowski and Sumi [Pawałowski&Sumi2010] confirm the Laitinen conjecture for any finite non-solvable group not isomorphic to or , and more generally, for any finite Oliver group satisfying the Nil-Condition.
[edit] 4 Further discussion
Summarizing the results of [Laitinen&Pawałowski1999], [Pawałowski&Solomon2002], [Morimoto2008], [Morimoto2010], and [Pawałowski&Sumi2010], one obtains the following theorem.
Theorem 4.1. For a finite non-solvable group not isomorphic to , the following two statements are true.
[edit] 5 References
- [Laitinen&Pawałowski1999] E. Laitinen and K. Pawałowski, Smith equivalence of representations for finite perfect groups, Proc. Amer. Math. Soc. 127 (1999), no.1, 297–307. MR1468195 (99b:57070) Zbl 0914.57025
- [Morimoto2008] M. Morimoto, Smith equivalent -representations are isomorphic, Proc. Amer. Math. Soc. 136 (2008), no.10, 3683–3688. MR2415055 (2009c:57054)
- [Morimoto2010] M. Morimoto, Nontrivial -matched -related pairs for finite gap Oliver groups, J. Math. Soc. Japan 62 (2010), no.2, 623–647. MR2662855 ()
- [Pawałowski&Solomon2002] K. Pawałowski and R. Solomon, Smith equivalence and finite Oliver groups with Laitinen number 0 or 1, Algebr. Geom. Topol. 2 (2002), 843–895 (electronic). MR1936973 (2003j:57057) Zbl 1022.57019
- [Pawałowski&Sumi2009] K. Pawałowski and T. Sumi, The Laitinen conjecture for finite solvable Oliver groups, Proc. Amer. Math. Soc. 137 (2009), no.6, 2147–2156. MR2480297 (2009k:57052) Zbl 1173.57015
- [Pawałowski&Sumi2010] K. Pawałowski and T. Sumi, Smith equivalence of representations and the Nil-Condition, to appear in the Proceedings of the Edinburgh Mathematical Society.
- [Smith1960] P. A. Smith, New results and old problems in finite transformation groups, Bull. Amer. Math. Soc. 66 (1960), 401–415. MR0125581 (23 #A2880) Zbl 0096.37501
Definition 1.1. Two real -modules and are called Smith equivalent if there exists a smooth action of on a homotopy sphere with exactly two fixed points and at which the tangent -modules are isomorphic to and , respectively, where the tangent -modules are determined on the tangent spaces and at and by taking the derivatives at and of the diffeomorphisms , considered for all .
- Let be the representation ring of , i.e., the Grothendieck ring of the differences of real -modules and . As a group, is a finitely generated free abelian group whose rank is the number of real conjugacy classes of elements . Recall that the real conjugacy class of is defined by . Hereafter, denotes the number of real conjugacy classes represented by elements not of prime power order.
- Let be the subgroup of consisting of the differences of two -matched real -modules and , where the -matched condition means that and are isomorphic when restricted to any prime power order subgroup of . The group is trivial, , if and only if . In the case , is a finitely generated free abelian group of rank , .
We shall make use of the notions of Smith set and pimary Smith set of .
- The Smith set of is the subset of consisting of the differences of two Smith equivalent real -modules and .
- The primary Smith set of is the subset of consisting of the differences of two -matched and Smith equivalent real -modules and .
If two real -modules and with are isomorphic, then and are Smith equivalent (the sphere admits the required action of , where acts trivially on and diagonally on ). Therefore, the sets and both contain the zero of . The following problems are motivated by the question of Paul A. Smith posed in 1960, in the article [Smith1960, the footenote on p. 406].
- Is it true that any two Smith equivalent real -modules are isomorphic, i.e., is it true that ?
- Is it true that any two -matched and Smith equivalent real -modules are isomorphic, i.e., is it true that ?
Let be the dimension homomorphism, i.e., for any two real -modules and .
Lemma 1.2 (see [Laitinen&Pawałowski1999]). For a finite group , the following two statements are true.
- The group is trivial, , if and only if or .
- If , is a finitely generated free abelian group of rank , .
As , Lemma 1.2 yields the following corollary.
Corollary 1.3. Let be a finite group with or . Then any two -matched and Smith equivalent real -modules are isomorphic, i.e., .
Problem 1.4 (Primary problem). For which finite groups , the following statement is true?
- Any two -matched and Smith equivalent real -modules are isomorphic if and only if or .
In order to answer the question in Problem 1.4, one shall check for which finite groups with , there exist two -matched and Smith equivalent real -modules that are not isomorphic (cf. Corollary 1.3). For a systematic discussion on the Smith equivalence of real -modules, we refer to the page Smith equivalence of reresentations. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. [Laitinen&Pawałowski1999, Appendix]).
