# Talk:Surgery obstruction map I (Ex)

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<wikitex>; | <wikitex>; | ||

− | + | If $X$ is a manifold, then the normal map $id_X$ gives the base point of $\mathcal N (X)$. | |

− | + | ||

An element of $[X,G/TOP]$ is given by a bundle $\xi$ together with a fiber homotopy trivialization $\phi$. | An element of $[X,G/TOP]$ is given by a bundle $\xi$ together with a fiber homotopy trivialization $\phi$. | ||

Under the isomorphism $\mathcal N (X) \cong [X,G/TOP]$, the pair $(\xi,\phi)$ corresponds to a normal map $M\to X$ covered by $\nu_M\to \nu_X\oplus \xi$. | Under the isomorphism $\mathcal N (X) \cong [X,G/TOP]$, 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 | The surgery obstruction of a normal map $M\to X$ covered by $\nu_M\to \eta$ equals | ||

− | $$ sign(M)-sign(X)=\langle L(-\eta),[X]\rangle - | + | $$ sign(M)-sign(X)=\langle L(-\eta),[X]\rangle - \langle L(TX),[X]\rangle,$$ |

so it depends only on the bundle over $X$. | so it depends only on the bundle over $X$. | ||

+ | 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. | ||

+ | Moreover | ||

+ | $$\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 $X=\mathbb H P^2$: | ||

+ | |||

There are fiber homotopically trivial bundles on $X$ corresponding to classes in $[X,G/TOP]$ | There are fiber homotopically trivial bundles on $X$ corresponding to classes in $[X,G/TOP]$ | ||

which restrict to any given class in $[S^4,G/Top]$, as follows from the Puppe sequence with $\pi_7(G/TOP)=0$. | which restrict to any given class in $[S^4,G/Top]$, as follows from the Puppe sequence with $\pi_7(G/TOP)=0$. | ||

From [[Fibre_homotopy_trivial_bundles_(Ex)|another exercise]] we know that on $S^4$ we have such vector bundles with first Pontryagin class $48k$ times the generator of $H^4(S^4)$. | From [[Fibre_homotopy_trivial_bundles_(Ex)|another exercise]] we know that on $S^4$ we have such vector bundles with first Pontryagin class $48k$ 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 $\phi$. | 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 $\phi$. | ||

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We compute | We compute | ||

− | $$ | + | $$ 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, $$ | = c \langle p_1(\xi)^2 ,[X] \rangle \ne 0, $$ | ||

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

## Revision as of 22:07, 29 May 2012

If is a manifold, then the normal map gives the base point of . An element of is given by a bundle together with a fiber homotopy trivialization . Under the isomorphism , the pair corresponds to a normal map covered by . The surgery obstruction of a normal map covered by equals

so it depends only on the bundle over . Now is the sum of and in with respect to the Whitney sum. Moreover

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

As an example take :

There are fiber homotopically trivial bundles on corresponding to classes in which restrict to any given class in , as follows from the Puppe sequence with . From another exercise we know that on we have such vector bundles with first Pontryagin class times the generator of . This means that on we have a vector bundle with whose sphere bundle is fiber homotopically trivial, by a fiber homotopy equivalence . We compute

where the constant can be computed from the L-genus to be .

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