# Laitinen conjecture

## 1 Primary problem

Let $G$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}G$ be a finite group. A real $G$$G$-module $V$$V$ is a finite dimensional real vector space with a linear action of $G$$G$, i.e., the action of $G$$G$ on $V$$V$ is given by a representation $G \to GL(V)$$G \to GL(V)$.

Definition 1.1. Two real $G$$G$-modules $U$$U$ and $V$$V$ are called Smith equivalent if there exists a smooth action of $G$$G$ on a homotopy sphere $\varSigma$$\varSigma$ with exactly two fixed points $x$$x$ and $y$$y$ at which the tangent $G$$G$-modules are isomorphic to $U$$U$ and $V$$V$, respectively, where the tangent $G$$G$-modules are determined on the tangent spaces $T_x(\varSigma)$$T_x(\varSigma)$ and $T_y(\varSigma)$$T_y(\varSigma)$ at $x$$x$ and $y$$y$ by taking the derivatives at $x$$x$ and $y$$y$ of the diffeomorphisms $\varSigma \to \varSigma$$\varSigma \to \varSigma$, $z \mapsto gz$$z \mapsto gz$ considered for all $g \in G$$g \in G$.

• Let $RO(G)$$RO(G)$ be the representation ring of $G$$G$, i.e., the Grothendieck ring of the differences $U - V$$U - V$ of real $G$$G$-modules $U$$U$ and $V$$V$. As a group, $RO(G)$$RO(G)$ is a finitely generated free abelian group whose rank is the number of real conjugacy classes of elements $g\in G$$g\in G$. Recall that the real conjugacy class of $g \in G$$g \in G$ is defined by $(g)^{\pm} := (g) \cup (g^{-1})$$(g)^{\pm} := (g) \cup (g^{-1})$. Hereafter, $r_G$$r_G$ denotes the number of real conjugacy classes $(g)^{\pm}$$(g)^{\pm}$ represented by elements $g \in G$$g \in G$ not of prime power order.
• Let $PO(G)$$PO(G)$ be the subgroup of $RO(G)$$RO(G)$ consisting of the differences $U - V$$U - V$ of two $\mathcal{P}$$\mathcal{P}$-matched real $G$$G$-modules $U$$U$ and $V$$V$, where the $\mathcal{P}$$\mathcal{P}$-matched condition means that $U$$U$ and $V$$V$ are isomorphic when restricted to any prime power order subgroup of $G$$G$. The group $PO(G)$$PO(G)$ is trivial, $PO(G) = 0$$PO(G) = 0$, if and only if $r_G = 0$$r_G = 0$. In the case $PO(G) \neq 0$$PO(G) \neq 0$, $PO(G)$$PO(G)$ is a finitely generated free abelian group of rank $r_G$$r_G$, $\operatorname{rank} PO(G) = r_G$$\operatorname{rank} PO(G) = r_G$.

We shall make use of the notions of Smith set and pimary Smith set of $G$$G$.

• The Smith set of $G$$G$ is the subset $Sm(G)$$Sm(G)$ of $RO(G)$$RO(G)$ consisting of the differences $U - V \in RO(G)$$U - V \in RO(G)$ of two Smith equivalent real $G$$G$-modules $U$$U$ and $V$$V$.
• The primary Smith set of $G$$G$ is the subset $PSm(G)$$PSm(G)$ of $RO(G)$$RO(G)$ consisting of the differences $U - V \in RO(G)$$U - V \in RO(G)$ of two $\mathcal{P}$$\mathcal{P}$-matched and Smith equivalent real $G$$G$-modules $U$$U$ and $V$$V$.

If two real $G$$G$-modules $U$$U$ and $V$$V$ with $\dim U^G = \dim V^G = 0$$\dim U^G = \dim V^G = 0$ are isomorphic, then $U$$U$ and $V$$V$ are Smith equivalent (the sphere $\varSigma = S(V \oplus \mathbb{R})$$\varSigma = S(V \oplus \mathbb{R})$ admits the required action of $G$$G$, where $G$$G$ acts trivially on $\mathbb{R}$$\mathbb{R}$ and diagonally on $V \oplus \mathbb{R}$$V \oplus \mathbb{R}$). Therefore, the sets $PSm(G)$$PSm(G)$ and $Sm(G)$$Sm(G)$ both contain the zero $0 = V - V$$0 = V - V$ of $RO(G)$$RO(G)$. 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 $G$$G$-modules are isomorphic, i.e., is it true that $Sm(G) = 0$$Sm(G) = 0$?
• Is it true that any two $\mathcal{P}$$\mathcal{P}$-matched and Smith equivalent real $G$$G$-modules are isomorphic, i.e., is it true that $PSm(G) = 0$$PSm(G) = 0$?

Let $d^G \colon RO(G) \to \mathbb{Z}$$d^G \colon RO(G) \to \mathbb{Z}$ be the dimension homomorphism, i.e., $d^G(U-V) = \dim U^G - \dim V^G$$d^G(U-V) = \dim U^G - \dim V^G$ for any two real $G$$G$-modules $U$$U$ and $V$$V$.