[edit] 2 Laitinen conjecture
Definition 2.1. Let be a finite group. Two real -modules and are called Laitinen-Smith equivalent if there exists a smooth action of on a homotopy sphere with exactly two fixed points and at which the tangent -modules are isomorphic to and , respectively (cf. Definition 1.1), and the action of on satisfies the Laitinen condition asserting that for any element of order for , the fixed point set is connected.
Definition 2.2. Let be a finite group. Two real -modules and are said to be -matched if and are isomorphic when restricted to any cyclic subgroup of of order for .
In Definition 2.1, the Laitinen condition implies that the two Smith equivalent real -modules and are -matched. As any two -matched and Smith equivalent real -modules and are -matched, Corollary 1.3 yields the next corollary.
Corollary 2.3. Let be a finite group with or . Then any two Laitinen-Smith equivalent real -modules are isomorphic.
In 1996, Erkki Laitinen ([Laitinen&Pawałowski1999, Appendix]) posed the following conjecture (cf. Problem 1.4).
Problem 2.4 (Laitinen conjecture). For a finite Oliver group , any two Laitinen-Smith equivalent real -modules are isomorphic if and only if or .
In order to prove that the Laitinen conjecture holds for a finite Oliver group , it sufficies to restrict attention to the case where , and to check that there exist two Laitinen-Smith equivalent real -modules that are not isomorphic (cf. Corollary 2.3).
[edit] 3 Results so far
- Laitinen and Pawałowski [Laitinen&Pawałowski1999] prove that the Laitinen conjecture holds for any finite (non-trivial) perfect group .
- Pawałowski and Solomon [Pawałowski&Solomon2002] prove that the Laitinen conjecture holds under either of the following condition:
- is a finite Oliver group of odd order (where always ).
- is a finite Oliver group with a quotient isomorphic to for two distinct odd primes and (where always ).
- is a finite non-solvable gap group not isomorphic to , the splitting extension of by the group of automorphism of the field .
- Morimoto [Morimoto2008] obtains the first counterexample to the Laitinen conjecture by proving that the Smith set and for .
- Pawałowski and Sumi [Pawałowski&Sumi2009] compute the primary Smith set for some finite solvable Oliver groups, to the effect that:
- and for , confirming the Latinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , and they prove that any element of is the difference of two Laitinen-Smith equivalent real -modules, confirming the Laitinen conjecture.
- Morimoto [Morimoto2010] checks that for , where , and he proves that any element of is the difference of two Laitinen-Smith equivalent real -modules, confirming the Laitinen conjecture.
- Pawałowski and Sumi [Pawałowski&Sumi2010] confirm the Laitinen conjecture for any finite non-solvable group not isomorphic to or , and more generally, for any finite Oliver group satisfying the Nil-Condition.
[edit] 4 Further discussion
Summarizing the results of [Laitinen&Pawałowski1999], [Pawałowski&Solomon2002], [Morimoto2008], [Morimoto2010], and [Pawałowski&Sumi2010], one obtains the following theorem.
Theorem 4.1. For a finite non-solvable group not isomorphic to , the following two statements are true.
[edit] 5 References
- [Laitinen&Pawałowski1999] E. Laitinen and K. Pawałowski, Smith equivalence of representations for finite perfect groups, Proc. Amer. Math. Soc. 127 (1999), no.1, 297–307. MR1468195 (99b:57070) Zbl 0914.57025
- [Morimoto2008] M. Morimoto, Smith equivalent -representations are isomorphic, Proc. Amer. Math. Soc. 136 (2008), no.10, 3683–3688. MR2415055 (2009c:57054)
- [Morimoto2010] M. Morimoto, Nontrivial -matched -related pairs for finite gap Oliver groups, J. Math. Soc. Japan 62 (2010), no.2, 623–647. MR2662855 ()
- [Pawałowski&Solomon2002] K. Pawałowski and R. Solomon, Smith equivalence and finite Oliver groups with Laitinen number 0 or 1, Algebr. Geom. Topol. 2 (2002), 843–895 (electronic). MR1936973 (2003j:57057) Zbl 1022.57019
- [Pawałowski&Sumi2009] K. Pawałowski and T. Sumi, The Laitinen conjecture for finite solvable Oliver groups, Proc. Amer. Math. Soc. 137 (2009), no.6, 2147–2156. MR2480297 (2009k:57052) Zbl 1173.57015
- [Pawałowski&Sumi2010] K. Pawałowski and T. Sumi, Smith equivalence of representations and the Nil-Condition, to appear in the Proceedings of the Edinburgh Mathematical Society.
- [Smith1960] P. A. Smith, New results and old problems in finite transformation groups, Bull. Amer. Math. Soc. 66 (1960), 401–415. MR0125581 (23 #A2880) Zbl 0096.37501
Definition 1.1. Two real -modules and are called Smith equivalent if there exists a smooth action of on a homotopy sphere with exactly two fixed points and at which the tangent -modules are isomorphic to and , respectively, where the tangent -modules are determined on the tangent spaces and at and by taking the derivatives at and of the diffeomorphisms , considered for all .
