Sphere bundles and spin (Ex)

• For $k\geq2$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}k\geq2$, determine the number of distinct linear $S^k$$S^k$-bundles over $S^2$$S^2$.
• Let $E_k\to S^2$$E_k\to S^2$ be the complex plane bundle with Euler number $k$$k$. Explain how to obtain the total space of the sphere-bundle $S(E_k)$$S(E_k)$ via surgery on $S^3$$S^3$.

For $m\geq 4$$m\geq 4$ and $M$$M$ a smooth closed manifold, let $S^1\hookrightarrow M^m$$S^1\hookrightarrow M^m$ be a nullhomotopic embedding. For $M$$M$ spin, show that there is more than one possible diffeormorphism type for the outcome of a surgery on this embedding.

Now suppose $M$$M$ is simply connected. For $M$$M$ not spin, show that the outcome of a surgery on this embedding is uniquely determined up to diffeomorphism (difficult!).

For hints, see section in [Gompf&Stipsicz1999] on surgery.

References

• [Gompf&Stipsicz1999] R. E. Gompf and A. I. Stipsicz, $4$$4$-manifolds and Kirby calculus, American Mathematical Society, 1999. MR1707327 (2000h:57038) Zbl 0933.57020