Talk:Surgery obstruction map I (Ex)

We take $X=\mathbb H P^2$$<wikitex>; We take $X=\mathbb H P^2$: The normal map $id_X$ gives the base point of $\mathcal N (X)\cong [X,G/TOP]$. An element of $[X,G/TOP]$ is given by a bundle $\xi$ together with a fiber homotopy trivialization $\phi$. Under the isomorphism $\mathcal N (X) \cong [X,G/O]$, the pair $(\xi,\phi)$ corresponds to a normal map $M\to X$ covered by $\nu_M\to \nu_X\oplus \xi$. The surgery obstruction of a normal map $M\to X$ covered by $\nu_M\to \eta$ equals $$ sign(M)-sign(X)=\langle L(-\eta),[X]\rangle -1,$$ so it depends only on the bundle over $X$. There are fiber homotopically trivial bundles on $X$ corresponding to classes in $[X,G/TOP]$ which restrict to any given class in $[S^4,G/Top]$, since the corresponding Atiyah-Hirzebruch spectral sequence collapses. From [[Fibre_homotopy_trivial_bundles_(Ex)|another exercise]] we know that on $S^4$ we have such vector bundles with first Pontryagin class k$ times the generator of $H^4(S^4)$. This means that on $X$ we have a vector bundle $\xi$ with $p_1(\xi)=48$ whose sphere bundle is fiber homotopically trivial, by a fiber homotopy equivalence $\phi$. Now $(\xi\oplus\xi,\phi * \phi)$ is the sum of $(\xi,\phi)$ and $(\xi,\phi)$ in $\mathcal N (X)=[X,G/O]$ with respect to the Whitney sum. We compute $$\sigma(-(\xi,-\phi))+\sigma(-(\xi,\phi)) - \sigma(-(\xi\oplus\xi,\phi * \phi)) = 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -1 = c \langle p_1(\xi)^2 ,[X] \rangle \ne 0, $$ and so the surgery obstruction is not a group homomorphism with respect to the Whitney sum. </wikitex>X=\mathbb H P^2$: The normal map $id_X$$id_X$ gives the base point of $\mathcal N (X)\cong [X,G/TOP]$$\mathcal N (X)\cong [X,G/TOP]$. An element of $[X,G/TOP]$$[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/O]$$\mathcal N (X) \cong [X,G/O]$, the pair $(\xi,\phi)$$(\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$. The surgery obstruction of a normal map $M\to X$$M\to X$ covered by $\nu_M\to \eta$$\nu_M\to \eta$ equals

$\displaystyle sign(M)-sign(X)=\langle L(-\eta),[X]\rangle -1,$ $$\displaystyle sign(M)-sign(X)=\langle L(-\eta),[X]\rangle -1,$$

so it depends only on the bundle over $X$$X$. There are fiber homotopically trivial bundles on $X$$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]$, since the corresponding Atiyah-Hirzebruch spectral sequence collapses. 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$$\phi$. Now $(\xi\oplus\xi,\phi * \phi)$$(\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. We compute

$\displaystyle \sigma(-(\xi,-\phi))+\sigma(-(\xi,\phi)) - \sigma(-(\xi\oplus\xi,\phi * \phi)) = 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -1 = c \langle p_1(\xi)^2 ,[X] \rangle \ne 0,$ $$\displaystyle \sigma(-(\xi,-\phi))+\sigma(-(\xi,\phi)) - \sigma(-(\xi\oplus\xi,\phi * \phi)) = 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -1 = c \langle p_1(\xi)^2 ,[X] \rangle \ne 0,$$

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