Research Projects

This project aims to investigate the interconnections between two classes of invariants associated with algebraic varieties: algebraic obstruction groups in the sense of M. V. Nori, S. M. Bhatwadekar, and R. Sridharan (such as the Euler class groups), and motivic cohomotopy groups (such as the Levine–Weibel Chow group and the Chow–Witt group defined by J. Barge–F. Morel). These groups arise in the study of the splitting behavior of top-rank (i.e., rank equal to the Krull dimension of the algebra) projective modules over affine algebras. The goal of this project is to establish concrete connections between these families of groups for affine real varieties, thereby bridging geometric, topological and algebraic approaches to the study of projective modules. In 1955, J.-P. Serre proved that a projective module over a ring splits off a free summand of rank one if its rank exceeds the dimension of the ring. However, examples show this bound is optimal. Motivated by topology, it is natural to ask whether one can develop a cohomology theory of Euler classes, taking values in a suitable group, that captures the primary obstruction to splitting a top-rank projective module. Nearly 40 years later, M. P. Murthy showed in 1994 that when the algebra is smooth over an algebraically closed field, the Chow group serves as the precise obstruction group. He conjectured this should also hold in the singular setting. It took 25 more years before A. Krishna, in 2019, proved this conjecture, settling the search for a cohomological description of the obstruction group for singular affine algebras over algebraically closed fields. Over the field of reals, Murthy’s conjecture must be modified, since here the Chow group fails to capture the obstruction, even for smooth varieties. In 1999, building on ideas of M. V. Nori, S. M. Bhatwadekar and R. Sridharan developed an algebraic approach to Euler classes and introduced the Euler class groups. They proved that over smooth real varieties, the weak Euler class group coincides with the Chow group, and conjectured that this correspondence extends to the singular case. This conjecture remains open. Recently, we settled a weaker version of it. A primary objective of this project is to give a complete solution to this conjecture of Bhatwadekar–Sridharan. This will also provide a method to compute these Euler class groups. Our strategy uses an induction argument, with my earlier result as the base case. In the top-1 case, we restrict our attention to smooth real varieties and study the discussed groups. The techniques here are primarily topological rather than algebraic, which is one of the key differences from my PhD work. Specifically, our strategy employs the motivic homotopy classification of projective modules via classifying spaces and analyzes them using the Moore–Postnikov tower.