On the complex index of sphere bundles

Abstract 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> $in{d}_{\mathbb{C}}\left(\alpha \right)$ , 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> ${\mathbb{S}}^{1}$ -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> ${\mathbb{S}}^{2k-1}$ to the sphere bundle S ( α ). We study basic properties of the complex index and provide sufficient conditions under which the equality <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> $in{d}_{\mathbb{C}}\left(\alpha \right)=\mathrm{dim}\left(\alpha \right)$ 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> ${I}_{\mathbb{C}}$ -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> $in{d}_{\mathbb{C}}\left(\alpha \right)=\mathrm{dim}\left(\alpha \right)$ . 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> ${I}_{\mathbb{C}}$ -triviality of spheres, iterated suspensions of complex and real projective spaces, and stunted real projective spaces.