Affirmative answers to the Smith question were obtained for the first time (under some restrictions imposed on the actions) by Atiyah and Bott [1], Milnor [4], and Sanchez [7], while the first negative answers (for specific groups G) go back to Cappell and Shaneson [2], and Petrie [6]. The paper [5] surveys related results obtained up to 2017.

The results of Atiyah and Bott [1] and Sanchez [7] yield the following theorem.

Atiyah-Bott-Sanchez Theorem.  Let G be a finite group acting smoothly on a sphere with exactly two fixed points. Then the two tangent G-modules are isomorphic as P-modules for any cyclic subgroup P of G of order 2, 4, or pk for an an odd prime p and k > 0.

For a finite group G, let Sm(G) denote the set of differences UV in the real representation ring RO(G) such that U and V occur as the tangent G-modules at x and y for a smooth action of G on a homotopy sphere with exactly two fixed points x and y. In the case where Sm(G) consists of just one element, we write Sm(G) = 0. The answer to the Smith question is affirmative if and only if Sm(G) = 0. In general, Sm(G) is not a subgroup of RO(G).

For a finite group G, let PSm(G) denote the subset of Sm(G) consisting of the differences UV such that U and V are isomorphic as P-modules for any subgroup P of G of prime power order. Equivalently, PSm(G) is the subset of Sm(G) consisting of the differences UV such that U and V are isomorphic as P-modules for any cyclic subgroup P of G of order 2k for k > 2 (see the Atiyah-Bott-Sanchez Theorem).

For a finite group G, by the Laitinen number of G we mean the number rc(G) of real conjugacy classes of elements of G not of prime power order. Here, by the real conjugacy class of an element g of G, we mean the union of conjugacy classes of g and the inverse g-1 of g.

Lemma. (Laitinen–Pawałowski [3]) Let G be a finite group with Laitinen number rc(G) = 0 or 1. Then PSm(G) = 0.

Theorem. (Laitinen-Pawałowski [3]) Let G be a finite perfect group. Then PSm(G) = 0 if and only if the Laitinen number rc(G) = 0 or 1.

Laitinen Conjecture (1999). PSm(G) is non-zero for any finite Oliver group G with rc(G) = 2 or rc(G) > 2.

In 1960, Paul Althaus Smith asked the following question. Assume that a finite group G acts smoothly on a sphere with exactly two fixed points. Is it true that the G-modules determined (via deriviation of the action) on the tangent spaces at the two fixed points are always isomorphic to each other?

According to Julius L. Shaneson (ICM 1983, Warsaw, Poland), Smith had raised a fundamental question on smooth transformation groups.

References

[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]  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.

[6]  Petrie, T., Three theorems in transformation groups,
Algebraic Topology, Aarhus 1978, LNM 763 (1979), 549–572.

[7]  Sanchez, C. U., Actions of groups of odd order on compact orientable manifolds,
Proc. Amer. Math. Soc. 54 (1976), 445–448.