“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 p^k for an odd prime p and k > 0, 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 4k 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.
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* 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 Sⁿ can be constructed in such a way that for any element g in G whose order is 2^k with k ≥ 3, the fixed point set (Sⁿ)^g is connected.

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 Sⁿ are isomorphic as 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.

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 ≠ PSL₂*(F₂₇) [7].
• G is a finite nonsolvable group with λ(G) ≥ 2 and G ≠ PSL₂*(F₂₇) and G ≠ Aut(A₆) [8].

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

• G = Aut(A₆), the group of automorphisms of the alternating group A₆ on six letters, and
• G = PSL₂*(F₂₇), the extension of the projective special linear group PSL₂(F₂₇) by Aut(F₂₇), where Aut(F₂₇) is the group of automorphism of the filed F₂₇ consisting of 27 elements and acting on PSL₂(F₂₇) in the canonical way.

For G = Aut(A₆), the Laitinen Conjecture is false by Morimoto’s work [5], and for G = PSL₂*(F₂₇), the Laitinen Conjecture is true by Morimoto’s work [6]. Therefore, except for G = Aut(A₆), 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.

[1]  Atiyah, M. F.; Bott, R., 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(A₆)-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.,