Laitinen conjecture
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{{beginrem|Definition}} | {{beginrem|Definition}} | ||
− | Let $G$ be a finite group. Two real $G$-modules $U$ and $V$ are called ''Laitinen-Smith equivalent'' if $U$ and $V$ are Smith equivalent in such a way that the corresponding action of $G$ on a homotopy sphere $\varSigma$ with $T_x(\varSigma) \cong U$ and $T_y(\varSigma) \cong V$ satisfies the ''Laitinen condition'' asserting that for any element $g \in G$ of order $2^a$ for $a \geq 3$, the fixed point set $\varSigma^g$ is connected. | + | Let $G$ be a finite group. Two real $G$-modules $U$ and $V$ are called ''Laitinen-Smith equivalent'' if $U$ and $V$ are Smith equivalent in such a way that the corresponding action of $G$ on a homotopy sphere $\varSigma$ with $T_x(\varSigma) \cong U$ and $T_y(\varSigma) \cong V$ as real $G$-modules, satisfies the ''Laitinen condition'' asserting that for any element $g \in G$ of order $2^a$ for $a \geq 3$, the fixed point set $\varSigma^g$ is connected. |
{{endrem}} | {{endrem}} | ||
{{beginthm|Proposition}}\label{pro:primary} | {{beginthm|Proposition}}\label{pro:primary} |
Revision as of 16:36, 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
.
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![\varSigma](/images/math/4/2/3/423f3285e9fa4860acd90c8f89b93728.png)
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![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
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![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![T_x(\varSigma)](/images/math/d/e/f/def848635e7040fa679cb61303d274c6.png)
![T_y(\varSigma)](/images/math/0/0/6/00689b4f6dd845036eff5f17cb66f0a2.png)
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![\varSigma \to \varSigma](/images/math/0/f/5/0f5bd1e2d542d57b9ce5792526c0952c.png)
![z \mapsto gz](/images/math/e/e/b/eeb173c0ebaf88034ca658b1ceceaea9.png)
![g \in G](/images/math/b/2/8/b28d61f8862ed5e922cb1baf53245863.png)
- 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
such that the real
-modules
and
are primary matched, i.e.,
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 Smith equivalent real
-modules
and
.
- The primary Smith set of
is the subset
of
consisting of the differences
of primary 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. 460].
- Is it true that any two Smith equivalent real
-modules are isomorphic, i.e., is it true that
?
- Is it true that any two primary 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 primary 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 primary 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 primary 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
and
are Smith equivalent in such a way that the corresponding action of
on a homotopy sphere
with
and
as real
-modules, satisfies the Laitinen condition asserting that for any element
of order
for
, the fixed point set
is connected.
Proposition 2.2.
Let be a finite group. Then any two Laitinen-Smith equivalent real
-modules
and
are primary matched, and therefore
.
In general, it is not known whether the converse statement is true, i.e., whether any element of is the difference of two Laitinen-Smith equivalent real
-modules.
Corollary 2.3.
Let be a finite group with
or
. Then any two Laitinen-Smith equivalent real
-modules are isomorphic.
Corollary 2.3 follows from Proposition 2.2 and Corollary 1.3. 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)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
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![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![T_x(\varSigma)](/images/math/d/e/f/def848635e7040fa679cb61303d274c6.png)
![T_y(\varSigma)](/images/math/0/0/6/00689b4f6dd845036eff5f17cb66f0a2.png)
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Tex syntax errorand
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![\varSigma \to \varSigma](/images/math/0/f/5/0f5bd1e2d542d57b9ce5792526c0952c.png)
![z \mapsto gz](/images/math/e/e/b/eeb173c0ebaf88034ca658b1ceceaea9.png)
![g \in G](/images/math/b/2/8/b28d61f8862ed5e922cb1baf53245863.png)
- 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
such that the real
-modules
and
are primary matched, i.e.,
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 Smith equivalent real
-modules
and
.
- The primary Smith set of
is the subset
of
consisting of the differences
of primary 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. 460].
- Is it true that any two Smith equivalent real
-modules are isomorphic, i.e., is it true that
?
- Is it true that any two primary 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 primary 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 primary 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 primary 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
and
are Smith equivalent in such a way that the corresponding action of
on a homotopy sphere
with
and
as real
-modules, satisfies the Laitinen condition asserting that for any element
of order
for
, the fixed point set
is connected.
Proposition 2.2.
Let be a finite group. Then any two Laitinen-Smith equivalent real
-modules
and
are primary matched, and therefore
.
