progress!
I don't usually post here, but as this is my most public outlet for my mathematical work. So here goes.
Today, I finally finished classifying all 1-connected 4-manifolds (up to diffeomorphism) which can be written as a biquotient G//H (this part has been done previously, I merely worked it out on my own). Further, given that a space CAN be written as a biquotient, I classified all the possible different ways of writing as a biquotient (under the assumption that no simple factor H' of H acts transtiively on any simple factor G' of G, and that G is simply connected - according to Totaro, this can be assumed wlog) (This is part is the new part).
w00t. Up next, dimension 5, then 6, then 7, then....a Ph. D.?
Summary of results:
S^4, CP^2, S^2 x S^2, CP^2 # CP^2 and CP^2 #-CP^2 all arise as biquotients. Any 4 manifold which can be written as a biquotient is diffeomorphic to one of these 5. Note that only S^4, CP^2, and S^2 x S^2 are homogeneous.
In the case of S^4, all possible choices of G and H look like the following: H is a product H_1 x SU(2) where each factor acts on precisely one side of G. G/H_1 = S^7 and the fibration SU(2) -> G/H_1 -> G/H_1 x H_2 is the usual hopf fibration. There are a total of 5 different ways of writing S^4. One is homogenous.
In the case of CP^2, there are 3 ways. One is homogenous.
In the case of S^2 x S^2, there are an infinite family of actions S^3 x S^3/ S^1 x S^1 which give this. Intuitively, one S^1 factor acts as the hopf map on one S^3 factor and trivial on the other. Then the other S^1 can be considered acting on S^3 x S^2. It acts as the usual hopf on S^3 and by rotating the S^2 an EVEN number of times. Note that this is a homogeneous action iff it rotates 0 times.
In the case of CP^2 # -CP^2, there are an infinite family of actions S^3 x S^3/ S^1 x S^1 which give this. Intuitive, one S^1 factor acts as the hopf map on one S^3 factor and trivial on the other. Then the other S^1 can be considered acting on S^3 x S^2. It acts as the usual hopf on S^3 and by rotating the S^2 an ODD number of times. (This is less surprising once you know the fact that CP^2 # - CP^2 is the unique nontrivial S^2 bundle over S^2)
Finally, in the case of CP^2 # CP^2, there is a unique action of S^1 x S^1 on S^3 x S^3.
PS - none of this has been checked by my advisor yet, so it could all end up false in the end ;-). Further, there is one detail in the S^2xS^2/CP^2 # +- CP^2 case I'm confident is true, but I have yet to work it out.
And for those who care (if anyone even glances at this), I've classified all the possible actions in the 5 d case. Here's what's known:
Every 1-connected 5 manifold which can be written as a biquotient is diffeomorphic to either S^5, the wu manifold SU(3)/SO(3), S^2 x S^3, or the unique nontrivial S^3 bundle over S^2. All 4 of these actually arise (the fact that these are all of them comes from the classification of 1-connected 5manifolds by smale and barden. The fact that all 4 arise is simply that the first 3 arise as homogeneous spaces and Pavlov showed the final space, which is NOT homogeneous, arises as well).
I have proven that all biquotient actions (same restrictions as above) are homogenous in the case of S^5 and SU(3)/SO(3).
I'm currently stuck: there are (countably) infintiely many nonequivalent actions of S^1 on S^3 x S^3 (the only groups I have left to consider).
It is know that all homogenous actions (which are a subclass of the actions i'm studying) only give rise to S^2 x S^3.
THe problem is simple: S^2 x S^3 and the unique nontrivial S^3 bundle over S^2 have hte same cohomology ring structures and can be distinguished by their second stieffel whitney class which is trivial for S^3 x S^2, and nontrivial for the other space.
I'm currently stuck trying to read a paper of Borel which shows how to crunch out stieffel whitney classes of homogeneous spaces as well as another paper by Singhoff extended this to biquotients.
Today, I finally finished classifying all 1-connected 4-manifolds (up to diffeomorphism) which can be written as a biquotient G//H (this part has been done previously, I merely worked it out on my own). Further, given that a space CAN be written as a biquotient, I classified all the possible different ways of writing as a biquotient (under the assumption that no simple factor H' of H acts transtiively on any simple factor G' of G, and that G is simply connected - according to Totaro, this can be assumed wlog) (This is part is the new part).
w00t. Up next, dimension 5, then 6, then 7, then....a Ph. D.?
Summary of results:
S^4, CP^2, S^2 x S^2, CP^2 # CP^2 and CP^2 #-CP^2 all arise as biquotients. Any 4 manifold which can be written as a biquotient is diffeomorphic to one of these 5. Note that only S^4, CP^2, and S^2 x S^2 are homogeneous.
In the case of S^4, all possible choices of G and H look like the following: H is a product H_1 x SU(2) where each factor acts on precisely one side of G. G/H_1 = S^7 and the fibration SU(2) -> G/H_1 -> G/H_1 x H_2 is the usual hopf fibration. There are a total of 5 different ways of writing S^4. One is homogenous.
In the case of CP^2, there are 3 ways. One is homogenous.
In the case of S^2 x S^2, there are an infinite family of actions S^3 x S^3/ S^1 x S^1 which give this. Intuitively, one S^1 factor acts as the hopf map on one S^3 factor and trivial on the other. Then the other S^1 can be considered acting on S^3 x S^2. It acts as the usual hopf on S^3 and by rotating the S^2 an EVEN number of times. Note that this is a homogeneous action iff it rotates 0 times.
In the case of CP^2 # -CP^2, there are an infinite family of actions S^3 x S^3/ S^1 x S^1 which give this. Intuitive, one S^1 factor acts as the hopf map on one S^3 factor and trivial on the other. Then the other S^1 can be considered acting on S^3 x S^2. It acts as the usual hopf on S^3 and by rotating the S^2 an ODD number of times. (This is less surprising once you know the fact that CP^2 # - CP^2 is the unique nontrivial S^2 bundle over S^2)
Finally, in the case of CP^2 # CP^2, there is a unique action of S^1 x S^1 on S^3 x S^3.
PS - none of this has been checked by my advisor yet, so it could all end up false in the end ;-). Further, there is one detail in the S^2xS^2/CP^2 # +- CP^2 case I'm confident is true, but I have yet to work it out.
And for those who care (if anyone even glances at this), I've classified all the possible actions in the 5 d case. Here's what's known:
Every 1-connected 5 manifold which can be written as a biquotient is diffeomorphic to either S^5, the wu manifold SU(3)/SO(3), S^2 x S^3, or the unique nontrivial S^3 bundle over S^2. All 4 of these actually arise (the fact that these are all of them comes from the classification of 1-connected 5manifolds by smale and barden. The fact that all 4 arise is simply that the first 3 arise as homogeneous spaces and Pavlov showed the final space, which is NOT homogeneous, arises as well).
I have proven that all biquotient actions (same restrictions as above) are homogenous in the case of S^5 and SU(3)/SO(3).
I'm currently stuck: there are (countably) infintiely many nonequivalent actions of S^1 on S^3 x S^3 (the only groups I have left to consider).
It is know that all homogenous actions (which are a subclass of the actions i'm studying) only give rise to S^2 x S^3.
THe problem is simple: S^2 x S^3 and the unique nontrivial S^3 bundle over S^2 have hte same cohomology ring structures and can be distinguished by their second stieffel whitney class which is trivial for S^3 x S^2, and nontrivial for the other space.
I'm currently stuck trying to read a paper of Borel which shows how to crunch out stieffel whitney classes of homogeneous spaces as well as another paper by Singhoff extended this to biquotients.