Talk:Sphere bundles and spin (Ex)
Patrickorson (Talk | contribs) |
Patrickorson (Talk | contribs) |
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$$ \begin{aligned}H_2(M;\Z_2)\cong H_2(M;\Z)\otimes\Z_2\qquad &\text{(Univ. coeff. thm.)}\\ | $$ \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)}\\ | H_2(M;\Z)\cong\pi_2(M)\qquad &\text{(Hurewicz)}\\ | ||
− | m>4\implies x\in\pi_2(M)\quad &\text{can be embedded}\end{aligned}$$Hence we take an embedded sphere $\Sigma$ on which $w_2(M)$ does not vanish and call this the ''weird sphere''. Then this $\Sigma$ must have twisted normal bundle. Consider our untwisted surgery | + | m>4\implies x\in\pi_2(M)\quad &\text{can be embedded}\end{aligned}$$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 that for our untwisted surgery, $S^1$ must initially bound a disk and we can extend the normal framing of $S^1$ to the normal 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 be 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 trivial 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 induces 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.<br /><br /> | To see this last part a different way, we may take a cylinder $M\times [0,1]$. This induces 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.<br /><br /> |
Revision as of 22:26, 29 March 2012
This is a standard clutching construction. Fix and suppose we have a linear bundle over the sphere . A fibre bundle over a contractible space is trivial (up to bundle isomorphism), so without loss of generality . We can now glue back the bundles via an automorphism of the fibre at every point on the boundary , varying continuously on the base i.e. a continuous map . In fact, a homotopic map will produce an isomorphic bundle so we are interested in a class of
Part 2
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
Part 3
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 induces 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.
This does not cover the case , I think it is still possible here but I don't have the solution.