Talk:Surgery obstruction map I (Ex)

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<wikitex>;
<wikitex>;
We take $X=\mathbb H P^2$:
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If $X$ is a manifold, then the normal map $id_X$ gives the base point of $\mathcal N (X)$.
The normal map $id_X$ gives the base point of $\mathcal N (X)$.
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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
$$ sign(M)-sign(X)=\langle L(-\eta),[X]\rangle -1,$$
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$$ sign(M)-sign(X)=\langle L(-\eta),[X]\rangle - \langle L(TX),[X]\rangle,$$
so it depends only on the bundle over $X$.
so it depends only on the bundle over $X$.
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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.
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Moreover
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$$\theta(-(\xi,-\phi))+\theta(-(\xi,\phi)) - \theta(-(\xi\oplus\xi,\phi * \phi))
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= 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -\langle L(TX),[X]\rangle $$
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If this is non-zero, then the surgery obstruction is not a group homomorphism with respect to the Whitney sum.
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As an example take $X=\mathbb H P^2$:
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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 $\phi$.
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
We compute
$$\theta(-(\xi,-\phi))+\theta(-(\xi,\phi)) - \theta(-(\xi\oplus\xi,\phi * \phi))
+
$$ 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -\langle L(TX),[X]\rangle
= 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, $$
= c \langle p_1(\xi)^2 ,[X] \rangle \ne 0, $$
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 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. 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

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

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

\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:

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. From 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. 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,

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

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


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