PROCEEDINGS OF THE INDIAN ACADEMY OF SCIENCES-MATHEMATICAL SCIENCES, vol.119, no.1, pp.101-108, 2009 (Journal Indexed in SCI)
Let G be a compact Lie group. In 1960, P A Smith asked the following question: "Is it true that for any smooth action of G on a homotopy sphere with exactly two fixed points, the tangent G-modules at these two points are isomorphic?" A result due to Atiyah and Bott proves that the answer is 'yes' for Z(P) and it is also known to be the same for connected Lie groups. In this work, we prove that two linear torus actions on S(n) which are c-cobordant (cobordism in which inclusion of each boundary component induces isomorphisms in Z-cohomology) must be linearly equivalent. As a corollary, for connected case, we prove a variant of Smith's question.