Laitinen conjecture
(→Laitinen conjecture) |
(→Primary problem) |
||
Line 6: | Line 6: | ||
{{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 $U - V$ of two $\mathcal{P} | + | * 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 two 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 two $\mathcal{P} | + | * 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. 405}}. | 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. 405}}. | ||
* ''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 $\mathcal{P} | + | * ''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. | ||
Line 24: | Line 24: | ||
{{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 $\mathcal{P} | + | * ''Any two $\mathcal{P}$-matched and Smith equivalent real $G$-modules are isomorphic, i.e., $PSm(G) = 0$, 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 $\mathcal{P} | + | 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 about the Smith equivalence relation, we refer to the page ''Smith equivalence of real $G$-modules''. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. {{cite|Laitinen&Pawałowski1999|Appendix}}). |
</wikitex> | </wikitex> | ||
Revision as of 21:00, 30 December 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
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. 405].
- 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
,
.
By definition and the Slice Theorem, . Hence, 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, i.e.,
, 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 about the Smith equivalence relation, we refer to the page Smith equivalence of real
-modules. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. [Laitinen&Pawałowski1999, Appendix]).
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).
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.
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.
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
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G \to GL(V)](/images/math/a/3/a/a3aa6f58a00196c544204217e31c98db.png)
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. 405].
- 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
,
.
By definition and the Slice Theorem, . Hence, 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, i.e.,
, 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 about the Smith equivalence relation, we refer to the page Smith equivalence of real
-modules. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. [Laitinen&Pawałowski1999, Appendix]).
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).
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.
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.
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
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G \to GL(V)](/images/math/a/3/a/a3aa6f58a00196c544204217e31c98db.png)
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. 405].
- 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
,
.
By definition and the Slice Theorem, . Hence, 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, i.e.,
, 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 about the Smith equivalence relation, we refer to the page Smith equivalence of real
-modules. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. [Laitinen&Pawałowski1999, Appendix]).
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).
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.
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.
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
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G \to GL(V)](/images/math/a/3/a/a3aa6f58a00196c544204217e31c98db.png)
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. 405].
- 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
,
.
By definition and the Slice Theorem, . Hence, 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, i.e.,
, 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 about the Smith equivalence relation, we refer to the page Smith equivalence of real
-modules. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. [Laitinen&Pawałowski1999, Appendix]).
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).
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.
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.
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
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G \to GL(V)](/images/math/a/3/a/a3aa6f58a00196c544204217e31c98db.png)
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. 405].
- 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
,
.
By definition and the Slice Theorem, . Hence, 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, i.e.,
, 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 about the Smith equivalence relation, we refer to the page Smith equivalence of real
-modules. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. [Laitinen&Pawałowski1999, Appendix]).
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).
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.
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.
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
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G \to GL(V)](/images/math/a/3/a/a3aa6f58a00196c544204217e31c98db.png)
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. 405].
- 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
,
.
By definition and the Slice Theorem, . Hence, 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, i.e.,
, 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 about the Smith equivalence relation, we refer to the page Smith equivalence of real
-modules. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. [Laitinen&Pawałowski1999, Appendix]).
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).
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.
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.
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
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G \to GL(V)](/images/math/a/3/a/a3aa6f58a00196c544204217e31c98db.png)
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. 405].
- 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
,
.
By definition and the Slice Theorem, . Hence, 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, i.e.,
, 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 about the Smith equivalence relation, we refer to the page Smith equivalence of real
-modules. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. [Laitinen&Pawałowski1999, Appendix]).
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).
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.
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.
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
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G \to GL(V)](/images/math/a/3/a/a3aa6f58a00196c544204217e31c98db.png)
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. 405].
- 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
,
.
By definition and the Slice Theorem, . Hence, 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, i.e.,
, 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 about the Smith equivalence relation, we refer to the page Smith equivalence of real
-modules. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. [Laitinen&Pawałowski1999, Appendix]).
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).
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.
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.
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
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G \to GL(V)](/images/math/a/3/a/a3aa6f58a00196c544204217e31c98db.png)
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. 405].
- 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
,
.
By definition and the Slice Theorem, . Hence, 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, i.e.,
, 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 about the Smith equivalence relation, we refer to the page Smith equivalence of real
-modules. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. [Laitinen&Pawałowski1999, Appendix]).
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).
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.
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.
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