- Let be the representation ring of , i.e., the Grothendieck ring of the differences of real -modules and . As a group, is a finitely generated free abelian group whose rank is the number of real conjugacy classes of elements . Recall that the real conjugacy class of is defined by . Hereafter, denotes the number of real conjugacy classes represented by elements not of prime power order.
- Let be the subgroup of consisting of the differences of two -matched real -modules and , where the -matched condition means that and are isomorphic when restricted to any prime power order subgroup of . The group is trivial, , if and only if . In the case , is a finitely generated free abelian group of rank , .
We shall make use of the notions of Smith set and pimary Smith set of .
- The Smith set of is the subset of consisting of the differences of two Smith equivalent real -modules and .
- The primary Smith set of is the subset of consisting of the differences of two -matched and Smith equivalent real -modules and .
If two real -modules and with are isomorphic, then and are Smith equivalent (the sphere admits the required action of , where acts trivially on and diagonally on ). Therefore, the sets and both contain the zero of . The following problems are motivated by the question of Paul A. Smith posed in 1960, in the article [Smith1960, the footenote on p. 406].
- Is it true that any two Smith equivalent real -modules are isomorphic, i.e., is it true that ?
- Is it true that any two -matched and Smith equivalent real -modules are isomorphic, i.e., is it true that ?
Let be the dimension homomorphism, i.e., for any two real -modules and .
Lemma 1.2 (see [Laitinen&Pawałowski1999]). For a finite group , the following two statements are true.
- The group is trivial, , if and only if or .
- If , is a finitely generated free abelian group of rank , .
As , Lemma 1.2 yields the following corollary.
Corollary 1.3. Let be a finite group with or . Then any two -matched and Smith equivalent real -modules are isomorphic, i.e., .
Problem 1.4 (Primary problem). For which finite groups , the following statement is true?
- Any two -matched and Smith equivalent real -modules are isomorphic if and only if or .
In order to answer the question in Problem 1.4, one shall check for which finite groups with , there exist two -matched and Smith equivalent real -modules that are not isomorphic (cf. Corollary 1.3). For a systematic discussion on the Smith equivalence of real -modules, we refer to the page Smith equivalence of reresentations. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. [Laitinen&Pawałowski1999, Appendix]).
[edit] 2 Laitinen conjecture
Definition 2.1. Let be a finite group. Two real -modules and are called Laitinen-Smith equivalent if there exists a smooth action of on a homotopy sphere with exactly two fixed points and at which the tangent -modules are isomorphic to and , respectively (cf. Definition 1.1), and the action of on satisfies the Laitinen condition asserting that for any element of order for , the fixed point set is connected.
Definition 2.2. Let be a finite group. Two real -modules and are said to be -matched if and are isomorphic when restricted to any cyclic subgroup of of order for .
In Definition 2.1, the Laitinen condition implies that the two Smith equivalent real -modules and are -matched. As any two -matched and Smith equivalent real -modules and are -matched, Corollary 1.3 yields the next corollary.
Corollary 2.3. Let be a finite group with or . Then any two Laitinen-Smith equivalent real -modules are isomorphic.
In 1996, Erkki Laitinen ([Laitinen&Pawałowski1999, Appendix]) posed the following conjecture (cf. Problem 1.4).
Problem 2.4 (Laitinen conjecture). For a finite Oliver group , any two Laitinen-Smith equivalent real -modules are isomorphic if and only if or .
In order to prove that the Laitinen conjecture holds for a finite Oliver group , it sufficies to restrict attention to the case where , and to check that there exist two Laitinen-Smith equivalent real -modules that are not isomorphic (cf. Corollary 2.3).
[edit] 3 Results so far
- Laitinen and Pawałowski [Laitinen&Pawałowski1999] prove that the Laitinen conjecture holds for any finite (non-trivial) perfect group .
- Pawałowski and Solomon [Pawałowski&Solomon2002] prove that the Laitinen conjecture holds under either of the following condition:
- is a finite Oliver group of odd order (where always ).
- is a finite Oliver group with a quotient isomorphic to for two distinct odd primes and (where always ).
- is a finite non-solvable gap group not isomorphic to , the splitting extension of by the group of automorphism of the field .
- Morimoto [Morimoto2008] obtains the first counterexample to the Laitinen conjecture by proving that the Smith set and for .
- Pawałowski and Sumi [Pawałowski&Sumi2009] compute the primary Smith set for some finite solvable Oliver groups, to the effect that:
- and for , confirming the Latinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , and they prove that any element of is the difference of two Laitinen-Smith equivalent real -modules, confirming the Laitinen conjecture.
- Morimoto [Morimoto2010] checks that for , where , and he proves that any element of is the difference of two Laitinen-Smith equivalent real -modules, confirming the Laitinen conjecture.
