On the complex index of sphere bundles

We consider the complex index, denoted by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>i</m:mi> <m:mi>n</m:mi> <m:msub> <m:mrow> <m:mi>d</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> indC(α ) , of a complex vector bundle α over a CW -complex, defined as the largest integer k for which there exists an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">S</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> S¹ -equivariant map from <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">S</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mi>k</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> S²ᵏ⁻¹ to the sphere bundle S ( α ). We study basic properties of the complex index and provide sufficient conditions under which the equality <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>i</m:mi> <m:mi>n</m:mi> <m:msub> <m:mrow> <m:mi>d</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>dim</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> indC(α )=dim(α ) holds. We also explore the stability properties of the complex index. A CW -complex B is said to be <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>I</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> </m:msub> </m:math> IC -trivial if every complex vector bundle α over B satisfies the equality <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>i</m:mi> <m:mi>n</m:mi> <m:msub> <m:mrow> <m:mi>d</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>dim</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> indC(α )=dim(α ) . We study the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>I</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> </m:msub> </m:math> IC -triviality of spheres, iterated suspensions of complex and real projective spaces, and stunted real projective spaces.