“*Smith had raised a fundamental question on smooth transformation groups*.”

— Julius L. Shaneson (ICM 1983, Warsaw, Poland)

In 1960, Paul Althaus Smith [12] asked the following question. If a finite group *G* acts smoothly on a sphere with exactly two fixed points, is it true that the representations of *G* determined on the tangent spaces at the two fixed points are *always* isomorphic to each other?**Smith equivalence. **For a finite group *G*, two real *G*-modules *U* and *V* are called *Smith equivalent* if there exists a smooth action of *G* on a sphere with exactly two fixed points at which *U* and *V* occur as the tangent *G*-modules.

Now, the Smith question can be restated as follows.

**Smith Question.** For a finite group *G*, is it true that two Smith equivalent real *G*-modules are *always* isomorphic to each other?

Under some restrictions imposed on the action in question, affirmative answers to the Smith question were obtained for the first time by Atiyah and Bott [1], Milnor [4], and Sanchez [11]. The results yield the following conclusion.

**Conclusion. *** Let G be a finite group acting smoothly on a sphere with exactly two fixed points. Then for any cyclic subgroup P of G of order *2, 4,

*or*> 0,

*p*^{k}for an odd prime p and k*the two tangent G-modules are isomorphic as P-modules.*

The first negative answers to the Smith Question go back to Cappell and Shaneson [2] in the case where *G* = *C*_{4k} , the cyclic group of order 4*k* with *k* ≥ 2, and to Petrie [10] for any finite abelian group *G* of odd order, with three or more noncyclic Sylow subgroups.

**Laitinen Conjecture **(1996). For a finite Oliver group *G* with two or more real conjugacy classes* of elements not of prime power order, the answer to the Smith question is negative, i.e., there exist Smith equivalent real *G*-modules which are not isomorphic.

________________

* By the *real conjugacy class* of an element *g* of *G*, we mean the union of the conjugacy classes (*g*) and (*g*^{-1}) of the element *g* and its inverse *g*^{-1}.

The *Laitinen number* of a finite group *G*, denoted by * λ*(

*G*), is the number of the real conjugacy classes of elements of

*G*which are not of prime power order. The Laitinen Conjecture predicts that for a finite Oliver group

*G*with

*(*

**λ***G*) ≥ 2, the answer to the Smith question is negative. Moreover, Laitinen expected also that the related action of

*G*on a sphere

*can be constructed in such a way that for any element*

*S*^{n}*g*in

*G*whose order is 2

*with*

^{k}*k*≥ 3, the fixed point set (

*)*

*S*^{n}*is connected.*

^{g}Two real

*G*-modules

*U*and

*V*are called

*primary Snith equivalent*if

*U*and

*V*are Smith equivalent and for any subgroup

*P*of

*G*of prime power order,

*U*and

*V*are isomorphic as

*P*-modules.

Once the Laitinen condition is satisfied, by the conclusion above, the tangent

*G*-modules at the two fixed points in

*are isomorphic as*

*S*^{n}*P*-modules for any subgroup

*P*of

*G*of prime power order.

**Lemma.**

*Let G be a finite group with*

*(*

**λ***G*) = 0

*or*1.

*Then any two primary Smith equivalent G-modules*

are isomorphic.

are isomorphic.

By the work of Laitinen–Pawałowski [3], Pawałowski–Solomon [7], and Pawałowski–Sumi [8], the Laitinen Conjecture is true in the following cases.

*G*is a finite perfect group with(**λ***G*) ≥ 2 [3].*G*is a finite Oliver group of odd order [7]. Here, always(**λ***G*) ≥ 2.- G is a finite Oliver group with quotient
*C*, (_{pq}*p*,*q*) =1,*p*,*q*odd [7]. Here, always(**λ***G*) ≥ 2. *G*is a finite nonsolvable gap group with(**λ***G*) ≥ 2 and*G*≠ PSL_{2}*(*F*_{27}) [7].*G*is a finite nonsolvable group with(**λ***G*) ≥ 2 and*G*≠ PSL_{2}*(*F*_{27}) and*G*≠ Aut(*A*_{6}) [8].

Above, two finite nonsolvable (and thus, Oliver) groups *G* with ** λ**(G) = 2 are mentioned:

*G*= Aut(*A*_{6}), the group of automorphisms of the alternating group*A*_{6}on six letters, and- G = PSL
_{2}*(*F*_{27}), the extension of the projective special linear group PSL_{2}(*F*_{27}) by Aut(*F*_{27}), where Aut(*F*_{27}) is the group of automorphism of the filed*F*_{27}consisting of 27 elements and acting on PSL_{2}(*F*_{27}) in the canonical way.

For *G* = Aut(*A*_{6}), the Laitinen Conjecture is false by Morimoto’s work [5], and for *G* = PSL_{2}*(*F*_{27}), the Laitinen Conjecture is true by Morimoto’s work [6]. Therefore, except for *G* = Aut(*A*_{6}), the Laitinen Conjecture is true for all finite nonsolvable groups *G* with * λ*(

*G*) ≥ 2.

A finite group *G* is nilpotent if and only if *G* is the product of its Sylow subgroups, and a finite nilpotent group *G* is an Oliver group if and only if *G* has at least three distinct Sylow subgroups. Therefore, if *G* is a finite nilpotent Oliver group, then *G* has a quotient isomorphic to *C _{pq}* for two relatively prime odd integers

*p*and

*q*.

The article [10] surveys the results (obtained up to 2017) related to the Smith Question and the Laitinen Conjecture.

**References**

*A Lefschetz fixed point formula for elliptic complexes: II. Applications*,

Annals of Math. 88 (1968), 451–491. [2] Cappell, S.; Shaneson, J. L.,

*Fixed points of periodic maps*,

Proc. Nat. Acad. Sci. USA 77 (1980), 5052–5054. [3] Laitinen, E.; Pawałowski, K.,

*Smith equivalence of representations for finite perfect groups*,

Proc. Amer. Math. Soc. 127 (1999), 297–307. [4] Milnor, J. W.,

*Whitehead torsion*,

Bull. Amer. Math. Soc. 72 (1966), 358–426. [5] Morimoto, M.,

*Smith equivalent*Aut(

*A*

_{6})-

*representations are isomorphic*,

Proc.Amer. Math. Soc. 136 (2008), 3683–3688. [6] Morimoto, M.,

*Nontrivial P(G)-matched S-related pairs for finite gap Oliver groups*,

J. Math. Soc. Japan 62, no. 2, (2010), 623–647 [7] Pawałowski, K.; Solomon, R.,

*Smith equivalence and finite Oliver groups,*

Algebraic & Geometric Topology 2 (2002), 843–895.

[8] Pawałowski, K.; Sumi, T.,

*The Laitinen Conjecture for finite non-solvable groups*,

Proc. Edinburgh Math. Soc. 56 (2013), 303–3368 [9] Pawałowski, K. M.,

*The Smith Equivalence Problem and the Laitinen Conjecture*,

Handbook of Group Actions, Vol. III, ALM 40, International Press of Boston (2018), 485–538. [10] Petrie, T.,

*Three theorems in transformation groups*,

Algebraic Topology, Aarhus 1978, LNM 763 (1979), 549–572. [11] Sanchez, C. U.,

*Actions of groups of odd order on compact orientable manifolds*,

Proc. Amer. Math. Soc. 54 (1976), 445–448. [12] Smith, P. A.,