- Pawałowski and Sumi [Pawałowski&Sumi2010] confirm the Laitinen conjecture for any finite non-solvable group not isomorphic to or , and more generally, for any finite Oliver group satisfying the Nil-Condition.
[edit] 4 Further discussion
Summarizing the results of [Laitinen&Pawałowski1999], [Pawałowski&Solomon2002], [Morimoto2008], [Morimoto2010], and [Pawałowski&Sumi2010], one obtains the following theorem.
Theorem 4.1. For a finite non-solvable group not isomorphic to , the following two statements are true.
[edit] 5 References
- [Laitinen&Pawałowski1999] E. Laitinen and K. Pawałowski, Smith equivalence of representations for finite perfect groups, Proc. Amer. Math. Soc. 127 (1999), no.1, 297–307. MR1468195 (99b:57070) Zbl 0914.57025
- [Morimoto2008] M. Morimoto, Smith equivalent -representations are isomorphic, Proc. Amer. Math. Soc. 136 (2008), no.10, 3683–3688. MR2415055 (2009c:57054)
- [Morimoto2010] M. Morimoto, Nontrivial -matched -related pairs for finite gap Oliver groups, J. Math. Soc. Japan 62 (2010), no.2, 623–647. MR2662855 ()
- [Pawałowski&Solomon2002] K. Pawałowski and R. Solomon, Smith equivalence and finite Oliver groups with Laitinen number 0 or 1, Algebr. Geom. Topol. 2 (2002), 843–895 (electronic). MR1936973 (2003j:57057) Zbl 1022.57019
- [Pawałowski&Sumi2009] K. Pawałowski and T. Sumi, The Laitinen conjecture for finite solvable Oliver groups, Proc. Amer. Math. Soc. 137 (2009), no.6, 2147–2156. MR2480297 (2009k:57052) Zbl 1173.57015
- [Pawałowski&Sumi2010] K. Pawałowski and T. Sumi, Smith equivalence of representations and the Nil-Condition, to appear in the Proceedings of the Edinburgh Mathematical Society.
- [Smith1960] P. A. Smith, New results and old problems in finite transformation groups, Bull. Amer. Math. Soc. 66 (1960), 401–415. MR0125581 (23 #A2880) Zbl 0096.37501
Definition 1.1. Two real -modules and are called Smith equivalent if there exists a smooth action of on a homotopy sphere with exactly two fixed points and at which the tangent -modules are isomorphic to and , respectively, where the tangent -modules are determined on the tangent spaces and at and by taking the derivatives at and of the diffeomorphisms , considered for all .
- Let be the representation ring of , i.e., the Grothendieck ring of the differences of real -modules and . As a group, is a finitely generated free abelian group whose rank is the number of real conjugacy classes of elements . Recall that the real conjugacy class of is defined by . Hereafter, denotes the number of real conjugacy classes represented by elements not of prime power order.
- Let be the subgroup of consisting of the differences of two -matched real -modules and , where the -matched condition means that and are isomorphic when restricted to any prime power order subgroup of . The group is trivial, , if and only if . In the case , is a finitely generated free abelian group of rank , .
We shall make use of the notions of Smith set and pimary Smith set of .
- The Smith set of is the subset of consisting of the differences of two Smith equivalent real -modules and .
- The primary Smith set of is the subset of consisting of the differences of two -matched and Smith equivalent real -modules and .
If two real -modules and with are isomorphic, then and are Smith equivalent (the sphere admits the required action of , where acts trivially on and diagonally on ). Therefore, the sets and both contain the zero of . The following problems are motivated by the question of Paul A. Smith posed in 1960, in the article [Smith1960, the footenote on p. 406].
- Is it true that any two Smith equivalent real -modules are isomorphic, i.e., is it true that ?
- Is it true that any two -matched and Smith equivalent real -modules are isomorphic, i.e., is it true that ?
Let be the dimension homomorphism, i.e., for any two real -modules and .
Lemma 1.2 (see [Laitinen&Pawałowski1999]). For a finite group , the following two statements are true.
- The group is trivial, , if and only if or .
- If , is a finitely generated free abelian group of rank , .
As , Lemma 1.2 yields the following corollary.
Corollary 1.3. Let be a finite group with or . Then any two -matched and Smith equivalent real -modules are isomorphic, i.e., .
Problem 1.4 (Primary problem). For which finite groups , the following statement is true?
- Any two -matched and Smith equivalent real -modules are isomorphic if and only if or .
In order to answer the question in Problem 1.4, one shall check for which finite groups with , there exist two -matched and Smith equivalent real -modules that are not isomorphic (cf. Corollary 1.3). For a systematic discussion on the Smith equivalence of real -modules, we refer to the page Smith equivalence of reresentations. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. [Laitinen&Pawałowski1999, Appendix]).
