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