Anant R. Shastri

Kinks in general relativity

The problem of classifying topologically distinct general relativistic metrics is discussed. For a wide class of parallelizable space-time manifolds it is shown that a certain integer-valued topological invariant n always exists, and that quantization when n is odd will lead to spinor wave functionals.

TYPE OF 3-MANIFOLDS AND ADDITION OF RELATIVISTIC KINKS

The type of a closed, connected, orientable 3-manifold M was first considered in the classification of relativistic kinks over the space-time 4-manifold M × R. In this paper theorems are developed relating the type of M to H1 (M; Z), which lead to the determination of the type of large families of 3-manifolds. The relation of type to connected sum is established, and the connected sum is also used to define addition of kinks. The kink addition is related to the sum of the kink numbers, as well as addition in the bordism group Ω3(P3).

A discriminant criterion for the two-dimensional Jacobian problem

2. Various equivalent formluations of this problem are known. We shall recall some of these results, relevant to our discussion, from [1]. A polynomial f^C[X, Y] is said to have r points at infinity, if its homogeneous component of maximal degree (i.e., the degree form) is a product of r coprime factors. If F(X, Y, Z) is the homogenization of f(X} Y), and δ: = {F(Xy Y, Z)=0} is the curve in P 2 then, 6 intersects the line at infinity, L:={Z=0} in precisely r distinct points. The total number of local branches of β at all of these r points taken together is called the number of places of f at infinity. Note that the number of points at infinity is not an automorphic invariant, whereas, the number of places at infinity of a nonconstant polynomial