[edit] 2 Laitinen conjecture
Definition 2.1. Let be a finite group. Two real -modules and are called Laitinen-Smith equivalent if there exists a smooth action of on a homotopy sphere with exactly two fixed points and at which the tangent -modules are isomorphic to and , respectively (cf. Definition 1.1), and the action of on satisfies the Laitinen condition asserting that for any element of order for , the fixed point set is connected.
Definition 2.2. Let be a finite group. Two real -modules and are said to be -matched if and are isomorphic when restricted to any cyclic subgroup of of order for .
In Definition 2.1, the Laitinen condition implies that the two Smith equivalent real -modules and are -matched. As any two -matched and Smith equivalent real -modules and are -matched, Corollary 1.3 yields the next corollary.
Corollary 2.3. Let be a finite group with or . Then any two Laitinen-Smith equivalent real -modules are isomorphic.
In 1996, Erkki Laitinen ([Laitinen&Pawałowski1999, Appendix]) posed the following conjecture (cf. Problem 1.4).
Problem 2.4 (Laitinen conjecture). For a finite Oliver group , any two Laitinen-Smith equivalent real -modules are isomorphic if and only if or .
In order to prove that the Laitinen conjecture holds for a finite Oliver group , it sufficies to restrict attention to the case where , and to check that there exist two Laitinen-Smith equivalent real -modules that are not isomorphic (cf. Corollary 2.3).
[edit] 3 Results so far
- Laitinen and Pawałowski [Laitinen&Pawałowski1999] prove that the Laitinen conjecture holds for any finite (non-trivial) perfect group .
- Pawałowski and Solomon [Pawałowski&Solomon2002] prove that the Laitinen conjecture holds under either of the following condition:
- is a finite Oliver group of odd order (where always ).
- is a finite Oliver group with a quotient isomorphic to for two distinct odd primes and (where always ).
- is a finite non-solvable gap group not isomorphic to , the splitting extension of by the group of automorphism of the field .
- Morimoto [Morimoto2008] obtains the first counterexample to the Laitinen conjecture by proving that the Smith set and for .
- Pawałowski and Sumi [Pawałowski&Sumi2009] compute the primary Smith set for some finite solvable Oliver groups, to the effect that:
- and for , confirming the Latinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , and they prove that any element of is the difference of two Laitinen-Smith equivalent real -modules, confirming the Laitinen conjecture.
- Morimoto [Morimoto2010] checks that for , where , and he proves that any element of is the difference of two Laitinen-Smith equivalent real -modules, confirming the Laitinen conjecture.
- Pawałowski and Sumi [Pawałowski&Sumi2010] confirm the Laitinen conjecture for any finite non-solvable group not isomorphic to or , and more generally, for any finite Oliver group satisfying the Nil-Condition.
[edit] 4 Further discussion
Summarizing the results of [Laitinen&Pawałowski1999], [Pawałowski&Solomon2002], [Morimoto2008], [Morimoto2010], and [Pawałowski&Sumi2010], one obtains the following theorem.
Theorem 4.1. For a finite non-solvable group not isomorphic to , the following two statements are true.
[edit] 5 References
- [Laitinen&Pawałowski1999] E. Laitinen and K. Pawałowski, Smith equivalence of representations for finite perfect groups, Proc. Amer. Math. Soc. 127 (1999), no.1, 297–307. MR1468195 (99b:57070) Zbl 0914.57025
- [Morimoto2008] M. Morimoto, Smith equivalent -representations are isomorphic, Proc. Amer. Math. Soc. 136 (2008), no.10, 3683–3688. MR2415055 (2009c:57054)
- [Morimoto2010] M. Morimoto, Nontrivial -matched -related pairs for finite gap Oliver groups, J. Math. Soc. Japan 62 (2010), no.2, 623–647. MR2662855 ()
- [Pawałowski&Solomon2002] K. Pawałowski and R. Solomon, Smith equivalence and finite Oliver groups with Laitinen number 0 or 1, Algebr. Geom. Topol. 2 (2002), 843–895 (electronic). MR1936973 (2003j:57057) Zbl 1022.57019
- [Pawałowski&Sumi2009] K. Pawałowski and T. Sumi, The Laitinen conjecture for finite solvable Oliver groups, Proc. Amer. Math. Soc. 137 (2009), no.6, 2147–2156. MR2480297 (2009k:57052) Zbl 1173.57015
- [Pawałowski&Sumi2010] K. Pawałowski and T. Sumi, Smith equivalence of representations and the Nil-Condition, to appear in the Proceedings of the Edinburgh Mathematical Society.
- [Smith1960] P. A. Smith, New results and old problems in finite transformation groups, Bull. Amer. Math. Soc. 66 (1960), 401–415. MR0125581 (23 #A2880) Zbl 0096.37501
Definition 1.1. Two real -modules and are called Smith equivalent if there exists a smooth action of on a homotopy sphere with exactly two fixed points and at which the tangent -modules are isomorphic to and , respectively, where the tangent -modules are determined on the tangent spaces and at and by taking the derivatives at and of the diffeomorphisms , considered for all .
