This is a quick note for my own benefit to record an idea I came across during my final year group project. We did this project on exotic spheres, which are manifolds that are homeomorphic to but are not diffeomorphic to . The theorem that we considered was the following:
Exotic Spheres There exist 7-manifolds that are homeomorphic but not diffeomorphic to the 7-sphere .
The part that I want to note here is the construction of the candidate manifolds for these Exotic Spheres. We did it using a technique called the ‘clutching construction’ and it is a way of construction vector bundles over spheres. Here is the construction in general:
Decompose into its upper and lower hemispheres, which are homeomorphic to disks: where the boundary is the intersection .
Suppose we have a map , then we define the quotient space where we identify with . Note that if we apply to we get another element of .
So the ‘gluing’ of these bundles is trivial on the boundary of the hemispheres but the map introduces a twist in the fibres . This gives us a dimensional vector bundle
where the map is projection onto the first coordinate, . We call the map a clutching function corresponding to the above vector bundle.
Here are a couple of definitions we will need.
Definition 1
Definition 2 is the isomorphism classes of oriented -dimensional vector bundles over the space
Proposition Two vector bundles, and constructed as above are isomorphic if and are homotopic
then restricts to over and to over . By Proposition in Hatcher’s ‘Vector Bundles and K-Theory’ these vectors bundles are then isomorphic.
This proposition gives us a bijection from the set of homotopy classes of maps which we denote by to . Denote this bijection by . We can simplify this further by considering the special orthogonal group which is a subgroup of . We also want to construct bundles over so lets consider the case . With these simplifications we get the following bijection: Now is the set of homotopy classes of maps from and this is exactly so in fact we have a bijection: Lets calculate . We need this propsition from Hatcher’s ‘Algebraic Topology’
Proposition 3 A covering space projection induces isomorphisms for all
Proposition 4
Proof By the previous proposition, it suffices to construct a covering space projection , as then we will have . Since we will then have .
The following is based on this paper.
Define a homomorphism which takes a pair of unit quaternions , which we identify with to the linear map defined by . Consider the kernel of this map, . Setting , we get . This implies that must be in the centre of , which is equal to . But since we are considering unit quaternions, this means that the centre is equal to . Therefore ker. So is a double cover of . We can conclude that is a covering map and apply the previous proposition to get that
Note that also acts on and . So, if instead of taking the clutching functions over we restrict to some discs or spheres, then we get a disc or sphere bundle associated to the vector bundle that arises via the clutching construction.
Therefore to construct candidates for exotic spheres, we will consider bundles over with structure group . We can classify these vector bundles using .
To construct these bundles, define a clutching function by where takes a unit quaternion (where we identify the unit quaternions with ) and sends it to the linear map as in this original paper by Milnor for some , where we use quaternionic multiplication on the right hand side. Then for each , we get a sphere bundlewhere the candidates for the exotic spheres are the total spaces of these bundles, .
We then went on to give conditions on such that is homeomorphic but not diffeomorphic to . I may go through these steps in further posts but for now I’ll spoil the surprise here in two propositions presented without proof.
Proposition 5 is homeomorphic to when .
Proposition 6 Let be a real, oriented, closed manifold as constructed above, with so that is a topological -sphere. Suppose is diffeomorphic to the -sphere. Then .