In general, it is not known whether the converse statement is true, i.e., whether any element of is the difference of two Laitinen-Smith equivalent real
-modules.
Corollary 2.3.
Let be a finite group with
or
. Then any two Laitinen-Smith equivalent real
-modules are isomorphic.
Corollary 2.3 follows from Proposition 2.2 and Corollary 1.3. 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)
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![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G \to GL(V)](/images/math/a/3/a/a3aa6f58a00196c544204217e31c98db.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
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![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
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![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
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Tex syntax errorand
Tex syntax errorof the diffeomorphisms
![\varSigma \to \varSigma](/images/math/0/f/5/0f5bd1e2d542d57b9ce5792526c0952c.png)
![z \mapsto gz](/images/math/e/e/b/eeb173c0ebaf88034ca658b1ceceaea9.png)
![g \in G](/images/math/b/2/8/b28d61f8862ed5e922cb1baf53245863.png)
- 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
such that the real
-modules
and
are primary matched, i.e.,
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 Smith equivalent real
-modules
and
.
- The primary Smith set of
is the subset
of
consisting of the differences
of primary 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. 460].
- Is it true that any two Smith equivalent real
-modules are isomorphic, i.e., is it true that
?
- Is it true that any two primary 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 primary 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 primary 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 primary 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
and
are Smith equivalent in such a way that the corresponding action of
on a homotopy sphere
with
and
as real
-modules, satisfies the Laitinen condition asserting that for any element
of order
for
, the fixed point set
is connected.
Proposition 2.2.
Let be a finite group. Then any two Laitinen-Smith equivalent real
-modules
and
are primary matched, and therefore
.
In general, it is not known whether the converse statement is true, i.e., whether any element of is the difference of two Laitinen-Smith equivalent real
-modules.
Corollary 2.3.
Let be a finite group with
or
. Then any two Laitinen-Smith equivalent real
-modules are isomorphic.
Corollary 2.3 follows from Proposition 2.2 and Corollary 1.3. 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
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- 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
such that the real
-modules
and
are primary matched, i.e.,
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 Smith equivalent real
-modules
and
.
- The primary Smith set of
is the subset
of
consisting of the differences
of primary 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. 460].
- Is it true that any two Smith equivalent real
-modules are isomorphic, i.e., is it true that
?
- Is it true that any two primary 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 primary 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 primary 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 primary 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
and
are Smith equivalent in such a way that the corresponding action of
on a homotopy sphere
with
and
as real
-modules, satisfies the Laitinen condition asserting that for any element
of order
for
, the fixed point set
is connected.
Proposition 2.2.
Let be a finite group. Then any two Laitinen-Smith equivalent real
-modules
and
are primary matched, and therefore
.
In general, it is not known whether the converse statement is true, i.e., whether any element of is the difference of two Laitinen-Smith equivalent real
-modules.
Corollary 2.3.
Let be a finite group with
or
. Then any two Laitinen-Smith equivalent real
-modules are isomorphic.
Corollary 2.3 follows from Proposition 2.2 and Corollary 1.3. 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
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![g \in G](/images/math/b/2/8/b28d61f8862ed5e922cb1baf53245863.png)
- 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
such that the real
-modules
and
are primary matched, i.e.,
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 Smith equivalent real
-modules
and
.
- The primary Smith set of
is the subset
of
consisting of the differences
of primary 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. 460].
- Is it true that any two Smith equivalent real
-modules are isomorphic, i.e., is it true that
?
- Is it true that any two primary 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 primary 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 primary 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 primary 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
and
are Smith equivalent in such a way that the corresponding action of
on a homotopy sphere
with
and
as real
-modules, satisfies the Laitinen condition asserting that for any element
of order
for
, the fixed point set
is connected.
Proposition 2.2.
Let be a finite group. Then any two Laitinen-Smith equivalent real
-modules
and
are primary matched, and therefore
.
In general, it is not known whether the converse statement is true, i.e., whether any element of is the difference of two Laitinen-Smith equivalent real
-modules.
Corollary 2.3.
Let be a finite group with
or
. Then any two Laitinen-Smith equivalent real
-modules are isomorphic.
Corollary 2.3 follows from Proposition 2.2 and Corollary 1.3. 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
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![g \in G](/images/math/b/2/8/b28d61f8862ed5e922cb1baf53245863.png)
- 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
such that the real
-modules
and
are primary matched, i.e.,
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 Smith equivalent real
-modules
and
.
- The primary Smith set of
is the subset
of
consisting of the differences
of primary 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. 460].
- Is it true that any two Smith equivalent real
-modules are isomorphic, i.e., is it true that
?