- Let be the representation ring of , i.e., the Grothendieck ring of the differences of real -modules and . As a group, is a finitely generated free abelian group whose rank is the number of real conjugacy classes of elements . Recall that the real conjugacy class of is defined by . Hereafter, denotes the number of real conjugacy classes represented by elements not of prime power order.
- Let be the subgroup of consisting of the differences of two -matched real -modules and , where the -matched condition means that and are isomorphic when restricted to any prime power order subgroup of . The group is trivial, , if and only if . In the case , is a finitely generated free abelian group of rank , .
We shall make use of the notions of Smith set and pimary Smith set of .
- The Smith set of is the subset of consisting of the differences of two Smith equivalent real -modules and .
- The primary Smith set of is the subset of consisting of the differences of two -matched and Smith equivalent real -modules and .
If two real -modules and with are isomorphic, then and are Smith equivalent (the sphere admits the required action of , where acts trivially on and diagonally on ). Therefore, the sets and both contain the zero of . The following problems are motivated by the question of Paul A. Smith posed in 1960, in the article [Smith1960, the footenote on p. 406].
- Is it true that any two Smith equivalent real -modules are isomorphic, i.e., is it true that ?
- Is it true that any two -matched and Smith equivalent real -modules are isomorphic, i.e., is it true that ?
Let be the dimension homomorphism, i.e., for any two real -modules and .
Lemma 1.2 (see [Laitinen&Pawałowski1999]). For a finite group , the following two statements are true.
- The group is trivial, , if and only if or .
- If , is a finitely generated free abelian group of rank , .
As , Lemma 1.2 yields the following corollary.
Corollary 1.3. Let be a finite group with or . Then any two -matched and Smith equivalent real -modules are isomorphic, i.e., .
Problem 1.4 (Primary problem). For which finite groups , the following statement is true?
- Any two -matched and Smith equivalent real -modules are isomorphic if and only if or .
In order to answer the question in Problem 1.4, one shall check for which finite groups with , there exist two -matched and Smith equivalent real -modules that are not isomorphic (cf. Corollary 1.3). For a systematic discussion on the Smith equivalence of real -modules, we refer to the page Smith equivalence of reresentations. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. [Laitinen&Pawałowski1999, Appendix]).
[edit] 2 Laitinen conjecture
Definition 2.1. Let be a finite group. Two real -modules and are called Laitinen-Smith equivalent if there exists a smooth action of on a homotopy sphere with exactly two fixed points and at which the tangent -modules are isomorphic to and , respectively (cf. Definition 1.1), and the action of on satisfies the Laitinen condition asserting that for any element of order for , the fixed point set is connected.
Definition 2.2. Let be a finite group. Two real -modules and are said to be -matched if and are isomorphic when restricted to any cyclic subgroup of of order for .
In Definition 2.1, the Laitinen condition implies that the two Smith equivalent real -modules and are -matched. As any two -matched and Smith equivalent real -modules and are -matched, Corollary 1.3 yields the next corollary.
Corollary 2.3. Let be a finite group with or . Then any two Laitinen-Smith equivalent real -modules are isomorphic.
In 1996, Erkki Laitinen ([Laitinen&Pawałowski1999, Appendix]) posed the following conjecture (cf. Problem 1.4).
Problem 2.4 (Laitinen conjecture). For a finite Oliver group , any two Laitinen-Smith equivalent real -modules are isomorphic if and only if or .
In order to prove that the Laitinen conjecture holds for a finite Oliver group , it sufficies to restrict attention to the case where , and to check that there exist two Laitinen-Smith equivalent real -modules that are not isomorphic (cf. Corollary 2.3).
[edit] 3 Results so far
- Laitinen and Pawałowski [Laitinen&Pawałowski1999] prove that the Laitinen conjecture holds for any finite (non-trivial) perfect group .
- Pawałowski and Solomon [Pawałowski&Solomon2002] prove that the Laitinen conjecture holds under either of the following condition:
- is a finite Oliver group of odd order (where always ).
- is a finite Oliver group with a quotient isomorphic to for two distinct odd primes and (where always ).
- is a finite non-solvable gap group not isomorphic to , the splitting extension of by the group of automorphism of the field .
- Morimoto [Morimoto2008] obtains the first counterexample to the Laitinen conjecture by proving that the Smith set and for .
- Pawałowski and Sumi [Pawałowski&Sumi2009] compute the primary Smith set for some finite solvable Oliver groups, to the effect that:
- and for , confirming the Latinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , and they prove that any element of is the difference of two Laitinen-Smith equivalent real -modules, confirming the Laitinen conjecture.
