In [1], Açkgöz et al. introduced and investigated the notions of w-I-continuous and w * -I-continuous functions in ideal topological spaces. In this paper, we investigate their relationships with continuous and θ-continuous functions.

We prove, under certain conditions, that if a solenoidal group (i.e. 1-dimensional compact connected abelian group) acts effectively on a compact space then the fixed point set is nonempty and H * G (X, Q) has a presentation similar to the presentation of H * (X, Q) as proven by Chang in the case of a circle group.

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.

No abstract