Talk:Representing homology classes by embedded 2-spheres (Ex)

1. Let $M=8\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$$\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}{^}M=8\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$. In the previous exercise, we showed that the intersection form $8\langle 1\rangle\oplus\langle-1\rangle$$8\langle 1\rangle\oplus\langle-1\rangle$ Of $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_wfON3K$$M$ is equivalent to $E_8\oplus\langle-1\rangle$$E_8\oplus\langle-1\rangle$ (using e.g. the classification of nondefinite even symmetric forms by rank and signature). Pick a basis $\langle y_1,\ldots, y_8, x\rangle$$\langle y_1,\ldots, y_8, x\rangle$ Of $H_2(M;\mathbb{Z})$$H_2(M;\mathbb{Z})$ so that the intersection form of the $y_i$$y_i$ is $E_8$$E_8$, $x\cdot y_i=0$$x\cdot y_i=0$ for each $i$$i$, and $x\cdot x=-1$$x\cdot x=-1$. Then $x$$x$ cannot be represented by a smoothly embedded $2$$2$-sphere, or else blowing down the $2$$2$-sphere would yield a smooth structure on a $4$$4$-manifold with intersection form $E_8$$E_8$ (violating Donaldson’s theorem).

2. Let $M=9\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$$M=9\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$. Similarly, the intersection form $9\langle 1\rangle\oplus\langle -1\rangle$$9\langle 1\rangle\oplus\langle -1\rangle$ is equivalent to $E_8\oplus\langle 1\rangle\oplus\langle -1\rangle$$E_8\oplus\langle 1\rangle\oplus\langle -1\rangle$. As in Part 1, let $x\in H_2(M;\mathbb{Z})$$x\in H_2(M;\mathbb{Z})$ with $x\cdot x=-1$$x\cdot x=-1$ represent the $\langle -1\rangle$$\langle -1\rangle$ of this decomposition. Then $x$$x$ cannot be represented by a smoothly embedded $2$$2$-sphere, or else blowing down the $2$$2$-sphere would yield a smooth structure of a $4$$4$-manifold with intersection form $E_8\oplus\langle 1\rangle$$E_8\oplus\langle 1\rangle$ (again violating Donaldson’s theorem; note that $E_8\oplus\langle 1\rangle$$E_8\oplus\langle 1\rangle$ is still positive-definite and non-diagonalizeable).

3. Let $M=8\mathbb{CP}^2\#\overline{\mathbb{CP}}^2\#(S^2\times S^2)$$M=8\mathbb{CP}^2\#\overline{\mathbb{CP}}^2\#(S^2\times S^2)$. The intersection form of $M$$M$ is $8\langle 1\rangle\oplus\langle-1\rangle\oplus\left(\begin{array}{cc}0&1\\1&0\end{array}\right)$$8\langle 1\rangle\oplus\langle-1\rangle\oplus\left(\begin{array}{cc}0&1\\1&0\end{array}\right)$, which similarly is equivalent to $E_8\oplus\langle-1\rangle\oplus\left(\begin{array}{cc}0&1\\1&0\end{array}\right)$$E_8\oplus\langle-1\rangle\oplus\left(\begin{array}{cc}0&1\\1&0\end{array}\right)$. As in parts 1 and 2, let $x\in H_2(M;\mathbb{Z})$$x\in H_2(M;\mathbb{Z})$ with $x\cdot x=-1$$x\cdot x=-1$ represent the $\langle-1\rangle$$\langle-1\rangle$ of this decomposition. Then $x$$x$ cannot be represented by a smoothly embedded $2$$2$-sphere, or else blowing down this sphere would yield a smooth manifold with intersection form $E_8\oplus \left(\begin{array}{cc}0&1\\1&0\end{array}\right)$$E_8\oplus \left(\begin{array}{cc}0&1\\1&0\end{array}\right)$ (violating Rokhlin’s theorem, as the signature of this form is $8\not\equiv 0\pmod{16}$$8\not\equiv 0\pmod{16}$).