Talk:Surgery obstruction map I (Ex)

If $X$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}X$ is a manifold, then the normal map $id_X$$id_X$ gives the base point of $\mathcal N (X)$$\mathcal N (X)$. An element of $[X,G/TOP]$$[X,G/TOP]$ is given by a bundle $\xi$$\xi$ together with a fiber homotopy trivialization $\phi$$\phi$. Under the isomorphism $\mathcal N (X) \cong [X,G/TOP]$$\mathcal N (X) \cong [X,G/TOP]$, the pair $(\xi,\phi)$$(\xi,\phi)$ corresponds to a normal map $M\to X$$M\to X$ covered by $\nu_M\to \nu_X\oplus \xi$$\nu_M\to \nu_X\oplus \xi$. Assume that the dimension is $4k$$4k$ and that $X$$X$ is simply connected. Then the surgery obstruction $\theta(f, \overline{f})$$\theta(f, \overline{f})$ of a normal map $f:M\to X$$f:M\to X$ covered by $\overline{f}:\nu_M\to \eta$$\overline{f}:\nu_M\to \eta$ equals

$\displaystyle \begin{array} {rl} \mathrm{sign}(M)-\mathrm{sign}(X) & = \langle L(TM), [M] \rangle - \langle L(TX),[X]\rangle \\ & = \langle L(\nu_M)^{-1}, [M] \rangle - \langle L(TX),[X]\rangle \\ & = \langle L(\overline{f}^*(\eta))^{-1}, f_*([X]) \rangle - \langle L(TX),[X]\rangle \\ & = \langle f^*L(\eta)^{-1}, f_*([X]) \rangle - \langle L(TX),[X]\rangle \\ & = \langle L(\eta)^{-1},[X]\rangle - \langle L(TX),[X]\rangle \end{array}$ $$\displaystyle \begin{array} {rl} \mathrm{sign}(M)-\mathrm{sign}(X) & = \langle L(TM), [M] \rangle - \langle L(TX),[X]\rangle \\ & = \langle L(\nu_M)^{-1}, [M] \rangle - \langle L(TX),[X]\rangle \\ & = \langle L(\overline{f}^*(\eta))^{-1}, f_*([X]) \rangle - \langle L(TX),[X]\rangle \\ & = \langle f^*L(\eta)^{-1}, f_*([X]) \rangle - \langle L(TX),[X]\rangle \\ & = \langle L(\eta)^{-1},[X]\rangle - \langle L(TX),[X]\rangle \end{array}$$

by the Hirzebruch signature theorem and several properties of the $L$$L$-genus. In particular the surgery obstruction depends only on the bundle over $X$$X$. Now $(\xi\oplus\xi,\phi * \phi)$$(\xi\oplus\xi,\phi * \phi)$ is the sum of $(\xi,\phi)$$(\xi,\phi)$ and $(\xi,\phi)$$(\xi,\phi)$ in $\mathcal N (X)=[X,G/O]$$\mathcal N (X)=[X,G/O]$ with respect to the Whitney sum. Moreover

$\displaystyle \theta(-(\xi,\phi))+\theta(-(\xi,\phi)) - \theta(-(\xi\oplus\xi,\phi * \phi)) = 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -\langle L(TX),[X]\rangle$ $$\displaystyle \theta(-(\xi,\phi))+\theta(-(\xi,\phi)) - \theta(-(\xi\oplus\xi,\phi * \phi)) = 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -\langle L(TX),[X]\rangle$$

If this is non-zero, then the surgery obstruction is not a group homomorphism with respect to the Whitney sum.

As an example take $X=\mathbb H P^2$$X=\mathbb H P^2$:

There are fiber homotopically trivial bundles on $X$$X$ corresponding to classes in $[X,G/TOP]$$[X,G/TOP]$ which restrict to any given class in $[S^4,G/Top]$$[S^4,G/Top]$, as follows from the Puppe sequence with $\pi_7(G/TOP)=0$$\pi_7(G/TOP)=0$. From another exercise we know that on $S^4$$S^4$ we have such vector bundles with first Pontryagin class $48k$$48k$ times the generator of $H^4(S^4)$$H^4(S^4)$. This means that on $X$$X$ we have a vector bundle $\xi$$\xi$ with $p_1(\xi)=48$$p_1(\xi)=48$ whose sphere bundle is fiber homotopically trivial, by a fiber homotopy equivalence $\phi$$\phi$. We compute

$\displaystyle 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -\langle L(TX),[X]\rangle = c \langle p_1(\xi)^2 ,[X] \rangle \ne 0,$ $$\displaystyle 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -\langle L(TX),[X]\rangle = c \langle p_1(\xi)^2 ,[X] \rangle \ne 0,$$

where the constant $c$$c$ can be computed from the L-genus to be $-1/9$$-1/9$.

So the surgery obstruction is not a group homomorphism with respect to the Whitney sum.