-
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
(as we are in the stable range). Hence there are two choices and so two bundles up to isomorphism. One the trivial bundle
and so the other will be called the
twisted bundle .
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
where the isomorphism is the winding number and this is the same as the Euler number of the resulting bundle. Now embed an
in
for surgery. We use the standard embedding of a sphere inside a larger sphere:
Form a framed embedding
in the first factor where the framing is given by twisting the meridian around
times as we pass around the
i.e. it is the element
represented by a map
. Now do surgery. The effect is the gluing
where
. This is now just the clutching construction as above.
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.
Hence we take an embedded sphere
on which
does not vanish and call this the
weird sphere. Then this
must have twisted normal bundle. Consider our untwisted surgery on
must bound a disk in a way that we can exten the normal framing of
to the framing on
(which is necessarily trivial. The twisted surgery must have the
other framing. Take the
bounded by
to inside a hemisphere of the weird sphere. Now form an isotopy moving
from one hemisphere to the other of the weird sphere. This must necessarily exchange the framing we have on
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.
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.