Talk:Surgery obstruction map I (Ex)
Line 2: | Line 2: | ||
If $X$ is a manifold, then the normal map $id_X$ gives the base point of $\mathcal N (X)$. | 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 $ | + | Under the isomorphism $\mathcal N (X) \cong [X,G/TOP]$, the pair $(\xi,\phi)$ corresponds to a normal map $M\to X$ covered by $\nu_M\to \nu_X\oplus \xi$. |
− | Assume that the dimension is $4k$ and that $X$ is simply connected. Then the surgery obstruction of a normal map $M\to X$ covered by $\nu_M\to \eta$ equals | + | Assume that the dimension is $4k$ and that $X$ is simply connected. Then the surgery obstruction $\theta(f, \overline{f})$ of a normal map $f:M\to X$ covered by $\overline{f}:\nu_M\to \eta$ equals |
$$ \begin{array} {rl} | $$ \begin{array} {rl} | ||
\mathrm{sign}(M)-\mathrm{sign}(X) & = \langle L(TM), [M] \rangle - \langle L(TX),[X]\rangle \\ | \mathrm{sign}(M)-\mathrm{sign}(X) & = \langle L(TM), [M] \rangle - \langle L(TX),[X]\rangle \\ |
Revision as of 18:26, 31 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 . Assume that the dimension is and that is simply connected. Then the surgery obstruction of a normal map covered by equals
by the Hirzebruch signature theorem and severeal property of the -genus. In particular the surgery obstruction 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.