| Using the ideas of Madhav Nori, there has been a program to develop an obstruction theory for projective modules whose rank equals the dimension of the ring over which they are defined. This theory is along the lines of the theory of vector bundles in topology. The groups where the obstructions take value are called the Euler class groups and were defined by Bhatwadekar and Raja Sridharan. This program has been pushed further, among others by Mrinal Kanti Das, Manoj Keshari and Satya Mandal. In particular, Bhatwadekar, Das and Mandal characterised the group in the case of smooth, affine varieties over R in topological terms.
In this talk, I will define the relevant algebraic notions and associated topological notions. I will mention some well-known results in the subject and finally state recent results about generalising the result of Bhatwadekar, Das and Mandal in the case where the base field is an arbitrary real closed field. |