Talk:Sphere bundles and spin (Ex)
Patrickorson (Talk | contribs) (Created page with "<wikitex>; This is a standard clutching construction. Fix $k\geq 2$ and suppose we have a linear $S^k$ bundle $A$ over the sphere $S^2=D_-^2\cup D_+^2$. A fibre bundle over a...") |
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The sphere bundle of a 2-plane bundle is an $S^1$-bundle, so the arguments from above carry through here as well. The sphere bundle of $E_k$ is given by the clutching construction above with the clutching map an element of $$[S^1,\text{Aut}(S^1)]=[S^1,SO(2)]=[S^1,S^1]=\Z,$$where the isomorphism is the winding number and this is the same as the Euler number of the resulting bundle. Now embed an $S^1$ in $S^3$ for surgery. We use the standard embedding of a sphere inside a larger sphere: $$S^3=\partial(D^2\times D^2)=S^1\times D^2\cup_{id}D^2\times S^1.$$Form a framed embedding $S^1\times D^2$ in the first factor where the framing is given by twisting the meridian around $k$ times as we pass around the $S^1$ i.e. it is the element $k\in[S^1,\text{Aut}(D^2)]=\Z$ represented by a map $\omega:S^1\to \text{Aut}(D^2)$. Now do surgery. The effect is the gluing $$D^2\times S^1\cup_{f}D^2\times S^1$$where $f(v,x)=(\omega(x)(v),x)$. This is now just the clutching construction as above.<br /><br /> | The sphere bundle of a 2-plane bundle is an $S^1$-bundle, so the arguments from above carry through here as well. The sphere bundle of $E_k$ is given by the clutching construction above with the clutching map an element of $$[S^1,\text{Aut}(S^1)]=[S^1,SO(2)]=[S^1,S^1]=\Z,$$where the isomorphism is the winding number and this is the same as the Euler number of the resulting bundle. Now embed an $S^1$ in $S^3$ for surgery. We use the standard embedding of a sphere inside a larger sphere: $$S^3=\partial(D^2\times D^2)=S^1\times D^2\cup_{id}D^2\times S^1.$$Form a framed embedding $S^1\times D^2$ in the first factor where the framing is given by twisting the meridian around $k$ times as we pass around the $S^1$ i.e. it is the element $k\in[S^1,\text{Aut}(D^2)]=\Z$ represented by a map $\omega:S^1\to \text{Aut}(D^2)$. Now do surgery. The effect is the gluing $$D^2\times S^1\cup_{f}D^2\times S^1$$where $f(v,x)=(\omega(x)(v),x)$. This is now just the clutching construction as above.<br /><br /> | ||
− | As $S^1$ is nullhomotopically embedded, we may consider this inside a contractible disk or, in the second summand of $M\cong M\# S^m$. Moreover we may embed it using the standard embedding $S^m=\partial(D^2\times D^m-1)$ as above. Hence the result will be $M'=M\#N$ where $N$ is either the trivial or twisted linear $(m-2)$-sphere bundle over $S^2$. As $M$ is spin, if $N$ is the trivial bundle then the effect of surgery is also spin. However, if $N$ is twisted then $M'$ cannot be spin as connect sum results in direct sum of second Stiefel-Whitney classes and $w_2(S^k\tilde{\times} S^2)$ is non-vanishing. To see this consider that the bundle over $S^2$ itself is not spin (that it is clutched by the non-trivial element of $\pi_1(SO(m-1))$ is more or less the definition of the obstruction to lifting to the spin group) and that this implies that the total space $S^k\tilde{\times} S^2$ is also spin. | + | As $S^1$ is nullhomotopically embedded, we may consider this inside a contractible disk or, in the second summand of $M\cong M\# S^m$. Moreover we may embed it using the standard embedding $S^m=\partial(D^2\times D^m-1)$ as above. Hence the result will be $M'=M\#N$ where $N$ is either the trivial or twisted linear $(m-2)$-sphere bundle over $S^2$. As $M$ is spin, if $N$ is the trivial bundle then the effect of surgery is also spin. However, if $N$ is twisted then $M'$ cannot be spin as connect sum results in direct sum of second Stiefel-Whitney classes and $w_2(S^k\tilde{\times} S^2)$ is non-vanishing. To see this consider that the bundle over $S^2$ itself is not spin (that it is clutched by the non-trivial element of $\pi_1(SO(m-1))$ is more or less the definition of the obstruction to lifting to the spin group) and that this implies that the total space $S^k\tilde{\times} S^2$ is also spin.<br /><br /> |
+ | Now the hard part! Assume $M$ is not spin, this means that there is a cocycle on which $w_2(M)$ does not vanish. | ||
+ | $$ \begin{aligned}H_2(M;\Z_2)\cong H_2(M;\Z)\otimes\Z_2\qquad &\text{(Univ. coeff. thm.)}\\ | ||
+ | H_2(M;\Z)\cong\pi_2(M)\qquad &\text{(Hurewicz)}\\ | ||
+ | n>4\implies x\in\pi_2(M)\quad &\text{can be embedded}\end{aligned}$$Hence we take an embedded sphere $\Sigma$ on which $w_2(M)$ does not vanish and call this the ''weird sphere''. Then this $\Sigma$ must have twisted normal bundle. Consider our untwisted surgery on $S^1$ must bound a disk in a way that we can exten the normal framing of $S^1$ to the framing on $D^2$ (which is necessarily trivial. The twisted surgery must have the ''other'' framing. Take the $D^2$ bounded by $S^1$ to inside a hemisphere of the weird sphere. Now form an isotopy moving $S^1$ from one hemisphere to the other of the weird sphere. This must necessarily exchange the framing we have on $S^1$ from tricial to twisted as the normal bundle of the weird sphere is twisted. However, the surgery at either end of an isotopy gives a diffeomorphic effect.<br /><br /> | ||
+ | |||
+ | To see this last part a different way, we may take a cylinder $M\times [0,1]$. This describes a (boring!) isotopy of the embedded $S^1$. At some $t\in[0,1]$, take a connect sum with the weird sphere and the isotopy $S^1\times[0,1]$. The $S^1$ now slides along the tube and then over the weird sphere. This is not an isotopy but is now a concordance. However, within codimension $>3$ we can improve a concordance to an isotopy. | ||
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Revision as of 22:05, 29 March 2012
The sphere bundle of a 2-plane bundle is an -bundle, so the arguments from above carry through here as well. The sphere bundle of is given by the clutching construction above with the clutching map an element of
As is nullhomotopically embedded, we may consider this inside a contractible disk or, in the second summand of . Moreover we may embed it using the standard embedding as above. Hence the result will be where is either the trivial or twisted linear -sphere bundle over . As is spin, if is the trivial bundle then the effect of surgery is also spin. However, if is twisted then cannot be spin as connect sum results in direct sum of second Stiefel-Whitney classes and is non-vanishing. To see this consider that the bundle over itself is not spin (that it is clutched by the non-trivial element of is more or less the definition of the obstruction to lifting to the spin group) and that this implies that the total space is also spin.
Now the hard part! Assume is not spin, this means that there is a cocycle on which does not vanish.
To see this last part a different way, we may take a cylinder . This describes a (boring!) isotopy of the embedded . At some , take a connect sum with the weird sphere and the isotopy . The now slides along the tube and then over the weird sphere. This is not an isotopy but is now a concordance. However, within codimension we can improve a concordance to an isotopy.