Lemma 1.2 (see [Laitinen&Pawałowski1999]). For a finite group $G$$G$, the following two statements are true.

• The group $PO(G) \cap \Ker \, d^G$$PO(G) \cap \Ker \, d^G$ is trivial, $PO(G) \cap \Ker \, d^G = 0$$PO(G) \cap \Ker \, d^G = 0$, if and only if $r_G = 0$$r_G = 0$ or $1$$1$.
• If $r_G \geq 2$$r_G \geq 2$, $PO(G) \cap \Ker \, d^G$$PO(G) \cap \Ker \, d^G$ is a finitely generated free abelian group of rank $r_G - 1$$r_G - 1$, $\operatorname{rank}(PO(G) \cap \Ker \, d^G) = r_G - 1$$\operatorname{rank}(PO(G) \cap \Ker \, d^G) = r_G - 1$.

As $PSm(G) = PO(G) \cap Sm(G) = PO(G) \cap \Ker \, d^G \cap Sm(G)$$PSm(G) = PO(G) \cap Sm(G) = PO(G) \cap \Ker \, d^G \cap Sm(G)$, Lemma 1.2 yields the following corollary.

Corollary 1.3. Let $G$$G$ be a finite group with $r_G = 0$$r_G = 0$ or $1$$1$. Then any two $\mathcal{P}$$\mathcal{P}$-matched and Smith equivalent real $G$$G$-modules are isomorphic, i.e., $PSm(G) = 0$$PSm(G) = 0$.

Problem 1.4 (Primary problem). For which finite groups $G$$G$, the following statement is true?

• Any two $\mathcal{P}$$\mathcal{P}$-matched and Smith equivalent real $G$$G$-modules are isomorphic if and only if $r_G = 0$$r_G = 0$ or $1$$1$.

In order to answer the question in Problem 1.4, one shall check for which finite groups $G$$G$ with $r_G \geq 2$$r_G \geq 2$, there exist two $\mathcal{P}$$\mathcal{P}$-matched and Smith equivalent real $G$$G$-modules that are not isomorphic (cf. Corollary 1.3). For a systematic discussion on the Smith equivalence of real $G$$G$-modules, we refer to the page Smith equivalence of reresentations. Here, we shall focus only on a related conjecture posed by Erkki Laitinen (cf. [Laitinen&Pawałowski1999, Appendix]).

## 2 Laitinen conjecture

Definition 2.1. Let $G$$G$ be a finite group. Two real $G$$G$-modules $U$$U$ and $V$$V$ are called Laitinen-Smith equivalent if there exists a smooth action of $G$$G$ on a homotopy sphere $\varSigma$$\varSigma$ with exactly two fixed points $x$$x$ and $y$$y$ at which the tangent $G$$G$-modules are isomorphic to $U$$U$ and $V$$V$, respectively (cf. Definition 1.1), and the action of $G$$G$ on $\varSigma$$\varSigma$ satisfies the Laitinen condition asserting that for any element $g \in G$$g \in G$ of order $2^a$$2^a$ for $a \geq 3$$a \geq 3$, the fixed point set $\varSigma^g = \{z \in \varSigma \ |\ gz = z\}$$\varSigma^g = \{z \in \varSigma \ |\ gz = z\}$ is connected.

Definition 2.2. Let $G$$G$ be a finite group. Two real $G$$G$-modules $U$$U$ and $V$$V$ are said to be $\mathcal{P}_2$$\mathcal{P}_2$-matched if $U$$U$ and $V$$V$ are isomorphic when restricted to any cyclic subgroup of $G$$G$ of order $2^a$$2^a$ for $a \geq 3$$a \geq 3$.

In Definition 2.1, the Laitinen condition implies that the two Smith equivalent real $G$$G$-modules $U$$U$ and $V$$V$ are $\mathcal{P}_2$$\mathcal{P}_2$-matched. As any two $\mathcal{P}_2$$\mathcal{P}_2$-matched and Smith equivalent real $G$$G$-modules $U$$U$ and $V$$V$ are $\mathcal{P}$$\mathcal{P}$-matched, Corollary 1.3 yields the next corollary.

Corollary 2.3. Let $G$$G$ be a finite group with $r_G = 0$$r_G = 0$ or $1$$1$. Then any two Laitinen-Smith equivalent real $G$$G$-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 $G$$G$, any two Laitinen-Smith equivalent real $G$$G$-modules are isomorphic if and only if $r_G = 0$$r_G = 0$ or $1$$1$.