- Is it true that any two primary 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 primary 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 primary 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 primary 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
and
are Smith equivalent in such a way that the corresponding action of
on a homotopy sphere
with
and
as real
-modules, satisfies the Laitinen condition asserting that for any element
of order
for
, the fixed point set
is connected.
Proposition 2.2.
Let be a finite group. Then any two Laitinen-Smith equivalent real
-modules
and
are primary matched, and therefore
.
In general, it is not known whether the converse statement is true, i.e., whether any element of is the difference of two Laitinen-Smith equivalent real
-modules.
Corollary 2.3.
Let be a finite group with
or
. Then any two Laitinen-Smith equivalent real
-modules are isomorphic.
Corollary 2.3 follows from Proposition 2.2 and Corollary 1.3. 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)
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Tex syntax errorand
Tex syntax errorof the diffeomorphisms
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![g \in G](/images/math/b/2/8/b28d61f8862ed5e922cb1baf53245863.png)
- 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
such that the real
-modules
and
are primary matched, i.e.,
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 Smith equivalent real
-modules
and
.
- The primary Smith set of
is the subset
of
consisting of the differences
of primary 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. 460].
- Is it true that any two Smith equivalent real
-modules are isomorphic, i.e., is it true that
?
- Is it true that any two primary 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 primary 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 primary 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 primary 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
and
are Smith equivalent in such a way that the corresponding action of
on a homotopy sphere
with
and
as real
-modules, satisfies the Laitinen condition asserting that for any element
of order
for
, the fixed point set
is connected.
Proposition 2.2.
Let be a finite group. Then any two Laitinen-Smith equivalent real
-modules
and
are primary matched, and therefore
.
In general, it is not known whether the converse statement is true, i.e., whether any element of is the difference of two Laitinen-Smith equivalent real
-modules.
Corollary 2.3.
Let be a finite group with
or
. Then any two Laitinen-Smith equivalent real
-modules are isomorphic.
Corollary 2.3 follows from Proposition 2.2 and Corollary 1.3. 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
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Tex syntax errorof the diffeomorphisms
![\varSigma \to \varSigma](/images/math/0/f/5/0f5bd1e2d542d57b9ce5792526c0952c.png)
![z \mapsto gz](/images/math/e/e/b/eeb173c0ebaf88034ca658b1ceceaea9.png)
![g \in G](/images/math/b/2/8/b28d61f8862ed5e922cb1baf53245863.png)
- 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
such that the real
-modules
and
are primary matched, i.e.,
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 Smith equivalent real
-modules
and
.
- The primary Smith set of
is the subset
of
consisting of the differences
of primary 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. 460].
- Is it true that any two Smith equivalent real
-modules are isomorphic, i.e., is it true that
?
- Is it true that any two primary 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 primary 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 primary 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 primary 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
and
are Smith equivalent in such a way that the corresponding action of
on a homotopy sphere
with
and
as real
-modules, satisfies the Laitinen condition asserting that for any element
of order
for
, the fixed point set
is connected.
Proposition 2.2.
Let be a finite group. Then any two Laitinen-Smith equivalent real
-modules
and
are primary matched, and therefore
.
In general, it is not known whether the converse statement is true, i.e., whether any element of is the difference of two Laitinen-Smith equivalent real
-modules.
Corollary 2.3.
Let be a finite group with
or
. Then any two Laitinen-Smith equivalent real
-modules are isomorphic.
Corollary 2.3 follows from Proposition 2.2 and Corollary 1.3. 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
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- 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
such that the real
-modules
and
are primary matched, i.e.,
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 Smith equivalent real
-modules
and
.
- The primary Smith set of
is the subset
of
consisting of the differences
of primary 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. 460].
- Is it true that any two Smith equivalent real
-modules are isomorphic, i.e., is it true that
?
- Is it true that any two primary 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 primary 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 primary 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 primary 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
and
are Smith equivalent in such a way that the corresponding action of
on a homotopy sphere
with
and
as real
-modules, satisfies the Laitinen condition asserting that for any element
of order
for
, the fixed point set
is connected.
Proposition 2.2.
Let be a finite group. Then any two Laitinen-Smith equivalent real
-modules
and
are primary matched, and therefore
.
In general, it is not known whether the converse statement is true, i.e., whether any element of is the difference of two Laitinen-Smith equivalent real
-modules.
Corollary 2.3.
Let be a finite group with
or
. Then any two Laitinen-Smith equivalent real
-modules are isomorphic.
Corollary 2.3 follows from Proposition 2.2 and Corollary 1.3. 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