Abstract We consider the complex index, denoted by i n d C (α) ind₂ (), of a complex vector bundle α over a CW -complex, defined as the largest integer k for which there exists an S 1 S^1 -equivariant map from S 2 k − 1 S^2k-1 to the sphere bundle S (α). We study basic properties of the complex index and provide sufficient conditions under which the equality i n d C (α) = dim (α) ind₂ () =dim () holds. We also explore the stability properties of the complex index. A CW -complex B is said to be I C I₂ -trivial if every complex vector bundle α over B satisfies the equality i n d C (α) = dim (α) ind₂ () =dim (). We study the I C I₂ -triviality of spheres, iterated suspensions of complex and real projective spaces, and stunted real projective spaces.
Singh et al. (Wed,) studied this question.