In order to prove that the Laitinen conjecture holds for a finite Oliver group $G$$G$, it sufficies to restrict attention to the case where $r_G \geq 2$$r_G \geq 2$, and to check that there exist two Laitinen-Smith equivalent real $G$$G$-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 $G$$G$.
• Pawałowski and Solomon [Pawałowski&Solomon2002] prove that the Laitinen conjecture holds under either of the following condition:
• $G$$G$ is a finite Oliver group of odd order (where always $r_G \geq 2$$r_G \geq 2$).
• $G$$G$ is a finite Oliver group with a quotient isomorphic to $\mathbb{Z}_{pq}$$\mathbb{Z}_{pq}$ for two distinct odd primes $p$$p$ and $q$$q$ (where always $r_G \geq 2$$r_G \geq 2$).
• $G$$G$ is a finite non-solvable gap group not isomorphic to $P\varSigma L_2(\mathbb{F}_{27}) := PSL_2(\mathbb{F}_{27}) \rtimes \operatorname{Aut}(\mathbb{F}_{27})$$P\varSigma L_2(\mathbb{F}_{27}) := PSL_2(\mathbb{F}_{27}) \rtimes \operatorname{Aut}(\mathbb{F}_{27})$, the splitting extension of $PSL_2(\mathbb{F}_{27})$$PSL_2(\mathbb{F}_{27})$ by the group $\operatorname{Aut}(\mathbb{F}_{27})$$\operatorname{Aut}(\mathbb{F}_{27})$ of automorphism of the field $\mathbb{F}_{27}$$\mathbb{F}_{27}$.
• Morimoto [Morimoto2008] obtains the first counterexample to the Laitinen conjecture by proving that the Smith set $Sm(G) = 0$$Sm(G) = 0$ and $r_G = 2$$r_G = 2$ for $G = \operatorname{Aut}(A_6)$$G = \operatorname{Aut}(A_6)$.
• Pawałowski and Sumi [Pawałowski&Sumi2009] compute the primary Smith set $PSm(G)$$PSm(G)$ for some finite solvable Oliver groups, to the effect that:
• $PSm(G) = Sm(G) = 0$$PSm(G) = Sm(G) = 0$ and $r_G = 1$$r_G = 1$ for $G = (\mathbb{Z}_3 \times A_4) \rtimes \mathbb{Z}_2$$G = (\mathbb{Z}_3 \times A_4) \rtimes \mathbb{Z}_2$, confirming the Latinen conjecture.
• $PSm(G) = Sm(G) = 0$$PSm(G) = Sm(G) = 0$ and $r_G = 2$$r_G = 2$ for $G = S_3 \times A_4$$G = S_3 \times A_4$, contrary to the Laitinen conjecture.
• $PSm(G) = Sm(G) = 0$$PSm(G) = Sm(G) = 0$ and $r_G = 2$$r_G = 2$ for $G = (\mathbb{Z}^2_2 \rtimes \mathbb{Z}_3)^2 \rtimes \mathbb{Z}_2$$G = (\mathbb{Z}^2_2 \rtimes \mathbb{Z}_3)^2 \rtimes \mathbb{Z}_2$, contrary to the Laitinen conjecture.
• $PSm(G) = Sm(G) = 0$$PSm(G) = Sm(G) = 0$ and $r_G = 2$$r_G = 2$ for $G = \operatorname{Aff}_2(\mathbb{F}_3)$$G = \operatorname{Aff}_2(\mathbb{F}_3)$, contrary to the Laitinen conjecture.
• $PSm(G) = Sm(G) = 0$$PSm(G) = Sm(G) = 0$ and $r_G = 3$$r_G = 3$ for $G = (A_4\times A_4) \rtimes \mathbb{Z}^2_2$$G = (A_4\times A_4) \rtimes \mathbb{Z}^2_2$, contrary to the Laitinen conjecture.
• $PSm(G) = Sm(G) = \mathbb{Z}$$PSm(G) = Sm(G) = \mathbb{Z}$ and $r_G = 3$$r_G = 3$ for $G = \mathbb{Z}_3 \times S_4$$G = \mathbb{Z}_3 \times S_4$, and they prove that any element of $PSm(G)$$PSm(G)$ is the difference of two Laitinen-Smith equivalent real $G$$G$-modules, confirming the Laitinen conjecture.
• Morimoto [Morimoto2010] checks that $PSm(G) = \mathbb{Z}$$PSm(G) = \mathbb{Z}$ for $G = P\varSigma L_2(\mathbb{F}_{27})$$G = P\varSigma L_2(\mathbb{F}_{27})$, where $r_G = 2$$r_G = 2$, and he proves that any element of $PSm(G)$$PSm(G)$ is the difference of two Laitinen-Smith equivalent real $G$$G$-modules, confirming the Laitinen conjecture.
• Pawałowski and Sumi [Pawałowski&Sumi2010] confirm the Laitinen conjecture for any finite non-solvable group $G$$G$ not isomorphic to $\operatorname{Aut}(A_6)$$\operatorname{Aut}(A_6)$ or $P\varSigma L_2(\mathbb{F}_{27})$$P\varSigma L_2(\mathbb{F}_{27})$, and more generally, for any finite Oliver group $G$$G$ 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 $G$$G$ not isomorphic to $\Aut(A_6)$$\Aut(A_6)$, the following two statements are true.

• Any two $\mathcal{P}$$\mathcal{P}$-matched and Smith equivalent real $G$$G$-modules are isomorphic if and only if $r_G = 0$$r_G = 0$ or $1$$1$ (cf. Problem 1.4).
• Any two Laitinen-Smith equivalent real $G$$G$-modules are isomorphic if and only if $r_G = 0$$r_G = 0$ or $1$$1$ (cf. Problem 2.4).