{"id":856,"date":"2020-09-20T12:23:34","date_gmt":"2020-09-20T10:23:34","guid":{"rendered":"https:\/\/kpa.faculty.wmi.amu.edu.pl\/?page_id=856"},"modified":"2023-05-16T18:52:20","modified_gmt":"2023-05-16T16:52:20","slug":"smith-problem","status":"publish","type":"page","link":"https:\/\/kpa.faculty.wmi.amu.edu.pl\/?page_id=856","title":{"rendered":"Laitinen Conjecture"},"content":{"rendered":"\n

“Smith had raised a fundamental question on smooth transformation groups<\/em>.”
— Julius L. Shaneson (ICM 1983, Warsaw, Poland)<\/p>\n\n\n\n

In 1960, Paul Althaus Smith [12] asked the following question. If a finite group G<\/em> acts smoothly on a sphere with exactly two fixed points, is it true that the representations of G<\/em> determined on the tangent spaces at the two fixed points are always<\/em> isomorphic to each other?

Smith equivalence. <\/strong>For a finite group G<\/em>, two real G<\/em>-modules U<\/em> and V<\/em> are called Smith equivalent<\/em> if there exists a smooth action of G<\/em> on a sphere with exactly two fixed points at which U<\/em> and V<\/em> occur as the tangent G<\/em>-modules. <\/p>\n\n\n\n

Now, the Smith question can be restated as follows. <\/p>\n\n\n\n

Smith Question.<\/strong> For a finite group G<\/em>, is it true that two Smith equivalent real G<\/em>-modules are always<\/em> isomorphic to each other?<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

Conclusion. <\/strong> 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 <\/em><\/em>2, 4, or pk<\/sup> for an odd prime p and k <\/em><\/em>> 0, the two tangent G-modules are isomorphic as P-modules.<\/em><\/p>\n\n\n\n

The first negative answers to the Smith Question go back to Cappell and Shaneson [2] in the case where G<\/em> = C<\/em>4k<\/em><\/sub> , the cyclic group of order 4k<\/em> with k<\/em> \u2265 2, and to Petrie [10] for any finite abelian group G<\/em> of odd order, with three or more noncyclic Sylow subgroups. <\/p>\n\n\n\n

Laitinen Conjecture <\/strong>(1996). For a finite Oliver group G<\/em> 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<\/em>-modules which are not isomorphic.
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* By the real conjugacy class<\/em> of an element g<\/em> of G<\/em>, we mean the union of the conjugacy classes (g<\/em>) and (g<\/em>-1<\/sup>) of the element g<\/em> and its inverse g<\/em>-1<\/sup>.<\/p>\n\n\n\n

The\u00a0Laitinen number<\/em>\u00a0of a finite group\u00a0G<\/em>, denoted by\u00a0\u03bb<\/strong><\/em>(G<\/em>), is the number of the real conjugacy classes of elements of\u00a0G<\/em>\u00a0which are not of prime power order.\u00a0The Laitinen Conjecture predicts that for a finite Oliver group G<\/em> with \u03bb<\/strong><\/em>(G<\/em>) \u2265 2, the answer to the Smith question is negative. Moreover, Laitinen expected also that the related action of G<\/em> on a sphere Sn<\/sup><\/em><\/em> can be constructed in such a way that for any element g<\/em> in G<\/em> whose order is 2k<\/sup><\/em><\/em> with k<\/em> \u2265 3, the fixed point set (Sn<\/sup><\/em><\/em>)g<\/sup><\/em><\/em> is connected.

Two real G<\/em>-modules U<\/em> and V<\/em> are called primary Snith equivalent<\/em> if U<\/em> and V<\/em> are Smith equivalent and for any subgroup P<\/em> of G<\/em> of prime power order, U<\/em> and V<\/em> are isomorphic as P<\/em>-modules.

Once the Laitinen condition is satisfied, by the conclusion above, the tangent G<\/em>-modules at the two fixed points in Sn<\/sup><\/em><\/em> are isomorphic as P<\/em>-modules for any subgroup P<\/em> of G<\/em> of prime power order.

Lemma.<\/strong> Let G be a finite group with <\/em>\u00a0\u03bb<\/strong><\/em>(G<\/em>) = 0 or<\/em> 1. Then any two primary Smith equivalent G-modules
are isomorphic. <\/em><\/p>\n\n\n\n

By the work of Laitinen–Pawa\u0142owski [3], Pawa\u0142owski–Solomon [7], and Pawa\u0142owski–Sumi [8], the Laitinen Conjecture is true in the following cases.<\/p>\n\n\n\n