- Morimoto [Morimoto2010] checks that for , where , and he proves that any element of is the difference of two Laitinen-Smith equivalent real -modules, confirming the Laitinen conjecture.
- Pawałowski and Sumi [Pawałowski&Sumi2010] confirm the Laitinen conjecture for any finite non-solvable group not isomorphic to or , and more generally, for any finite Oliver group satisfying the Nil-Condition.
[edit] 4 Further discussion
Summarizing the results of [Laitinen&Pawałowski1999], [Pawałowski&Solomon2002], [Morimoto2008], [Morimoto2010], and [Pawałowski&Sumi2010], one obtains the following theorem.
Theorem 4.1. For a finite non-solvable group not isomorphic to , the following two statements are true.
[edit] 5 References
- [Laitinen&Pawałowski1999] E. Laitinen and K. Pawałowski, Smith equivalence of representations for finite perfect groups, Proc. Amer. Math. Soc. 127 (1999), no.1, 297–307. MR1468195 (99b:57070) Zbl 0914.57025
- [Morimoto2008] M. Morimoto, Smith equivalent -representations are isomorphic, Proc. Amer. Math. Soc. 136 (2008), no.10, 3683–3688. MR2415055 (2009c:57054)
- [Morimoto2010] M. Morimoto, Nontrivial -matched -related pairs for finite gap Oliver groups, J. Math. Soc. Japan 62 (2010), no.2, 623–647. MR2662855 ()
- [Pawałowski&Solomon2002] K. Pawałowski and R. Solomon, Smith equivalence and finite Oliver groups with Laitinen number 0 or 1, Algebr. Geom. Topol. 2 (2002), 843–895 (electronic). MR1936973 (2003j:57057) Zbl 1022.57019
- [Pawałowski&Sumi2009] K. Pawałowski and T. Sumi, The Laitinen conjecture for finite solvable Oliver groups, Proc. Amer. Math. Soc. 137 (2009), no.6, 2147–2156. MR2480297 (2009k:57052) Zbl 1173.57015
- [Pawałowski&Sumi2010] K. Pawałowski and T. Sumi, Smith equivalence of representations and the Nil-Condition, to appear in the Proceedings of the Edinburgh Mathematical Society.
- [Smith1960] P. A. Smith, New results and old problems in finite transformation groups, Bull. Amer. Math. Soc. 66 (1960), 401–415. MR0125581 (23 #A2880) Zbl 0096.37501
Definition 1.1. Two real -modules and are called Smith equivalent if there exists a smooth action of on a homotopy sphere with exactly two fixed points and at which the tangent -modules are isomorphic to and , respectively, where the tangent -modules are determined on the tangent spaces and at and by taking the derivatives at and of the diffeomorphisms , considered for all .
- Let be the representation ring of , i.e., the Grothendieck ring of the differences of real -modules and . As a group, is a finitely generated free abelian group whose rank is the number of real conjugacy classes of elements . Recall that the real conjugacy class of is defined by . Hereafter, denotes the number of real conjugacy classes represented by elements not of prime power order.
- Let be the subgroup of consisting of the differences of two -matched real -modules and , where the -matched condition means that and are isomorphic when restricted to any prime power order subgroup of . The group is trivial, , if and only if . In the case , is a finitely generated free abelian group of rank , .
We shall make use of the notions of Smith set and pimary Smith set of .
- The Smith set of is the subset of consisting of the differences of two Smith equivalent real -modules and .
- The primary Smith set of is the subset of consisting of the differences of two -matched and Smith equivalent real -modules and .
If two real -modules and with are isomorphic, then and are Smith equivalent (the sphere admits the required action of , where acts trivially on and diagonally on ). Therefore, the sets and both contain the zero of . The following problems are motivated by the question of Paul A. Smith posed in 1960, in the article [Smith1960, the footenote on p. 406].
- Is it true that any two Smith equivalent real -modules are isomorphic, i.e., is it true that ?
- Is it true that any two -matched and Smith equivalent real -modules are isomorphic, i.e., is it true that ?
Let be the dimension homomorphism, i.e., for any two real -modules and .
Lemma 1.2 (see [Laitinen&Pawałowski1999]). For a finite group , the following two statements are true.
- The group is trivial, , if and only if or .
- If , is a finitely generated free abelian group of rank , .
As , Lemma 1.2 yields the following corollary.
Corollary 1.3. Let be a finite group with or . Then any two -matched and Smith equivalent real -modules are isomorphic, i.e., .
Problem 1.4 (Primary problem). For which finite groups , the following statement is true?
- Any two -matched and Smith equivalent real -modules are isomorphic if and only if or .
In order to answer the question in Problem 1.4, one shall check for which finite groups with , there exist two -matched and Smith equivalent real -modules that are not isomorphic (cf. Corollary 1.3). For a systematic discussion on the Smith equivalence of real -modules, we refer to the page Smith equivalence of reresentations. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. [Laitinen&Pawałowski1999, Appendix]).
[edit] 2 Laitinen conjecture
Definition 2.1. Let be a finite group. Two real -modules and are called Laitinen-Smith equivalent if there exists a smooth action of on a homotopy sphere with exactly two fixed points and at which the tangent -modules are isomorphic to and , respectively (cf. Definition 1.1), and the action of on satisfies the Laitinen condition asserting that for any element of order for , the fixed point set is connected.
Definition 2.2. Let be a finite group. Two real -modules and are said to be -matched if and are isomorphic when restricted to any cyclic subgroup of of order for .
In Definition 2.1, the Laitinen condition implies that the two Smith equivalent real -modules and are -matched. As any two -matched and Smith equivalent real -modules and are -matched, Corollary 1.3 yields the next corollary.
Corollary 2.3. Let be a finite group with or . Then any two Laitinen-Smith equivalent real -modules are isomorphic.
In 1996, Erkki Laitinen ([Laitinen&Pawałowski1999, Appendix]) posed the following conjecture (cf. Problem 1.4).
Problem 2.4 (Laitinen conjecture). For a finite Oliver group , any two Laitinen-Smith equivalent real -modules are isomorphic if and only if or .
In order to prove that the Laitinen conjecture holds for a finite Oliver group , it sufficies to restrict attention to the case where , and to check that there exist two Laitinen-Smith equivalent real -modules that are not isomorphic (cf. Corollary 2.3).
[edit] 3 Results so far
- Laitinen and Pawałowski [Laitinen&Pawałowski1999] prove that the Laitinen conjecture holds for any finite (non-trivial) perfect group .
- Pawałowski and Solomon [Pawałowski&Solomon2002] prove that the Laitinen conjecture holds under either of the following condition:
- is a finite Oliver group of odd order (where always ).
- is a finite Oliver group with a quotient isomorphic to for two distinct odd primes and (where always ).
- is a finite non-solvable gap group not isomorphic to , the splitting extension of by the group of automorphism of the field .
- Morimoto [Morimoto2008] obtains the first counterexample to the Laitinen conjecture by proving that the Smith set and for .
- Pawałowski and Sumi [Pawałowski&Sumi2009] compute the primary Smith set for some finite solvable Oliver groups, to the effect that:
- and for , confirming the Latinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , contrary to the Laitinen conjecture.
- and for , and they prove that any element of is the difference of two Laitinen-Smith equivalent real -modules, confirming the Laitinen conjecture.
- Morimoto [Morimoto2010] checks that for , where , and he proves that any element of is the difference of two Laitinen-Smith equivalent real -modules, confirming the Laitinen conjecture.
- Pawałowski and Sumi [Pawałowski&Sumi2010] confirm the Laitinen conjecture for any finite non-solvable group not isomorphic to or , and more generally, for any finite Oliver group satisfying the Nil-Condition.
[edit] 4 Further discussion
Summarizing the results of [Laitinen&Pawałowski1999], [Pawałowski&Solomon2002], [Morimoto2008], [Morimoto2010], and [Pawałowski&Sumi2010], one obtains the following theorem.
Theorem 4.1. For a finite non-solvable group not isomorphic to , the following two statements are true.
[edit] 5 References
- [Laitinen&Pawałowski1999] E. Laitinen and K. Pawałowski, Smith equivalence of representations for finite perfect groups, Proc. Amer. Math. Soc. 127 (1999), no.1, 297–307. MR1468195 (99b:57070) Zbl 0914.57025
- [Morimoto2008] M. Morimoto, Smith equivalent -representations are isomorphic, Proc. Amer. Math. Soc. 136 (2008), no.10, 3683–3688. MR2415055 (2009c:57054)
- [Morimoto2010] M. Morimoto, Nontrivial -matched -related pairs for finite gap Oliver groups, J. Math. Soc. Japan 62 (2010), no.2, 623–647. MR2662855 ()
- [Pawałowski&Solomon2002] K. Pawałowski and R. Solomon, Smith equivalence and finite Oliver groups with Laitinen number 0 or 1, Algebr. Geom. Topol. 2 (2002), 843–895 (electronic). MR1936973 (2003j:57057) Zbl 1022.57019
- [Pawałowski&Sumi2009] K. Pawałowski and T. Sumi, The Laitinen conjecture for finite solvable Oliver groups, Proc. Amer. Math. Soc. 137 (2009), no.6, 2147–2156. MR2480297 (2009k:57052) Zbl 1173.57015
- [Pawałowski&Sumi2010] K. Pawałowski and T. Sumi, Smith equivalence of representations and the Nil-Condition, to appear in the Proceedings of the Edinburgh Mathematical Society.
- [Smith1960] P. A. Smith, New results and old problems in finite transformation groups, Bull. Amer. Math. Soc. 66 (1960), 401–415. MR0125581 (23 #A2880) Zbl 0096.37501